Cosmic Ray Muon Production Altitude is not right

[quark.expr]

Cosmic Ray Muon Production Altitude is not right!

Hi there,

I wrote a Programm to calculate the cosmic ray muon intensity on different altitudes. I dedicated this programm from the intensity of the primary cosmic rays and the intercations in the atmosphere.

First a Proton with the Energy E1 enters the atmosphere and after the mean interaction length after 18km above sea level collides with a Proton of an atomic nucleus which is not in motion. For a pp collision like that the center of mass energy is given by:

Sqrt(s) = Sqrt(2 * E1 *mp)

Where mp is the mass of the Proton of the nucleus. This collision produces +/- and 0 Pions each one 33%, the number of Pions produced can be read out of a plot from Review on Particle Physics (see attachment) and depends on the cms Energy. The kinetic Energy of the Produced particle can be calculated by:

Ekin=(Sqrt(s)-N*m)/N

where N is the number of Pions produced and m the mass of a Pion.

Using the same algorithm I can calculate the other collisions in the atmosphere, now between Pions +/- and nuclear Protons. The altitudes of such collisions would be 12 km, then 9 km then 7.7km, calculated with the mean interaction lenght.

The Problem is the pions do not reach the second collision because of decaying into muons. The result up to Energies of the primary Proton of 100TeV all the pions decay at about 14 to 12.5km altitude in Muons.
In fact this is different we do have certain energetic Pions which decay on 7km altitude.

Why could my programm fail in calculating the muon production altitude? Can you help me?

 

[f95toli]

How fast are the particles moving? Compared to, let's say, c?

 

[Norman]

Just as a side note:

You may want to rethink the idea that all types of pions are created equally in the atmosphere. You should have a look here. You can plainly see that for a given incident momentum a positive pion is 2-4 times as likely to be produced as a negative pion from a PP collision.

This is intuitive from the point of view of conservation of charge. The initial state is a +2 so when you make a negative pion you must create 3 other positive charged particles. Not so with a positive pion you only need one more charged particle- don't forget conservation of baryon number either :).

 

[quark.expr]

Depends of the Energy of the primary cosmic ray Proton. The velocity of the particles is about 0.9999..c meanging a relativistic calculation is needed which i included. Of course it is decreasing while interacting whit the atmosphere.

 

[Piewie]

Does your simulation include pion decays with a decay time much longer than the mean life?

 

[Timo]

A natural start to find errors is double- and triple-checking all (and in your case, where I have trouble understanding some of your steps, I think I should put emphasis on "all") equations, their applicability and their correct implementation in the code with easy tests. I don't exactly understand what your program does, but the statement

First a Proton with the Energy E1 enters the atmosphere and after the mean interaction length after 18km above sea level collides with a Proton of an atomic nucleus which is not in motion. For a pp collision like that the center of mass energy is given by:

Sqrt(s) = Sqrt(2 * E1 *mp)

is wrong, for example. The cms-energy is $$\left| \left| \left( \begin{array}{cc}E_1 + m_P \\ \vec p_1 \end{array} \right) \right| \right| = \dots = \sqrt{ 2m_P^2 + 2 E_1 m_P }$$.

 

[quark.expr]

Thanks good idea,

I could include the exponential distribution of the life time of a Pion which would give me a the number of Pions that reach the other altitude of interaction from the number of Pions produced.

But another reason why the calculation could be wrong could be, that the Pions after a Pi P collision do not all have the same kinetic energy. Is there a distribution which could describe this?

 

[Norman]

Let me try to understand your basic premise:

1.) You have incident proton spectrum (flux and energy).
2.) You take the mean free path of the proton flux and have it collide with "free" protons (which are actually contained in nuclei) using the cross section data you force them to create Pions inelastically. (there is still a chance of elastic scattering at intermediate energies).
3.) You have the 3 types of pions created symmetrically
4.) These pions travel their mfp and interact again with free protons (not nuclei?)
5.) You also allow for decay of the pions into muons, the decay length is well known.


Why are you not using the Boltzman Equation for flux transport. I understand why you would not want to do this with a Monte Carlo (probabilistic) code due to resources. I believe the only semi-valid way (not completely valid because you need to use certain approximations) of deterministic transport is the Boltzman Equation.

And this problem has already been worked on. See this paper for instance: MESTRN: A Deterministic Meson-Muon Transport Code for Space Radiation

 

[mormonator_rm]

Just as a side note:

You may want to rethink the idea that all types of pions are created equally in the atmosphere. You should have a look here. You can plainly see that for a given incident momentum a positive pion is 2-4 times as likely to be produced as a negative pion from a PP collision.

