What is the Maximum Mass of X0 in Electron-Proton Collisions?

[Matt atkinson]

Homework Statement

Electrons of energy 5.00 GeV are incident upon target protons and a search is made for neutral particles X0 produced in the reaction

e− + p → e− + p + X0

What is the maximum mass of X0 which can be produced? Use any reasonable approximations, but justify your arguments.

[Mass of electron is 0.511 MeV/c2; that of proton is 938.3 MeV/c2.]
[hint: Consider C.O.M frame]

Homework Equations

$$W^2=(∑E_i)^2-(∑p_i)^2$$
Fact that the effective mass $$W^2$$ is invarient.
also $$E=(∑m_i)^2+(∑T_i)^2$$

The Attempt at a Solution

As the question asks for the maximum mass of X0 so I assumed all particles at rest after the collision occured.
So before the collision in the COM frame:
$$W=(m_e+m_p)+500GeV$$
As I assumed particles at rest after the collision then:
$$W^2=E^2-0$$
but from here I just get that $$m_{X^o}=500GeV$$ which Seems too easy... and wrong.
Could anyone just point me in the right direction?

 

[Orodruin]

also $$E=(∑m_i)^2+(∑T_i)^2$$

This relation is not correct. It is only correct if you remove the squares on the right hand side, otherwise it even has the wrong dimensions.

$$W=(m_e+m_p)+500GeV$$

Where did this relation come from? I suggest you use the definition of W from your relevant equations. What is the momentum of an electron of energy 5 GeV? (It is also unclear from the question whether the 5 GeV are the total or kinetic energy of the electron.)

 

[Matt atkinson]

This relation is not correct. It is only correct if you remove the squares on the right hand side, otherwise it even has the wrong dimensions.
Where did this relation come from? I suggest you use the definition of W from your relevant equations. What is the momentum of an electron of energy 5 GeV? (It is also unclear from the question whether the 5 GeV are the total or kinetic energy of the electron.)

About the energy it just says that, I copied the question exactly.
sorry about the energy relation it was a typo.
Should be:
$$E=∑m_i+∑T_i$$ Because I did this from the COM frame in which $$∑p_i=0$$
Then my equation for W should simplify to, where I assumed 500GeV was the kinetic.
$$W=(m_e+m_p)+500GeV$$
As I see it now, there is a side note saying if the question doesn't explicitly say kinetic energy then It should be assumed to be total energy.
so that means:
$$W=m_p+500GeV$$
for
$$∑p_i=0$$
Is this correct?

 

[Orodruin]

How do you know the kinetic energy is 500 GeV in the CoM frame? Note that the given energy is not in the CoM frame (and it is 5 GeV rather than 500 GeV).

I find the hint slightly misleading. I would not suggest doing everything in the CoM frame.

 

[Matt atkinson]

How do you know the kinetic energy is 500 GeV in the CoM frame? Note that the given energy is not in the CoM frame (and it is 5 GeV rather than 500 GeV).

I find the hint slightly misleading. I would not suggest doing everything in the CoM frame.

Okay, that's why i was confused seemed like it could be much easier in the frame where the proton is at rest, and use conservation of energy and momentum to work the mass of the X_o.
Thanks ill have a go at doing it now and post my results.

 

[Orodruin]

Note: I am not saying the CoM frame is useless in the problem ... My hint would have been that invariants are nice things.

 

[Matt atkinson]

I tried this:
$$p_e^2=E^2-m_e^2$$
$$m_e=0.511MeV/c^2$$
$$p_e=4.999GeV\approx 5GeV$$
so then before
$$\Sigma p=0+5GeV$$
$$\Sigma E=5GeV+m_p$$
and after..
$$\Sigma p=0$$
$$\Sigma E=m_{X^o}+m_e+m_p$$
I just can't seem to see where I am going wrong

 

[Orodruin]

Some comments so far:

The electron energy is 5 GeV so the total energy before is 5 GeV + proton mass. Now the electron mass is negligible as you have seen so you will anyway not make a big error adding it in.

For your equations for after collision, you have used the CoM system so you cannot relate your before and after directly. Do you know some quantity that is frame independent?

 

[Matt atkinson]

I could use the effective mass W squared?
and work it out for both before and after?

 

[Orodruin]

Yes, this would be my suggestion.

 

[Matt atkinson]

okay this is what I did (b denotes before, a denotes after):
$$W_b^2=E_b^2-p_b^2=E_a^2-p_a^2$$
$$as \ W_a=W_b$$
$$W=\sqrt{(5GeV+938.3MeV)2-(5Gev)^2}=3204MeV$$
neglecting the mass of the electron as m_e << m_p;
$$W^2=(m_p+m_{X^o})^2=m_{X^o}^2+2m_{X^o}m_p+m_p^2$$
I had to use quadratic formula? and got:
$$m_{X^o}=(-(2*938.3) \pm 6404 )MeV$$
but this doesn't seem right

 

[Orodruin]

And what possible roots does this correspond to? Which one is physical?

 

[mfb]

There is no need for solving a quadratic formula. If W^2 = x^2 for some x and you know the sign of x, then you can simply take the root on both sides.

 

[Matt atkinson]

the + root as negative mass is not physical, I see now! thanks :D

 

[Matt atkinson]

Oh so i should just have done:
$$W=m_X+m_p$$
by square rooting both sides

 

[mfb]

Yes.
The other approach is possible but way too complicated (and your result was not right).

 

[Matt atkinson]

Yes.
The other approach is possible but way too complicated (and your result was not right).

Thankyou! i got:
$$m_X=2270MeV$$