Why is the decay channel $H \to \gamma\gamma$ direct evidence that the spin of the Higgs must be different from one?

Question

The title says it all really, I searched this website and came across a post with a question titled Why is the Higgs boson spin 0?. But it doesn't really answer my question in the title.

But this next post I found on this site is much more strongly related to my question.

In fact, it is one of the answers to the question in that post which I am essentially questioning. Here is a direct quote for the answer given by @Chiral Anomaly:

The Higgs boson has spin $0$. A photon has spin $1$. The total angular momentum cannot change in the decay, so a Higgs boson cannot decay into a single photon, regardless of the energy. But the total angular momentum of two photons can be zero (because their spins can be oriented in opposite directions), so this decay mode can conserve angular momentum.

The problem with this quote is that it just seems to be just concerned with conserving the spin projection quantum number - which I denote here by $s_z$ and $s$ for the spin quantum number. The $z$-direction in $s_z$ is arbitrary and could be any direction.

As I understand it, photons always have spin, $s=1$. Just like electrons always have a spin of $s=1/2$, but can have $2s+1 (= 2)$ spin projections, namely, $s_z=\pm 1/2$.

So the only way to conserve spin projection would be to have one photon with $-1$ spin projection and the other photon with $+1$ spin projection. Or, have both photons with zero spin projection. That is the only way to conserve spin projection in the $H \to \gamma\gamma$ decay mode. But, the problem is we are not trying to conserve spin projection, but rather spin, $s$, itself.

I tried searching the internet for explanations of why that decay shows the spin of the Higgs boson is different from one, but I cannot get a straight answer. It must be something quite trivial as the original discovery paper in July 2012 simply stated in its abstract that "The decay to two photons indicates that the new particle is a boson with spin different from one" which can be found here after selecting the appropriate title paper which looks like:

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So, just to summarize, why does the Higgs diphoton decay prove that the spin of the Higgs is different from one?

Answer 1

You surely learned at the beginning of your elementary particles' course that the two photons are bosons, so the γγ state is symmetric in them.

Addition of two spin 1 states yields a spin s=1 only when they are in an antisymmetric state, $$ |1, 0\rangle= \frac{1}{\sqrt{2}}\bigl (|\uparrow\downarrow\rangle -|\downarrow\uparrow\rangle \bigr ), $$ but in which they cannot be, by boson symmetry, above. (Recall the cross product of two 3-vectors is a 3-pseudovector, antisymmetric in the two constituent vectors!) Spin projection and spin are preserved in this expression, but the antisymmetry of the wavefunction kills it for symmetric photons.

So the spin of the Higgs must be 0 or 2. Spin 2 is firmly excluded from production estimates, cf Fig 2 in ATLAS Aad, Georges, Tatevik Abajyan, Brad Abbott, Jalal Abdallah, S. Abdel Khalek, Rosemarie Aben, B. Abi et al. "Evidence for the spin-0 nature of the Higgs boson using ATLAS data" Physics Letters B 726 (2013) 120-144 .

Answer 2

In simple terms :

Spin had to be invented and assigned studying the interactions of elementary particles because of conservation of angular momentum. There are three laws which go through all physics theories/models as axioms, conservation of energy, conservation of momentum, and conservation of angular momentum.

Conservation of energy led to special relativity for the study of nuclear and elementary particle interactions , special relativity reduces to Galilean mathematically smoothly . Conservation of momentum at the center of mass of a particle forces it to decay to two particles.

For conservation of angular momentum:

The pi0 decays to two photons and the decay was seen long before a quark model was proposed for the theory of elementary particles.

The discovery of particles is a process a century long, and started with cosmic rays, where the pi+ and pi- were seen , and the pi0 inferred.

The existence of the neutral pion was inferred from observing its decay products from cosmic rays, a so-called "soft component" of slow electrons with photons. The π0 was identified definitively at the University of California's cyclotron in 1950 by observing its decay into two photons. Later in the same year, they were also observed in cosmic-ray balloon experiments at Bristol University.

The symmetry models developed at the time gave the pion spin zero, and this has not been falsified. I use the pi0 decay into two photons to give an example of entanglement, that if the spin projection of one photon is identified as +1, the other has -1, because of conservation of angular momentum, spin is part of the angular momentum of a system of particles.

The studies led to the eightfold way of symmetries for the particles discovered,

where the pi0 has to have spin zero from the symmetry.

The eightfold way eventually led to the symmetries of the present standard model

So the decay of Higgs to two photons, within our current model of particle physics with its symmetries , by conservation of angular momentum algebraically leads to the conclusion that it either has spin 0 or spin 2.