The pseudoscalar Yukawa theory has Lagrangian \begin{equation*} \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 + i g \phi \bar \psi \gamma^5 \psi \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 $\phi$.
Decay rate of boson
If $M > 2 m$, then the boson can decay into an electron and a positron. At tree level, the decay rate is [1] \begin{equation*} \Gamma = \frac{g^2}{8 \pi} \sqrt{M^2 - 4m^2} \end{equation*}
For example, if $ g = 1 $, $m = 1\ \text{GeV}$ and $M = 4\ \text{GeV}$, then the lifetime of the boson is $4.8\times 10^{-24} s$.
Scattering at tree level
The spin averaged amplitude for the scattering $e^+ e^- \to e^+ e^-$ is [2] \begin{equation}\label{eq:20160713c} \langle | \mathcal{M} |^2 \rangle = g^4 \left[ \frac{ s^2 }{(M^2 -s)^2} + \frac{s t}{(M^2 - s) (M^2 - t)} + \frac{ t^2}{(M^2 -t)^2} \right] \end{equation} From \eqref{eq:20160713c} 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} \qquad\text{with}\qquad \frac{d\sigma}{d\Omega} = \left( \frac{1}{ 8 \pi E} \right)^2 \langle | \mathcal{M} |^2 \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 again $ g = 1 $, $m = 1\ \text{GeV}$ and $M = 4\ \text{GeV}$. The Mathematica code can be found at the bottom of this post.
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| Total cross section of $e^+ e^- \to e^+ e^-$ in pseudoscalar Yukawa theory |
In the rest of the post, I calculate the loop correction to the boson propagator if $M > 2 m$. I decided to do this calculation because Srednicki only calculates the loop correction in the case $M < 2 m$. I find the case $M > 2m$ more interesting, because adding loops to the boson propagator resolves the singularity in the total cross section.
Loop correction to the boson propagator
If $M > 2 m$, the sum of the 1PI diagrams of the boson propagator is [3] \begin{equation*} \Pi(p) = \frac{g^2}{4 \pi^2} \left( \int_0^1\!\!\! dx \left( m^2 - 3 x (1-x) p^2 \right) \log \frac{D(x,p)}{| D(x,M) |} + \kappa\ (M^2 - p^2)\right) \end{equation*} with \begin{equation}\label{eq:20160715a} \kappa = \int_0^1\!\!\! dx\ x (1-x) \frac{3 x (1-x) M^2 - m^2}{D(x,M)} \end{equation} and \begin{equation*} D(x,p) = - x(1-x) p^2 + m^2 \end{equation*}
Scattering at one loop
The spin averaged amplitude for the scattering $e^+ e^- \to e^+ e^-$, including the loops in the boson propagator, is \begin{equation}\label{eq:20160713e} \langle | \mathcal{M} |^2 \rangle = g^4 \left[ s^2 | \Delta(p_1 + p_2)|^2 + \frac{1}{2} s t \left( \Delta(p_1 -p_3)\overline{\Delta(p_1 + p_2)} +\Delta(p_1 + p_2)\overline{\Delta(p_1 - p_3)}\right)+ t^2 | \Delta(p_1 - p_3) |^2 \right] \end{equation} with \begin{equation*} \Delta(p) = \frac{1}{p^2 - M^2 + \Pi(p)} \end{equation*} I again plot the cross section for the same values $ g = 1 $, $m = 1\ \text{GeV}$ and $M = 4\ \text{GeV}$, now using \eqref{eq:20160713e}
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| Total cross section of $e^+ e^- \to e^+ e^-$ in pseudoscalar Yukawa theory black line: tree level; red line: with loop correction in boson propagator |
It is also known that the width of the resonance is related to the lifetime of the particle. The shorter the particle lives, the wider the resonance. If $M$ increases, $\Gamma$ increases, thus the lifetime decreases, thus the resonance becomes wider. This is illustrated in the next graph.
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| Total cross section of $e^+ e^- \to e^+ e^-$ for $M=4\ \text{GeV}$ and $M = 10\ \text{GeV}$ |
[1] This is exercise 48.4 in Srednicki. The answer can be checked against notes by A. George.
[2] I obtained this formula as follows. Srednicki calculates the amplitude for the same process in the Yukawa theory (not the pseudoscalar Yukawa theory) in chapter 48. I repeated his calculation, changing $g$ to $i g \gamma^5$ at the appropriate places. This leads to extra factors of $\gamma^5$ in the trace formulas as compared to the Yukawa theory, finally giving formula \eqref{eq:20160713c}.