This is intuitive from the point of view of conservation of charge. The initial state is a +2 so when you make a negative pion you must create 3 other positive charged particles. Not so with a positive pion you only need one more charged particle- don't forget conservation of baryon number either :).

Sir, we are here referring to the Drell-Yan process of pion production. I do see you checked the proton-proton listing, but I fear you have interpreted the data incorrectly: the fact that pp-->pi+X is three times as likely as pp-->pi-X does not mean that charged pion production in cosmic ray showers follows the same trend. pp-->pi+X is not equivalent to pp-->pppi+X, nor is pp-->pi-X equivalent to pp-->pppi-X. In fact, the Clebsh-Gordon Coefficients clearly state that pi+/- will occur twice as frequently as pi0, giving the standard ratio of pi+/-:pi0 = 2:1. Furthermore, since pi+ and pi- must occur together to conserve charge in the reaction pp-->pppi+pi-pi0, quantities of pi+ and pi- will be equal. Hence the ratio pi+:pi-:pi0 = 1:1:1.

 

[Norman]

why would you restrict yourselves to pp->pppiX only? Isn't the inclusive cross section pp->piX what you truly want since you do not care what else is made since you are only concerned with the decay of the pion into the muon?

I agree with you completely if those are the cross sections you actually want. But the fact that you are trying to simulate an air shower with only a proton cloud for an atmosphere seemed unlikely to me... I guess I was wrong.

your statement concerning the ratio of pions may be correct, but your argument is flawed:

for 3 pion production pp->pppi+pi-pi0 is just as likely as pp->pnpi+pi+pi- for all the data I was able to reference.

 

[Sean Torrebadel]

What if I told you that a pion is the first stage of photon decay... and by the way, a pion has a half life, which means that its lifespan is statistically determined as the average of a wide range of values, even if they only last a billionth of a second...

 

[malawi_glenn]

What if I told you that a pion is the first stage of photon decay... and by the way, a pion has a half life, which means that its lifespan is statistically determined as the average of a wide range of values, even if they only last a billionth of a second...

Don't mix in your own theories here. Can you post your theory in its full form here and we have a look at it. Quite cool that a guy has a new theory for QED...

 

[mormonator_rm]

why would you restrict yourselves to pp->pppiX only? Isn't the inclusive cross section pp->piX what you truly want since you do not care what else is made since you are only concerned with the decay of the pion into the muon?

I agree with you completely if those are the cross sections you actually want. But the fact that you are trying to simulate an air shower with only a proton cloud for an atmosphere seemed unlikely to me... I guess I was wrong.

your statement concerning the ratio of pions may be correct, but your argument is flawed:

for 3 pion production pp->pppi+pi-pi0 is just as likely as pp->pnpi+pi+pi- for all the data I was able to reference.

Your reasoning on the first point is flawed, as the "X" must have a baryon number of 2, and ultimately describes the two recoiling nucleons and a further item "X". Charge conservation will require the "X" to cancel the charge of the produced pion and any transition in nucleon charge. The incoming high-energy proton will always be impinging on either a proton or a neutron (as pointed out in your second point, which is indeed valid), and the energy will likely correspond to a three-prong Drell-Yan production (although I will include two-prong for lower energy). Hence;

pp --> pi+X1 --> pi+ppX2, X2 = pi- (two-prong) OR pi-pi0 (three-prong)
pp --> pi-X1 --> pi-ppX2, X2 = pi+ (two-prong) OR pi+pi0 (three-prong)
pp --> pi0X1 --> pi0ppX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong) OR pi0pi0 (non-observed)
^
set pp --> pp; pi+:pi-:pi0 = 1:1:1 (two-prong) OR 1:1:1 (three-prong)

pp --> pi+X1 --> pi+pnX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong) OR pi0pi0 (three-prong)
pp --> pi-X1 --> pi-pnX2, X2 = pi+pi+ (three-prong only)
pp --> pi0X1 --> pi0pnX2, X2 = pi+ (two-prong) OR pi+pi0 (three-prong)
^
set pp --> pn; pi+:pi-:pi0 = 1:0:1 (two-prong) OR 3/2:1/2:1 (three-prong)

pp --> pi+X1 --> pi+nnX2, X2 = pi+ (two-prong) OR pi+pi0 (three-prong)
pp --> pi-X1 --> pi-nnX2 is not allowed
pp --> pi0X1 --> pi0nnX2, X2 = pi+pi+ (three-prong only)
^
set pp --> nn; pi+:pi-:pi0 = 1:0:0 (two-prong) OR 1:0:1/2 (three-prong)

