Spin-parity of two photons

[tg85]

I am trying to understand how charmonium states can be produced in electron-positron collisions, and which quantum numbers are possible for each process. I am having trouble understanding the quantum numbers that are possible for the two-photon process, e⁺e^- → e⁺e^- γγ → e⁺e^- cc̅.

I have read in several places [1-3] that the C-parity for the final states from these reactions must be +1. This I get, since both photons have C-parity -1. However, J^P seems to be restricted to 0^±, 2^±, 3⁺, 4^±, 5⁺, ... This I don't get.

The spins of the two photons couple to 0 or 2, correct? I tried to come up with allowed quantum numbers by assuming different orbital angular momenta, starting from 0, and then trying to combine them (somewhat naively) with the 0/2 state. I tried to get the final state's parity using (-1)(-1)(-1)^L. Obviously, this leads to completely different values, including, for example, J^PC=1⁺⁺.

What would be the right way to understand the allowed quantum numbers? Is it possible to calculate them in such a simplistic way?

[1] http://www-conf.kek.jp/qwg08/session3_3/uehara.pdf
[2] http://inspirehep.net/record/1257857/files/Beauty 2013_048.pdf
[3] http://arxiv.org/pdf/1311.0968v1

 

[AndyW]

.pdf
Thank you for your question about charmonium states in electron-positron collisions. I will try my best to explain the possible quantum numbers for this process.

First of all, it is important to understand that the quantum numbers for a particle describe its intrinsic properties such as spin, parity, and charge. In the case of charmonium states, we are dealing with a system of a charm quark and an anti-charm quark bound together by the strong nuclear force. This means that the quantum numbers of the charmonium states are determined by the properties of the charm quark and the anti-charm quark.

Now, let's look at the two-photon process e+e- → e+e- γγ → e+e- cc̅. In this process, two photons are produced from the initial electron-positron collision. These photons then interact with each other to create a charm-anticharm pair. The quantum numbers of the final state are determined by the quantum numbers of the photons and the charm-anticharm pair.

As you correctly stated, the C-parity of the final state must be +1 because both photons have C-parity -1. This means that the charm-anticharm pair must also have C-parity +1. The possible quantum numbers for the charm-anticharm pair can be determined by considering the possible combinations of spin, parity, and charge for a charm and an anti-charm quark.

The spin of a charm quark is 1/2, while the spin of an anti-charm quark is -1/2. This means that the total spin of the charm-anticharm pair can be either 0 or 1. The parity of a charm quark is -1, while the parity of an anti-charm quark is +1. This means that the total parity of the charm-anticharm pair can be either -1 or +1.

Combining these possibilities, we can see that the possible quantum numbers for the charm-anticharm pair are 0-, 1-, 0+, or 1+. These correspond to the possible charmonium states J/ψ, ψ(2S), χc0, and χc1, respectively. Other states with higher quantum numbers are also possible, but the ones listed here are the most commonly studied in electron-positron collisions