[3] I obtained this formula as follows. Srednicki calculates the loop correction in the case $M < 2 m$ in chapter 51. The case $M > 2 m$ is the same apart from the on-shell conditions. In the case $M < 2m$ one can impose $\Pi(M) = \Pi'(M)=0$. In the case $M > 2 m$ one can only impose $\Re ( \Pi(M) ) = \Re ( \Pi'(M) )=0$. This is explained on page 152 in Srednicki.
[4] As written in \eqref{eq:20160715a}, the integral for $\kappa$ diverges. The integral should be interpreted as \begin{equation}\label{eq:20160715b} \kappa = \int_0^1\!\!\! dx\ x (1-x) \frac{3 x (1-x) M^2 - m^2 + i \epsilon}{- x (1-x) M^2 + m^2 - i \epsilon} \end{equation} with $\epsilon$ a small positive real number. The integral can then be calculated for non-zero $\epsilon$. After taking $\epsilon$ to zero, the result is \begin{equation*} \kappa = -\frac{1}{2} - \frac{2 m^2}{M^2} + \frac{m^4}{M^4} \frac{1}{W} \log\frac{1- W}{1+W} \end{equation*} with $W = \sqrt{1 - 4 m^2 / M^2}$
Mathematica code for graph at tree level
dim = 0.389; (* GeV^2 mbarn from http://pdg.lbl.gov/1998/consrpp.pdf*)
pseudoM2[g_,m_,M_,s_,t_,u_]:= g^4 ( s^2/(s - M^2)^2+ s t /((s - M^2) (t - M^2)) + t^2/(t - M^2)^2 )
pseudod\[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 pseudoM2[g,m,M,s,t,u]]
pseudo\[Sigma][g_?NumericQ,m_?NumericQ,M_?NumericQ,EE_?NumericQ]:= 2 Pi NIntegrate[pseudod\[Sigma][g,m,M,EE,\[Theta]] Sin[\[Theta]],{\[Theta],0,Pi}]
Mathematica code for graph at loop level
\[CapitalDelta][x_,m_,p2_]:= m^2 - x( 1-x) p2mylog[z_]/; z \[Element] Reals:= If[z>0,Log[z],If[z<0, Log[Abs[z]] - I Pi]]
I include the singularities in the integration specification to help NIntegrate, see Mathematica help pages
\[CapitalPi][g_?NumericQ,m_?NumericQ,M_?NumericQ,p2_?NumericQ] /;M > 2 m:= Module[{term1, term2, term3,x1,x2,x,W},
x1 = 1/2 ( 1 - Sqrt[1 - 4 m^2/p2]);
x2 = 1/2 ( 1 + Sqrt[1 - 4 m^2/p2]);
term1 = If[p2 < 4 m^2, NIntegrate[ ( m^2 - 3 x (1-x) p2) mylog[\[CapitalDelta][x,m,p2]] ,{x,0,1},PrecisionGoal->3], NIntegrate[ ( m^2 - 3 x (1-x) p2) mylog[\[CapitalDelta][x,m,p2]] ,{x,0,x1,x2,1},PrecisionGoal->3]];
x1 = 1/2 ( 1 - Sqrt[1 - 4 m^2/M^2]);
x2 = 1/2 ( 1 + Sqrt[1 - 4 m^2/M^2]);
term2 = NIntegrate[ ( m^2 - 3 x (1-x) p2) mylog[Abs[\[CapitalDelta][x,m,M^2]]] ,{x,0,x1,x2,1},PrecisionGoal->3];
W = Sqrt[1 - 4 m^2/M^2];
term3 = -(1/2)-2 m^2/ M^2+4 m^4 / (M^4 W) Log[(1 - W)/(1 + W)];
g^2/(4 Pi^2)(term1 - term2 + term3 (M^2 - p2))]
PropLoop[g_,m_,M_,p2_]:= 1 / (p2 - M^2 + \[CapitalPi][g,m,M,p2])
pseudoM2WithLoop[g_,m_,M_,s_,t_,u_]:= g^4 ( s^2 Abs[PropLoop[g,m,M,s]]^2+ 1/2 s t (PropLoop[g,m,M,s] Conjugate [PropLoop[g,m,M,t]] + PropLoop[g,m,M,t] Conjugate [PropLoop[g,m,M,s]]) + t^2Abs[PropLoop[g,m,M,t]]^2 )
pseudod\[Sigma]WithLoop[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 pseudoM2WithLoop[g,m,M,s,t,u]]
pseudo\[Sigma]WithLoop[g_?NumericQ,m_?NumericQ,M_?NumericQ,EE_?NumericQ]:= 2 Pi NIntegrate[pseudod\[Sigma]WithLoop[g,m,M,EE,\[Theta]] Sin[\[Theta]],{\[Theta],0,Pi},PrecisionGoal->2]
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