W Decays and Color

Tommaso Dorigo posted three physics questions on his blog. They’re rather easy and I hope any particle physics student could answer them correctly. His third question touches upon a favorite bit of phenomenology, so let me expand upon it a bit here.

The W boson decays to a pair of fermions nearly all of the time. I will not worry about radiative corrections – i.e., the “extra” photons and gluons that may be emitted in the process W→f+fbar (where “fbar” means an anti-fermion). As Tommaso points out, the weak interactions are universal – the probability of the W to decay into one particular f+fbar pair is the same as the probability for any other f+fbar pair. More precisely, the coupling constant is the same – the phase space will be smaller for heavier fermions than for lighter fermions, which reduces the likelihood that heavier fermions will materialize in a W decay. I will neglect these mass effects here.

So predicting a branching ratio such as BR(W→e+ν) amounts to counting all of the possible f+fbar pairs that a W boson can decay to. The branching ratio is then just one over that total number of possible final states.

How many such states are there, in the standard model – i.e., in the real world as we know it? For leptonic final states, we have (e,ν_e), (μ,ν_μ) and (τ,ν_τ) – this is quite clear. (Forgive me for not putting bars where they belong – it is hard to do it with this editor and it does not matter for the present discussion.) The quark final states require a little more care. Clearly, the top quark (mass = 172 GeV) is too heavy, but the other five quarks are not. You might think you have these six states: (u,d), (c,s), (u,s), (c,d), (u,b) and (c,b), based on electric charge. But the last four of these six states are not weak doublets. More to the point, the CKM matrix, which allows quarks from different weak doublets to couple to the W boson, is nearly diagonal, meaning that the (u,b) and (c,b) final states make a very small, even negligible contribution. Furthermore, the 2×2 sub-matrix which governs the (u,d) and (c,s) couplings is nearly unitary, so whatever part of (u,d) is reduced is picked up by (u,s), so to speak. In the end, because of this important and unexplained feature of the standard model, it is fine to just take the naive set (u,d) and (c,s) and ignore the mixing of weak doublets allowed by the CKM matrix.

If you’re quick and not careful, you’ll conclude that the W can decay only to the five states (e,ν_e), (μ,ν_μ), (τ,ν_τ), (u,d) and (c,s), and you would predict that BR(W→e+ν) = 1/5 = 0.2. This prediction is wrong, as measurements give BR(W→e+ν) = (10.75±0.13)% (see the Particle Data Group web page).

Color is the key to the calculation. Remember that quarks come in three colors (the conserved charge of the strong interaction), so when we consider W→u+dbar, there are three distinct channels, corresponding to u(red)+dbar(anti-red), u(blue)+dbar(anti-blue) and u(green)+dbar(anti-green). Notice that the W boson is a color singlet, so if we choose the color of the u-quark, then the color of the d-anti–quark is determined.

Revising our calculation, we have three leptonic states plus six quark states, so the naive prediction is BR(W→e+ν) = 1/9 = 11%, which is quite good indeed. The agreement with the experimental value is clear proof that there are three colors of quarks, and that the W couples to all fermion doublets with equal strength, modulo the factors incorporated in the CKM matrix. I find this a really very nice piece of physics.

This kind of simple phenomenological calculation is at the heart of basic experimental particle physics. It is nice to cast it as an exercise for the student, but in truth we do this kind of work whenever a new particle is observed. For example, a crude measurement of BR(W→e+ν) told us in the 1980s that the top quark mass must be at least M_W, else a smaller BR would have been observed. (What is that number, by the way? Take a look again at Tommao’s post.) It came as a bit of a surprise that M_t≈172 GeV, which of course is much too heavy to allow W→t+b, which is part of the reason why single-top production is so interesting.

In the 1990s, a parallel line of reasoning led to the conclusion that there are only three species of light neutrinos, through measurement of Z→ν+νbar. This is one of the most important and most beautiful of the results from LEP 1.

In the spirit of Tommaso’s post, let me pose a question for the reader. Suppose there were a hidden lepton charge, similar to color, so that there were two kinds of electrons, muons and taus. What would be the prediction for BR(W→e+ν_e), and to what degree is this excluded by the measured value?

An additional question: what can we conclude about W decays to exotic particles this way? I stated that W bosons decay only decay to fermion pairs. Why not to boson pairs??