6 Weak interactions and invariance principles
6.1 Characteristics of weak interactions
In the previous sections we talked about interactions that are mediated by photons (QED)
and gluons (QCD). In terms of relative strength we said that strong interactions are
“stronger” than electromagnetic ones. In the same spirit we introduce weak interaction
which, as the word says, have a weaker relative strength than the electromagnetic ones.
Weak interactions are responsible for radioactive beta decays in nuclear physics, as well as
many other phenomena in particle physics, which are the subject of this unit.
As previously mentioned, weak interactions are mediated by massive bosons, the 𝑊𝑊 ± and
the 𝑍𝑍 0 with a mass of approximately 80 GeV/c2 and 91 GeV/c2, respectively. All fermions in
the standard model interact via the exchange of weak bosons, this includes the leptons
(both charged and neutrinos) and the quarks. Therefore neutrinos can only interact via
weak interactions, which makes very difficult to detect, as their interaction cross section is
very small.
There are two types of weak interactions, based on the boson being exchanged
1) Charged-current interactions are mediated by the exchange of 𝑊𝑊 ± bosons with
diagrams like the one in Fig 6.1(a) and (b). These interactions are between particles
at the top/bottom of the leptons and quarks pairs. Interactions between different
lepton family are forbidden, while interactions between different quark families are
allowed. Hence the conservation of the 3 lepton numbers and one baryon number.
Charged-current interactions are responsible for the nuclear 𝛽𝛽-decays, and the muon
decay, see Fig 6.2 (right), and for neutrino interactions, see Fig 6.2 (left)
2) Neutral current interactions are mediated by the exchange of 𝑍𝑍 0 bosons and
interactions are with leptons or quarks of the same type (or flavour).
Fig 6.1 Weak interactions vertexes mediated by the W boson coupling leptons (left) and
quarks (right)
49
, Fig 6.2. (a) Neutrino-neutron interactions mediated by the charged current. (b) Muon decay.
From Perkins fig 2.6.
6.2 Comparing interaction types
Now that we introduced the three fundamental interactions in particle physics, we can
compare them and review how we evaluate their Feynman diagrams to study physical
processes. Figure 6.3 shows Feynman diagrams comparing simple 2 → 2 processes
mediated by electromagnetism, strong and weak forces.
In Unit 2 we discussed the Fermi golden rule and we said that the rate of a process is
proportional to the square of the matrix element. We defined the matrix element as the
Fourier transform of the potential and described it using a Feynman diagram. The matrix
element is proportional to
- The product of the “charges” at the vertexes. In the case of electromagnetism, that
is the electric charge, or the constants 𝑔𝑔𝑠𝑠 , 𝑔𝑔𝑤𝑤 in the case of the strong and weak
interactions, respectively
1
- The distance 1/r in the case of massless boson mediating the force or 𝑒𝑒 −𝑟𝑟/𝑅𝑅 , where
𝑟𝑟
the range R is related to the mass of the mediator 𝑅𝑅 = ℏ/𝑚𝑚𝑚𝑚 as shown in Exercise A.
We express the interaction rates in terms of proportionality to the factors above, so we
have
- For the electromagnetic interaction, the potential is proportional to
𝑈𝑈 ∝ 𝑒𝑒 2 /𝑟𝑟
therefore the matrix element is proportional to its Fourier transform, which we
calculated as
𝑀𝑀𝑓𝑓𝑓𝑓 ∝ 𝑒𝑒 2 /𝑞𝑞 2
and the rate is proportional to
𝑒𝑒 2 𝛼𝛼𝑒𝑒𝑒𝑒2
𝑊𝑊 ∝ 4 ∝ 4
𝑞𝑞 𝑞𝑞
Which we showed leads to an expression for the Rutherford cross section and its
dependency on sin2 θ/2.
- Likewise for the strong interaction at short distances (≪ 1 fm) , we can evaluate a
rate of interaction that depends on the strong constant 𝛼𝛼𝑆𝑆
𝑔𝑔𝑆𝑆4 𝛼𝛼𝑠𝑠
𝑊𝑊 ∝ 4 ∝ 4
𝑞𝑞 𝑞𝑞
50
6.1 Characteristics of weak interactions
In the previous sections we talked about interactions that are mediated by photons (QED)
and gluons (QCD). In terms of relative strength we said that strong interactions are
“stronger” than electromagnetic ones. In the same spirit we introduce weak interaction
which, as the word says, have a weaker relative strength than the electromagnetic ones.