pn --> pi+X1 --> pi+ppX2, X2 = pi-pi- (three-prong only)
pn --> pi-X1 --> pi-ppX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong) OR pi0pi0 (three-prong)
pn --> pi0X1 --> pi0ppX2, X2 = pi- (two-prong) OR pi-pi0 (three-prong)
^
set pn --> pp; pi+:pi-:pi0 = 0:1:1 (two-prong) OR 1/2:3/2:1 (three-prong)

pn --> pi+X1 --> pi+pnX2, X2 = pi- (two-prong) OR pi-pi0 (three-prong)
pn --> pi-X1 --> pi-pnX2, X2 = pi+ (two-prong) OR pi+pi0 (three-prong)
pn --> pi0X1 --> pi0pnX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong) OR pi0pi0 (three-prong)
^
set pn --> pn; pi+:pi-:pi0 = 1:1:1 (two-prong) OR 1:1:1 (three-prong)

pn --> pi+X1 --> pi+nnX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong) OR pi0pi0 (three-prong)
pn --> pi-X1 --> pi-nnX2, X2 = pi+pi+ (three-prong only)
pn --> pi0X1 --> pi0nnX2, X2 = pi+ (two-prong) OR pi+pi0 (three-prong)
^
set pn --> nn; pi+:pi-:pi0 = 1:0:1 (two-prong) OR 3/2:1/2:1 (three-prong)

nn --> pi+X1 --> pi+ppX2 is not allowed
nn --> pi-X1 --> pi-ppX2, X2 = pi- (two-prong) OR pi-pi0 (three-prong)
nn --> pi0X1 --> pi0ppX2, X2 = pi-pi- (three-prong only)
^
set nn --> pp; pi+:pi-:pi0 = 0:1:0 (two-prong) OR 0:1:1/2 (three-prong)

nn --> pi+X1 --> pi+pnX2, X2 = pi-pi- (three-prong only)
nn --> pi-X1 --> pi-pnX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong only) OR pi0pi0 (three-prong)
nn --> pi0X1 --> pi0pnX2, X2 = pi- (two-prong) OR pi-pi0 (three-prong)
^
set nn --> pn; pi+:pi-:pi0 = 0:1:1 (two-prong) OR 1/2:3/2:1 (three-prong)

nn --> pi+X1 --> pi+nnX2, X2 = pi- (two-prong) OR pi-pi0 (three-prong)
nn --> pi-X1 --> pi-nnX2, X2 = pi+ (two-prong) OR pi+pi0 (three-prong)
nn --> pi0X1 --> pi0nnX2, X2 = pi0 (two-prong) OR pi+pi- (three-prong) OR pi0pi0 (non-observed)
^
set nn --> nn; pi+:pi-:pi0 = 1:1:1 (two-prong) OR 1:1:1 (three-prong)

From the above summary, one can see that using the entire list of allowed transitions shows the equal production of pi+ and pi- over the entire set, and also shows that pi+/-:pi0 naturally becomes 2:1, confirming the corresponding Clebsh-Gordon Coefficient. This shows that, when deduced over the whole, the pi+ and pi- are produced equally, and the neutral pion is produced half as frequently as a charged pion. Note that the sum of pp-->pi+X, when taken alone, does occur more often than all pp-->pi-X alone; in fact, the ratio;

(pp-->pi+X):(pp-->pi-X) = 13:5

falls naturally out of the list. This supports the data listed for pp-->piX which you initially found, but still allows for the symmetric production of charged pions on the whole.

However, the condition requires the incident particle to be a proton (since only a charged proton could be accelerated sufficiently, while a neutron would not be a good candidate for interstellar acceleration), which cuts out all initial states with "nn". If we are allowed to assume that, on average, the gasses in our atmosphere contain equal parts of protons and neutrons, then the number of target protons should roughly equal the number of target neutrons. Although atmospheric target types are balanced, the selection of the incident particle creates an asymetry to the problem, which is further exacerbated by restricting the data to three-prong type (for higher-energy collisions). The end result gives the ratio;

pi+:pi-:pi0 = 13:9:11

Since the incident protons are constantly adding to the charge of the whole, and the chemistry of the atmosphere remains fairly stable in spite of this, then we must presume that the bulk of the added charge is carried in the produced pions, as evidenced by the asymetry in favor of positive charged pion production. This should lead to a significant abundance of positive charged muons over negative charged muons in cosmic ray showers, though not more than 50% difference. Interestingly, the sum of positive and negative pions is still double the figure for neutral pions, which conserves the Clebsh-Gordon Coefficient.

So... you are right to say that pp-->pi+X is seen more often than pp-->pi-X, but this fact does not truly suggest that pi+ is produced more often in nature than pi- over all. Charge conservation requires that pi+ does not occur any more often than pi-, and baryon number conservation alone results in the same thing. Hopefully this observation is helpful.