4 momentum is particle physics

In summary, the conversation discusses the creation of pions through the collision of a positron and an electron. The minimum kinetic energy required for the electrons to produce the pions is determined and it is noted that the pions will be created at relative rest, moving with the same velocity. The conservation of momentum and energy are also considered, with the calculation of invariant mass being a simpler method. It is also noted that the pions do not need to have equal velocities, as long as there is enough energy available. The conversation concludes with a discussion on the relationship between energy equivalence and invariant mass.
  • #1
fengqiu
19
1
Okay so i have a question for you guys
if I have a positron striking an electron at rest to create 2 pions( + and -) and I want to calculate the minimum kinnetic energy that the electrons can possesses to create these pions... then the created pions will be at rest correct?
so this gives me two four vectors
[Ee++Ee-,Pe+C,0,0)
the other one being
[Epi++Epi-,0,0,0]
now what really confuses me is.
1) momentum isn't conserved? we're still in the same frame of refernce ie lab frame, shouldn't momentum be conserved?
2) is energy conserved? if so when i equate the energies and equate the four vector length (invariant mass) i get different answers?

Thanks for your help!

Adam
 
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  • #2
fengqiu said:
then the created pions will be at rest correct?
Incorrect, this would - as you have noticed - violate conservation of momentum.

The pions will be created at relative rest, i.e., they will be moving with the same velocity.
 
  • #3
Orodruin said:
Incorrect, this would - as you have noticed - violate conservation of momentum.

The pions will be created at relative rest, i.e., they will be moving with the same velocity.
oh okay... so if they have the same velocity... that makes a bit more sense...
OKAY I'm still confused about the energies. could i not just solve the equation by equating the energies?

Cheers\

Adam
 
  • #4
Yes. You can do either, equating the invariant masses is simpler.

In order for us to figure out where you go wrong, you must provide us with your actual computations, i.e., you must show us what you do, not only tell us what you do.
 
  • #5
i'm just crunching through the calculations now, I'll get back to you if i get the right answer, sorry one more question is that how do you know the pions are going at equal velocities?
 
  • #6
fengqiu said:
i'm just crunching through the calculations now, I'll get back to you if i get the right answer, sorry one more question is that how do you know the pions are going at equal velocities?

They do not need to do so if there is more energy available. The invariant mass of the two pions is the smallest when they do and therefore it represents the threshold value, i.e., when the electrons need the least invariant mass - translating to the lowest energy possible.
 
  • #7
Orodruin said:
The invariant mass of the two pions is the smallest when they do

To specify, call the pion 4-momenta ##p_1## and ##p_2##, respectively. This leads to
$$
s = (p_1+p_2)^2 = 2 m_\pi^2 + 2 p_1\cdot p_2.
$$
You can compute the inner product ##p_1 \cdot p_2## in any frame since it is Lorentz invariant. Computing it in the rest frame of the first pion gives ##p_1 = (m_\pi,0)## and ##p_2 = (m_\pi + T_\pi,p_\pi)##, where ##T_\pi## and ##p_\pi## are the kinetic energy and momentum of the second pion in the rest frame of the first. Hence
$$
s = 4 m_\pi^2 + 2m_\pi T_\pi \geq 4 m_\pi^2,
$$
with equality when ##T_\pi = 0##, i.e., when the pions are at relative rest.
 
  • #8
you, Orodruin are the man!
thanks for your help
it turns out that we get the same equation for energy equivalence and invariant mass
that makes ALOT more sense
 
  • #9
fengqiu said:
you, Orodruin are the man!

Not really, I have just been teaching this stuff for six years and make a living as a particle physicist ...
 
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What is 4 momentum in particle physics?

4 momentum is a physical quantity that describes the motion of a particle in four-dimensional space-time. It includes the particle's energy, momentum, and direction of motion.

How is 4 momentum calculated?

4 momentum is calculated by multiplying the particle's energy by its velocity in each of the three spatial dimensions, and then adding the particle's mass multiplied by the speed of light in the time dimension.

Why is 4 momentum important in particle physics?

4 momentum is important in particle physics because it allows scientists to describe and predict the behavior of particles in collisions and interactions. It is also a fundamental concept in special relativity and quantum mechanics.

What is the difference between 4 momentum and 3 momentum?

4 momentum includes the particle's energy and momentum in all four dimensions, while 3 momentum only includes the particle's momentum in the three spatial dimensions. 4 momentum is a more complete and accurate description of a particle's motion.

How is 4 momentum conserved in particle interactions?

In a closed system, where no external forces are acting, the total 4 momentum before and after a particle interaction must be equal. This is known as the conservation of 4 momentum and is a fundamental principle in particle physics.

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