Weak interactions are responsible for radioactive beta decays in nuclear physics, as well as
many other phenomena in particle physics, which are the subject of this unit.
As previously mentioned, weak interactions are mediated by massive bosons, the 𝑊𝑊 ± and
the 𝑍𝑍 0 with a mass of approximately 80 GeV/c2 and 91 GeV/c2, respectively. All fermions in
the standard model interact via the exchange of weak bosons, this includes the leptons
(both charged and neutrinos) and the quarks. Therefore neutrinos can only interact via
weak interactions, which makes very difficult to detect, as their interaction cross section is
very small.
There are two types of weak interactions, based on the boson being exchanged
1) Charged-current interactions are mediated by the exchange of 𝑊𝑊 ± bosons with
diagrams like the one in Fig 6.1(a) and (b). These interactions are between particles
at the top/bottom of the leptons and quarks pairs. Interactions between different
lepton family are forbidden, while interactions between different quark families are
allowed. Hence the conservation of the 3 lepton numbers and one baryon number.
Charged-current interactions are responsible for the nuclear 𝛽𝛽-decays, and the muon
decay, see Fig 6.2 (right), and for neutrino interactions, see Fig 6.2 (left)
2) Neutral current interactions are mediated by the exchange of 𝑍𝑍 0 bosons and
interactions are with leptons or quarks of the same type (or flavour).
Fig 6.1 Weak interactions vertexes mediated by the W boson coupling leptons (left) and
quarks (right)
49
, Fig 6.2. (a) Neutrino-neutron interactions mediated by the charged current. (b) Muon decay.
From Perkins fig 2.6.
6.2 Comparing interaction types
Now that we introduced the three fundamental interactions in particle physics, we can
compare them and review how we evaluate their Feynman diagrams to study physical
processes. Figure 6.3 shows Feynman diagrams comparing simple 2 → 2 processes
mediated by electromagnetism, strong and weak forces.
In Unit 2 we discussed the Fermi golden rule and we said that the rate of a process is
proportional to the square of the matrix element. We defined the matrix element as the
Fourier transform of the potential and described it using a Feynman diagram. The matrix
element is proportional to
- The product of the “charges” at the vertexes. In the case of electromagnetism, that
is the electric charge, or the constants 𝑔𝑔𝑠𝑠 , 𝑔𝑔𝑤𝑤 in the case of the strong and weak
interactions, respectively
1
- The distance 1/r in the case of massless boson mediating the force or 𝑒𝑒 −𝑟𝑟/𝑅𝑅 , where
𝑟𝑟
the range R is related to the mass of the mediator 𝑅𝑅 = ℏ/𝑚𝑚𝑚𝑚 as shown in Exercise A.
We express the interaction rates in terms of proportionality to the factors above, so we
have
- For the electromagnetic interaction, the potential is proportional to
𝑈𝑈 ∝ 𝑒𝑒 2 /𝑟𝑟
therefore the matrix element is proportional to its Fourier transform, which we
calculated as
𝑀𝑀𝑓𝑓𝑓𝑓 ∝ 𝑒𝑒 2 /𝑞𝑞 2
and the rate is proportional to
𝑒𝑒 2 𝛼𝛼𝑒𝑒𝑒𝑒2
𝑊𝑊 ∝ 4 ∝ 4
𝑞𝑞 𝑞𝑞
Which we showed leads to an expression for the Rutherford cross section and its
dependency on sin2 θ/2.
- Likewise for the strong interaction at short distances (≪ 1 fm) , we can evaluate a
rate of interaction that depends on the strong constant 𝛼𝛼𝑆𝑆
𝑔𝑔𝑆𝑆4 𝛼𝛼𝑠𝑠
𝑊𝑊 ∝ 4 ∝ 4
𝑞𝑞 𝑞𝑞
50
