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Each of the following Feynman diagrams shows how the Higgs boson can decay, apart from one.
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Which diagram shows a decay of the Higgs boson that is not possible?
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The Higgs boson can decay in a number of different ways.
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So to identify the decay that is not possible, we’ll need to identify which decay violates some rule of physics.
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All of the possible decays of the Higgs boson follow the conservation rules.
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So let’s find the diagram where the products don’t follow conservation rules like charge, lepton number, and baryon number.
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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.
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Recall that we write the symbol for the Higgs boson with a superscript zero because the charge is zero.
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Furthermore, since the Higgs boson is a boson, not a lepton or a quark, its lepton number and baryon number are both zero.
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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.
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In this first diagram, the proposed products are a pair of neutral Z bosons.
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Since these are neutral particles, their total charge is zero.
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And since they are bosons, their lepton number and baryon number are also zero.
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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.
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So in this interaction, all three quantities are conserved.
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Therefore, at least according to these three measures, this diagram does show a possible decay of the Higgs boson.
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We have another diagram where the decay products are a pair of bosons.
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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.
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Here again, since we’re dealing with bosons, total lepton and total baryon number are zero.
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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.
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So this diagram also shows a decay that conserves their three quantities.
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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.
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So just like the W plus and W minus bosons, the total charge is zero and is conserved.
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The 𝜏 and anti-𝜏 are also both leptons, not baryons.
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So the total baryon number is simply zero.
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Finally, the 𝜏 contributes a lepton number of one.
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The anti-𝜏 contributes a lepton number of negative one.
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And one minus one is zero.
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So lepton number is still conserved.
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That is, 𝜏–anti-𝜏 pair has a total charge, lepton number, and baryon number of zero is not surprising.
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The 𝜏 and anti-𝜏 are antiparticles of each other.
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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.
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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.
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So the charge, lepton number, and baryon number for the pair will be zero.
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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.
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Furthermore, neither of these are leptons, so the total lepton number is still zero.
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Finally, the baryon number is a function of the difference between the number of quarks and the number of antiquarks.
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Since there’s exactly one quark and one antiquark, this difference is zero and the baryon number is zero.
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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.
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At this point, the only diagram left that we haven’t verified conserves charge, lepton number, and baryon number is this one.
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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.
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In this diagram, the two proposed products are a bottom quark and a charm antiquark.
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Neither of these is a lepton, so lepton number is conserved.
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And just like for the bottom quark–antiquark pair, one of these is a quark and one is an antiquark.
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So the baryon number is also conserved.
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As for relative charge, recall that the bottom quark has a relative charge of negative one-third.
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The charm quark has a relative charge of positive two-thirds.
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Since the charm antiquark is the antiparticle of the charm quark, it has a charge of negative two-thirds.
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However, when we add up this total charge, negative one-third plus negative two-thirds is negative one.
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Negative one is not zero.
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So this interaction doesn’t conserve total charge.
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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.
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So the Higgs boson cannot decay into a single bottom quark and a single charm antiquark.
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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.
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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.