Decay into electron-positron pair in Yukawa theory

[Alhaurin]

Homework Statement

I have a question regarding exercise 48.4-b in Srednicki's QFT book (the chapter is related to Yukawa theories). I have the official solution + explanation to the problem but I still do not fully understand the reasoning used in it, so perhaps you can help me.

In the excersise we are asked to calculate the probability amplitude for decay of a scalar into an electron-positron pair, focusing on the conditions that make the amplitude zero. The pair's three momentum goes along the Z axis and spin is quantized on the X axis.

According to the solution, we get zero when the two fermion spins are opposite. The reason for this would be that parity forces the orbital angular momentum "l" of the pair to be odd (l=1): since the original particle had zero spin then necessarily the particles must have spins aligned on X to overcome the non-zero orbital angular momentum.

Homework Equations

The Attempt at a Solution

What I do not understand about that answer is that, as far as I know, the orbital angular momentum referred to in the problem is the quantum number "l" and not the value of angular momentun on the X direction. This means that theoretically the electron-positron pair could be in a |l,m_l> = |1,0> orbital, have spins in opposite directions and produce a zero spin value in the X direction.

In fact, the resulting state after the decay should be a combination of states with orbital numbers |l,m_l> = |1,-1>,|1,0>,|1,1> to preserve total angular momentum. That would include the aforementioned case where |l,m_l> = |1,0> and |s,m_s> = |1,0> (the latter representing the overall spin state of both particles).

My only guess is that perhaps the |l,m_l> = |1,0> orbital, when expressed in momentum space, has no momentum aligned in the Z axis and so for the purposes of this problem we should only focus on |l,m_l> = |1,-1>,|1,1>. Unfortunately, I have not been able to find useful information on the momentum space properties of orbitals and so I cannot verify this idea.

 

[Orodruin]

It does not matter which direction you select for the spins or for the direction. The only thing which is important from a symmetry point of view is that the final state transforms in the same way as the initial state. If the orbital state has l = 1, then the electron spins must align symmetrically into a l=1 state which in turn anti-aligns with the orbital state to give a total angular momentum of zero. This is the only way in which you can construct an overall singlet state.

 

[Alhaurin]

It does not matter which direction you select for the spins or for the direction. The only thing which is important from a symmetry point of view is that the final state transforms in the same way as the initial state. If the orbital state has l = 1, then the electron spins must align symmetrically into a l=1 state which in turn anti-aligns with the orbital state to give a total angular momentum of zero. This is the only way in which you can construct an overall singlet state.

Thank you very much for your answer.

Unfortunately I still do not understand why the pair cannot be in a |10> orbital state. According to Clebsh-Gordan coefficients, |00> = C1*|11>*|1-1> + C2*|10>*|10> + C3*|1-1>|11>, where the first |xy> state in |xy>*|ab> is the orbital wavefunction and |ab> the total spin state. Wouldn't the presence of C2*|10>*|10> mean that a quantum state with both spins antialigned (resulting in a total spin state |10>) be possible?

 

[Orodruin]

Thank you very much for your answer.

Unfortunately I still do not understand why the pair cannot be in a |10> orbital state. According to Clebsh-Gordan coefficients, |00> = C1*|11>*|1-1> + C2*|10>*|10> + C3*|1-1>|11>, where the first |xy> state in |xy>*|ab> is the orbital wavefunction and |ab> the total spin state. Wouldn't the presence of C2*|10>*|10> mean that a quantum state with both spins antialigned (resulting in a total spin state |10>) be possible?

The |10> state has the electron spins aligned, not anti-aligned. It is only the anti-symmetric |00> state which has the electron spins anti-aligned.

In other words, it is the total angular momentum l which tells you whether the spins are aligned or anti-aligned, not the component in a particular direction. You can get a state with a particular component equal to zero by simply rotating another state (given integer spin). Only the total angular momentum is invariant under rotations.