Quantum field theory 1, lecture 26
Higgs/Yukawa theory
In the following two lectures we will discuss a quantum field theoretic model that extends somewhat beyond quantum electrodynamics. We add to the theory a neutral massive scalar field that couples to the fermions through a Yukawa interaction. One may see that additional massive scalar particle as an analog of the Higgs boson, even though our model reflects only a few of the properties of the real electroweak standard model.
We discuss the model as a further example for an interesting quantum field theory and because we can nicely study there decay processes.
A massive Higgs boson can decay into two fermions through the Yukawa interaction. This is a tree level process and rather easy to calculate.
Interestingly a neutral and massive Higgs boson can also decay into two photons. This process is not allowed at tree level (because the Higgs boson is neutral), but it is induced by loop diagrams. This will be the first loop diagram we will calculate in detail.
In the second part of the lecture course loop diagrams and their physical consequences will be studied in much more detail. For the Higgs decay into photons we do not need renormalization yet, which simplifies the discussion. Nevertheless there will be some new elements to be discussed.
Action for Higgs/Yukawa theory and fermion mass
Let us consider the following extension of QED by a neutral scalar field (with \(m=gv\)) \[S[\bar{\psi},\psi, A, \phi] = \int_x \left\{ -\bar{\psi} \gamma^\mu\left(\partial_\mu - ieA_\mu \right)\psi - m\bar{\psi} \psi -\frac{1}{4} F^{\mu\nu}F_{\mu\nu} -\frac{1}{2} \phi \left(-\partial_\mu\partial^\mu + M^2 \right)\phi - g \phi\bar{\psi}\psi \right\}.\] Note that a constant (homogeneous) scalar field \(\phi\) modifies the fermion mass according to \[m_\text{eff} = m+ g\phi = g(v+\phi)\] In fact, one can understand the massses of elementary fermions (leptons and quarks) in the standard model of elementary particle physics as being due to such a scalar field expectation value for the Higgs field.
Propagators and vertices
In the theory above we have now different propagators
with scalar propagator \[\Delta(x-y) = \int_p e^{ip(x-y)} \frac{1}{p^2 + M^2}.\] The vertices are
Higgs decay into fermions
Higgs decay to fermions
Let us discuss first the process \(\phi \to f^{-}f^{+}.\) The fermions could be leptons (\(e\), \(\mu\), \(\tau\)) or quarks (\(u\), \(d\), \(s\), \(c\), \(b\), \(t\)). The Feynman diagram for the decay is simply
According to the Feynman rules we obtain \[\mathcal{T} = g \; \bar{u}_{s_1}(q_1) v_{s_2}(q_2).\] For the absolute square one finds \[|\mathcal{T}|^2 = g^2 \, \bar{u}_{s_1}(q_1) v_{s_2}(q_2) \, \bar{v}_{s_2}(q_2) u_{s_1}(q_1).\]
Spin sums and Dirac traces
We will assume that the final spins are not observed and sum them \[\sum_\text{spins} |\mathcal{T}|^2 = g^2 \; \text{tr}\left\{(-i\slashed{q}_2 - m)(-i\slashed{q}_1 + m) \right\}\] We used here again the spin sum formula \[\sum_s v_s(p) \bar{v}_s(p) = -i\slashed{p}-m,\quad\quad\quad \sum_s u_s(p) \bar{u}_s(p) = -i\slashed{p}+m.\] Performing also the Dirac traces gives \[\sum_{\text{spins}} |\mathcal{T}|^2 = g^2 \left(-4 q_1 \cdot q_2 - 4m^2 \right).\]
Kinematics in the Higgs boson rest frame
Let us now go into the rest frame of the decaying particle where \[p=(M,0,0,0), \quad\quad\quad q_1 = \left(\tfrac{M}{2}, \mathbf{q} \right),\quad\quad\quad q_2= \left(\tfrac{M}{2}, -\mathbf{q} \right),\] with \[\mathbf{q}^2 = -m^2 + \tfrac{M^2}{4}, \quad\quad q_1 \cdot q_2= -\frac{M^2}{4} - \mathbf{q}^2 = -\frac{M^2}{2}+m^2,\] and \[\sum_{\text{spins}}|\mathcal{T}|^2 = 2 \, g^2 M^2 \left(1-4\frac{m^2}{M^2}\right).\] Note that the decay is kinematically possible only for \(M>2m\) so that the bracket is always positive.
Decay rate
For the particle decay rate we get \[\frac{d \Gamma}{d\Omega} = \frac{|\mathbf{q}_1|}{32 \pi^2 M^2} \sum_{\text{spins}}|\mathcal{T}|^2 =\frac{g^2 M}{32 \pi^2}\left(1-4\frac{m^2}{M^2}\right)^{3/2}.\] Because this is independent of the solid angle \(\Omega\) one can easily integrate to obtain the decay rate \[\Gamma = \frac{g^2 M}{8\pi}\left(1-4\frac{m^2}{M^2}\right)^{3/2}.\]
Dependence on fermion mass
If the scalar boson \(\phi\) is the Higgs boson, the Yukawa coupling is in fact proportional to the fermion mass \(m\), \[g = \frac{m}{v}.\] One has then \[\Gamma = \frac{M^3}{32 \pi v^2} \, f\left(\frac{2m}{M} \right)\] where \[f(x) = x^2 (1-x^2)^{3/2}\] Decay into light fermions is suppressed because of small coupling while decay into very heavy fermions is suppressed by small phase space or even kinematically excluded for \(2m > M\).
For Higgs boson mass of \(M= 125\) GeV the largest decay rate to fermions is to \(b\bar{b}\) (bottom quark and anti-quark). This corresponds to \(m = 4.18\) GeV. The top quark would have larger coupling but is in fact too massive (\(m = 172\) GeV). (The lepton with largest mass is the tauon \(\tau\) with \(m = 1.78\) GeV.)