# Chirality vs Helicity in Top Quark Decay

I still have not had a good explanation of how a Right Handed Top Quark decays.

As I understand it, helicity and chirality are both a part of spin. Does this mean that if either is left handed, the quark decays? What if neither are left handed? Is this possible?

• Helicity is intrinsic in origin caused by the spin, chirality is not but a relativistic effect of helicity. You could say chirality is relativistic helicity. Commented Feb 20, 2022 at 23:25

Required reading. Recall the top is heavy, $$m_t\sim 2m_W$$, so there is no point in talking about its helicity. You go to its rest frame, where it has spin 1/2 along an axis; and moving up or down along that will result in positive or negative helicity. So, top helicity is out of the discussion when it comes to decays.

Being so heavy, its chirality is easily reversed by the huge mass term, so if you have the wrong chirality all you have to do is wait a little... very little. In its rest frame, the top is equal parts left and right chirality. Chirality is not a constant of the motion and if flips on time scales of $$O(1/m_t)$$

Nevertheless, chirality matters, since, nevertheless, the weak interaction only involves left chirality in its decay, period: $$g\overline{ b_L} ~~\gamma^\mu W^-_\mu t_L + \hbox{h.c}.$$ A left-chiral t decays to a $$W^+$$ and a left-chiral b (with a smattering of s and d).

Thus, formally, a right-chiral top, or a left-chiral antitop do not decay. But they can propagate to their enantiomorphs in no time!

The Ws produced are thus polarized either with negative helicity, or longitudinal! No positive helicity, W helicity states of the top quark decays. The red arrows indicate the spin direction, while the yellow and blue arrows indicate the direction of the momenta of the respective particles. The green top quark is taken to be at rest (not moving).

In theory, you may estimate their polarization ratio, $$\frac{\hbox{longitudinal}}{\hbox{total}}\sim \frac{1}{2(m_W/m_t)^2+1} \sim 0.7,$$ verified experimentally to high precision!

Angular distributions from the decay leptons of W bosons in top quark decays at three individual helicities, compared to the standard model mixture. The helicity fractions are extracted from fits to angular distributions of the decay leptons of the respective W bosons. As an example, above are shown the results from 8 TeV data taken by CMS: F0=0.681±0.012 (stat)±0.023 (syst), FL=0.323±0.008 (stat)±0.014 (syst), FR=−0.004±0.005 (stat)±0.014 (syst).

• thank you.. I am not a physicist or student.. Just an engineer with a taste for particle physics. How does the rest mass of the particle change its spin? How long does it take to change chirality? It has to be shorter than the time it takes to bond with a Left Handed -T quark.... (t meson)
– Rick
Commented Feb 20, 2022 at 23:03
• The rest mass does not change spin. Helicity is the spin projected on the direction of the momentum. So at rest, there is no such thing. If I move in the direction of spin, i have positive helicity. If in the opposite direction, then negative. Helicity is not Lorentz invariant for a massive particle, so, here, useless. Chirality is Lorentz invariant, but mass terms flip it easily. Quantifying the frequency of chirality oscillations is a messy and counterproductive/academic endeavor, probably worth its own question. Commented Feb 20, 2022 at 23:08

The decay of a top quark into a $$W^+$$ and a bottom quark can easily be understood when considerin the top as a combination of two 1/3 charged and one non-charged particle.

Top quark:

$$CCU$$

$$W^+$$:

$$CCCUUU$$

bottom quark:

$$\overline{C}\overline{U}\overline{U}$$

The $$W^+$$ and bottom quark contain exactly the same particles as the top quark, after cancelation of the particles and anti particles. The $$W^+$$ decays to an anti muon and an neutrino, as can be seen below. The $$W^+$$ is a fluctuation and the $$W^-$$ part delivers the anti bottom, anti charm quark, and a bottom quark (see picture below), all with the fitting chiralities and particle $$C$$ and $$U$$ particle content (a $$W^-$$ contains the same particles as the anti charm and bottom quark). Because neutrinos ($$UUU$$) are left handed only, so are the $$W^+$$, as $$UUU$$ is a part of them and they are vector particles (suggesting that electrons, $$\overline{C}\overline{C}\overline{C}$$ were all right handed only, when they hadn't yet interacted). So