Helicity combinations in $gg\rightarrow t\bar{t}$

So, I am trying to understand what helicity combinations can occur in the outgoing top-antitop pair in the tree-level scattering $$gg\rightarrow t\bar{t}$$. There are 3 diagrams to consider (see below), an s-, t- and a u-channel.

$gg\rightarrow t\bar{t}$">

I started thinking about the s-channel. Since the gluon is spin 1 and massless, it can have $$s_z=\pm1$$, thus the top and antitop must have $$s_z=1/2$$ or $$s_z=-1/2$$ each. Is that correct?

However, by that logic I do not understand how the two incoming gluons with $$s_z=\pm1$$ can fuse to the intermediate gluon with $$s_z=\pm1$$.

As for the helicities of the outgoing quarks in their COM frame, in this case it would mean that we either have a LH top and RH antitop or the other way around. Right?

As for the t-channel, if I apply my reasoning to the upper vertex, this means that for an incoming $$s_z=+1$$ gluon, the adjacent fermion lines must have $$s_z=1/2$$ each, forcing the lower gluon line to have $$s_z=+1$$ as well and the outgoing antitop must hence have $$s_z=1/2$$ too. In the $$t\bar{t}$$ COM frame we then have the same situation as for the s-channel. If the first incoming gluon has instead $$s_z=-1$$, we then have the exact opposite situation.

So, in the COM frame of the outgoing quarks we always have a combination of LH and RH helicities.

Is my reasoning sound? If not, where did I go wrong?

• That's true but I am asking about helicity, not chirality. – Thomas Wening Jan 27 at 17:23
• All virtual and all off shell. – Cosmas Zachos Jan 27 at 17:40
• Could you please elaborate that? I am not aware of any connection between a particle being off-shell, its chirality and its helicity. – Thomas Wening Jan 27 at 21:28
• Okay, then to circumvent the problematics of confinement, let us consider the following process instead $\gamma\gamma\rightarrow f\bar{f}$, where $f$ is any charged lepton. Chirality is evidently still conserved at all vertices but which helicities can the outgoing leptons have in their COM frame? Since we fix the frame, the helicities should be well-defined. Is my argumentation from above sound, when applied to this scenario? And does the answer indeed depend on the magnetic spin numbers of the incoming photons? – Thomas Wening Jan 28 at 11:49
• – Cosmas Zachos Jan 28 at 15:29

1 Answer

I'll only answer the restricted modified question. As per your comment:

... let us consider the following process instead, $$γγ→f\bar f$$, where f is any charged lepton.

The amp will preserve helicity and violate chirality, which gets easier and easier as the mass grows bigger, as is the case for your top. The products will have either both positive, or both negative helicity.

In the cm frame, your two hypothetical photons with helicities ± 1, will add to a total $$J_z$$ of 0 or ±2 along their direction of motion. Their symmetric combination is spinless, just like the charged pion; that is, they must preserve helicity, so the two fermion decay products must have identical helicities : +1/2 both, or else -1/2 both.

(This is what happens in the decay of $$\pi^+$$, where the product neutrino has helicity -1/2 and the $$\mu^+$$ has perforce helicity -1/2: it has violated its WI-dictated Right chirality, via its mass, to a "Left" helicity state. Of course, L and R helicity labels are inappropriate, but, as always, are meant to evoke the associated chiralities. But this is why decay to $$\mu$$ is preferred over decay to e.)

At the same time, by dint of the vector nature of the interactions, the fermion line will have the same chirality throughout, so each particle in the $$f\bar f$$ product pair will have one L and one R. This mismatch between helicity and chirality will ensure that if f is massless, when helicity coincides with chirality, the decay won't go: helicity wins. But, the larger the fermion mass, the more chirality may be violated; e.g., a L-chiral vertex state can propagate to a R-chiral component, and develop the requisite helicity.

• To be sure, gluon helicity amps are covered in Chapter 27 of M Schwartz's QFT and the SM standard text, but it might well be a bit out of my depth (and yours?) to adapt the question to it...
• Just to make sure: Chirality is conserved by the interaction term in the QED Lagrangian but the mass term causes the chiralities to couple, right? Shouldn't we then rather say QED conserves chirality but its conservation is violated by the mass term? – Thomas Wening Jan 29 at 13:31
• Right. And right. QED charcterizes the vertex. The mass term moots that. The bigger the mass, the sloppier the conservation principle. – Cosmas Zachos Jan 29 at 14:54