### Video Transcript

Each of the following Feynman
diagrams shows how the Higgs boson can decay, apart from one. Which diagram shows a decay of the
Higgs boson that is not possible?

The Higgs boson can decay in a
number of different ways. So to identify the decay that is
not possible, we’ll need to identify which decay violates some rule of physics. All of the possible decays of the
Higgs boson follow the conservation rules. So let’s find the diagram where the
products don’t follow conservation rules like charge, lepton number, and baryon
number. Since each of these interactions
starts with a single Higgs boson, the overall charge, lepton number, and baryon
number of the products will need to be the same as the charge, lepton number, and
baryon number of the Higgs boson.

Recall that we write the symbol for
the Higgs boson with a superscript zero because the charge is zero. Furthermore, since the Higgs boson
is a boson, not a lepton or a quark, its lepton number and baryon number are both
zero. Let’s now go diagram by diagram and
check that the total charge, lepton number, and baryon number of the proposed
products are all zero.

In this first diagram, the proposed
products are a pair of neutral Z bosons. Since these are neutral particles,
their total charge is zero. And since they are bosons, their
lepton number and baryon number are also zero. So a pair of Z bosons has a total
charge of zero, a total lepton number of zero, and a total baryon number of
zero. So in this interaction, all three
quantities are conserved. Therefore, at least according to
these three measures, this diagram does show a possible decay of the Higgs
boson.

We have another diagram where the
decay products are a pair of bosons. In this case, it’s a W plus boson
with a relative charge of positive one and a W minus boson with a relative charge of
negative one, where the sign of the boson corresponds to the sign of the charge. Here again, since we’re dealing
with bosons, total lepton and total baryon number are zero. To find the total charge, we add
the positive one charge from the W plus boson and the negative one charge from the W
minus boson to get one minus one equals zero. So this diagram also shows a decay
that conserves their three quantities.

Moving on to the decay into a 𝜏
and an anti-𝜏 lepton, the relative charge of the 𝜏 is negative one and the
relative charge of the anti-𝜏 is positive one. So just like the W plus and W minus
bosons, the total charge is zero and is conserved. The 𝜏 and anti-𝜏 are also both
leptons, not baryons. So the total baryon number is
simply zero. Finally, the 𝜏 contributes a
lepton number of one. The anti-𝜏 contributes a lepton
number of negative one. And one minus one is zero. So lepton number is still
conserved. That is, 𝜏–anti-𝜏 pair has a
total charge, lepton number, and baryon number of zero is not surprising. The 𝜏 and anti-𝜏 are
antiparticles of each other. And for any pair of antiparticles,
even if those particles are themselves composite, the charge, lepton number, and
baryon number of one particle will be exactly canceled by the presence of its
antiparticle.

With this in mind, we can also see
without calculation that the bottom quark–antiquark pair is a possible decay of the
Higgs boson, since again we have a particle and its antiparticle. So the charge, lepton number, and
baryon number for the pair will be zero. If we were to do the calculation,
we’d find that the negative one-third relative charge of the bottom quark is exactly
canceled by the positive one-third relative charge of the bottom antiquark. Furthermore, neither of these are
leptons, so the total lepton number is still zero.

Finally, the baryon number is a
function of the difference between the number of quarks and the number of
antiquarks. Since there’s exactly one quark and
one antiquark, this difference is zero and the baryon number is zero. Thus, by calculation, we verified
what we already knew from the fact that the bottom quark and the bottom antiquark
are antiparticles of each other.

At this point, the only diagram
left that we haven’t verified conserves charge, lepton number, and baryon number is
this one. Let’s make sure that this diagram
actually does violate one of these three conservation rules, since there are
technically other rules we haven’t considered, like energy and momentum, that we
simply don’t have enough information to apply in this question.

In this diagram, the two proposed
products are a bottom quark and a charm antiquark. Neither of these is a lepton, so
lepton number is conserved. And just like for the bottom
quark–antiquark pair, one of these is a quark and one is an antiquark. So the baryon number is also
conserved. As for relative charge, recall that
the bottom quark has a relative charge of negative one-third. The charm quark has a relative
charge of positive two-thirds. Since the charm antiquark is the
antiparticle of the charm quark, it has a charge of negative two-thirds. However, when we add up this total
charge, negative one-third plus negative two-thirds is negative one. Negative one is not zero. So this interaction doesn’t
conserve total charge.

Since every possible decay of the
Higgs boson must follow conservation rules, including charge, this must not be a
possible decay of the Higgs boson. So the Higgs boson cannot decay
into a single bottom quark and a single charm antiquark.

Finally, it’s worth noting that the
Z boson is its own antiparticle and the W plus boson is the antiparticle of the W
minus boson. So the qualitative argument that we
developed for the 𝜏–anti-𝜏 pair that we then applied to the bottom quark–antiquark
pair could’ve also been applied to these two diagrams as well.