Friday, July 8, 2016

Scattering in Yukawa theory

I illustrate the cross section of the scattering $e^+ e^- \to e^+ e^-$ in Yukawa theory.
The Yukawa theory has Lagrangian \begin{equation}\label{eq:20160707a} \mathcal{L} = \frac{1}{2} \left( \partial_{\mu} \phi \right)^2 - \frac{1}{2} M^2 \phi^2 + \bar \psi \left( i \gamma^{\mu} \partial_{\mu} \psi - m \right) \psi + \frac{\lambda}{4!} \phi^4 + g \bar \psi \psi \phi \end{equation}

The physical interpretation of this Lagrangian is as follows. $\phi$ is a real scalar field describing a boson with spin $0$ and mass $M$. $\psi$ is a Dirac spinor describing an electron and a positron with mass $m$. These particles interact with one another by exchanging the boson $\phi$. It can be shown that in this theory the force between electrons and electrons, between electrons and positrons and between positrons and positrons is always positive. Furthermore $\phi$ can interact with itself via the quartic coupling $\phi^4$. This Lagrangian is well-known in QFT. It was originally proposed to describe the strong nuclear force [1], but later it was found that quantum chromo dynamics gives a more fundamental description of the nuclear force. Yukawa interactions are important in the Standard Model: there are Yukawa interactions between the Higgs field and massless quark and lepton fields [2].

In chapter 48 in his book, Srednicki calculates the spin averaged amplitude for the scattering $e^+ e^- \to e^+ e^-$. At tree level this is \begin{equation}\label{eq:20160707b} \langle | \mathcal{M} |^2 \rangle = g^4 \left[ \frac{ (s - 4m^2)^2}{(M^2 -s)^2} + \frac{s t - 4 m^2 u}{(M^2 - s) (M^2 - t)} + \frac{ (t - 4m^2)^2}{(M^2 -t)^2} \right] \end{equation} with $s$, $t$ and $u$ the Mandelstam variables. From \eqref{eq:20160707b} one can calculate the total cross section. In the center of mass frame this is \begin{equation*} \sigma = 2 \pi \int_0^{\pi}\!\! d\theta\ \sin\theta \frac{d\sigma}{d\Omega} \end{equation*} with \begin{equation*} \frac{d\sigma}{d\Omega} = \left( \frac{1}{ 8 \pi E} \right)^2 \langle | \mathcal{M} | \rangle \end{equation*} Here $E$ is the center of mass energy, i.e. the sum of the energy of the incoming electron and positron. I plot $\sigma$ as function of $E$. I take $g = 0.01$, $m = 1\ \text{GeV}$, $M = 4\ \text{GeV}$. The Mathematica code can be found at the bottom of this post.

Total cross section of $e^+ e^- \to e^+ e^-$ in Yukawa theory
There is a spike at $E = 4 \text{GeV}$. At this energy, the electron and positron have enough energy to produce a boson, which then decays into an electron and positron. This is called a resonance. Resonances are one way to find new particles in colliders. For example, here is a graph [3] of scattering experiments that shows a resonance at 91 GeV, which is the mass of the $Z$ boson.

In the next post, I study the effect of loops on the shape of the resonance.

[1] Yukawa's pion
[2] Yukawa interaction on Wikipedia
[3] The W and Z at LEP, CERN COURIER, May 4, 2004

Mathematica code
dim = 0.389; (* GeV^2 mbarn from*) 

M2[g_,m_,M_,s_,t_,u_]:= g^4 ( ( s - 4 m^2)^2/(s - M^2)^2+ (s t - 4 m^2 u)/((M^2 - s) (M^2 - t)) + ( t - 4 m^2)^2/(t - M^2)^2 ) 

d\[Sigma][g_,m_,M_,EE_,\[Theta]_]:= Module[{s = (2 EE)^2, t = - 4 (EE^2 - m^2) Sin[\[Theta]/2]^2, u = - 4 (EE^2 - m^2) Cos[\[Theta]/2]^2},dim /(16 Pi EE)^2 M2[g,m,M,s,t,u]]

[Sigma][g_,m_,M_,EE_]:= 2 Pi NIntegrate[d\[Sigma][g,m,M,EE,\[Theta]] Sin[\[Theta]],{\[Theta],0,Pi}] 

(* x axis in GeV, y axis in mbarn *) 
LogPlot[\[Sigma][0.1,1,4,EE/2],{EE,1.01,10},PlotRange->All,AxesLabel->{"E in GeV","\[Sigma] in mbarn"}]

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