Rules for Hamiltonians

Rules for $H_ {3}^{(-)}$

Fig 1. Rules for various components of $H_3$. I regard them as vertices. The meaning of graphical elements are explain later.

Rules for $H_ {4}^{(-)}$

Fig 2. Rules for various components of $H_4$.

Rules for $H_ {5}^{(-)}$

Fig 3. Rules for various components of $H_5$. The coefficients $C_ {5,3}$ and $C_ {5,1}$ are made of counter terms. In specific we have $C_ {5,3}= \frac{m\delta g}{\sqrt{2}} +\frac{g \delta m_ {1}^{2}}{2\sqrt{2}m} - \frac{m^{2}\delta m_ {1}^{4}}{2\sqrt{2}g}$ and $C_ {5,1}= m^{2}\delta v_ {3}-\frac{3gm\mathcal{I}_ {1}}{\sqrt{2}}+\frac{m^{3}\delta g^{2}}{\sqrt{2}g^{3}}-\frac{m \delta g\delta m_ {1}^{2}}{2\sqrt{2}g^{2}} -\frac{\delta m_ {1}^{4}}{8\sqrt{2}gm}$.

Diagrams for States

Leading Order

At leading order there is one vacuum state correction and two momentum eigenstate corrections, their diagrammatic expressions are given in the figure below.

Fig 4. States at leading order. The first panel gives the leading order correction to the vacuum state, while the second and third panel gives the leading order correction to momentum states.

Next Leading Order

The fundamental diagrams for $\left\lvert \vec{p} \right\rangle_ {2}^{(-)}$ are shown in the below.

Fig 5. Fundamental diagrams of $g^2$ corrections to $\left\lvert \vec{p} \right\rangle_ {2}^{(-)}$ States. By fundamental I mean the diagrams directly built up block-by-block by Hamiltonians and lower order states. Fundamental diagrams are good for study the cancelation between loops and counter terms. After the renormalization procedure is done, based on the number of extarnal legs, the fundamental diagrams can ben summed over to give more concise, summarized diagrams.

Let’s number the diagrams left to right, top to bottom. The contributions are

\[\begin{align*} (1) :\;& H_ {4}^{(4)} \left\lvert \vec{p} \right\rangle_ {0}^{(1)} = \frac{g^{2}}{4} \int \frac{d^{3}p_ {1,2,3}}{(2\pi)^{9}} \, \left\lvert \vec{p},\vec{p}_ {1,2,3,-1-2-3} \right\rangle_ {0} , \\ (2) :\;& H_ {3}^{(3)} \left\lvert \vec{p} \right\rangle_ {1}^{(2)} = \frac{3g^{2}m^{2}}{4\omega_ {p}}\int \frac{d^{3}p_ {1,2,3}}{(2\pi)^{9}} \, \frac{\left\lvert \vec{p}-\vec{p}_ {3}, \vec{p}_ {1,2,3,-1-2} \right\rangle_ {0}}{\omega _ {p}-\omega_ {3}-\omega_ {p-p_ {3}}} , \\ (3) :\;& H_ {3}^{(3)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)} = -\frac{m^{2}g^{2}}{2} \int \frac{d^{3}p_ {1,2,3,4}}{(2\pi)^{12}} \, \frac{\left\lvert \vec{p},\vec{p}_ {1,2,3,4,-1-2,-3-4} \right\rangle_ {0}}{\omega_ {1}+\omega_ {2}+\omega_ {1+2}}, \\ (4) :\;& H_ {3}^{(1)} \left\lvert \vec{p} \right\rangle_ {1}^{(4)} \supset - \frac{3g^{2}m^{2}}{4} \int \frac{d^{3}p_ {1,2,3}}{(2\pi)^{9}} \, \frac{3\left\lvert \vec{p},\vec{p}_ {1,2,3,-1-2-3} \right\rangle_ {0}}{\omega_ {2+3}(\omega_ {1}+\omega_ {2+3}+\omega_ {1+2+3})} , \\ (5) :\;& H_ {3}^{(1)}\left\lvert \vec{p} \right\rangle_ {1}^{(2)} = - \frac{9g^{2}m^{2}}{4\omega_ {p}} \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{\left\lvert \vec{p}-\vec{p}_ {1}-\vec{p}_ {2},\vec{p}_ {1,2} \right\rangle_ {0}}{\omega_ {1+2}(-\omega _ {p} +\omega_ {p-p_ {1}-p_ {2}}+\omega_ {1+2})} , \\ (6) :\;& H_ {3}^{(1)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)} \supset - \frac{3g^{2}m^{2}}{4} \int \frac{d^{3}p_ {1,2,3}}{(2\pi)^{9}} \, \frac{\left\lvert \vec{p}_ {1,2,3,-1-2},\vec{p}-\vec{p}_ {3} \right\rangle_ {0}}{\omega_ {p}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} , \\ (7) :\;& H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {1}^{(2)} = -\frac{9g^{2}m^{2}}{8\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{\left\lvert \vec{p} \right\rangle_ {0}}{\omega_ {1}\omega_ {p+p_ {1}}(-\omega _ {p} +\omega_ {1}+\omega_ {p+p_ {1}})}, \\ (8) :\;& H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)} \supset - \frac{9m^{2}g^{2}}{4\omega _ {p} } \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{\left\lvert \vec{p}-\vec{p}_ {1}-\vec{p}_ {2},\vec{p}_ {1,2} \right\rangle_ {0}}{\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} \\ (9) :\;& H_ {3}^{(-3)}\left\lvert p \right\rangle_ {1}^{(4)} \supset - \frac{3}{8}g^{2}m^{2}\left\lvert \vec{p} \right\rangle_ {0}\int d^{3}x \, \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} \\ (10) :\;& - H_ {3}^{(-3)}\left\lvert p \right\rangle_ {1}^{(4)} \supset \frac{9g^{2}m^{2}\left\lvert \vec{p} \right\rangle_ {0}}{8\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1}\omega_ {p+p_ {1}}(\omega _ {p} +\omega_ {1}+\omega_ {p+p_ {1}})} \\ (11) :\;& H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)} \supset - \frac{9m^{2}g^{2}}{4} \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{\left\lvert \vec{p},\vec{p}_ {1,-1} \right\rangle_ {0}}{\omega_ {2}\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} \\ (12) :\;& H_ {4}^{(0)}\left\lvert p \right\rangle_ {0}^{(1)} \supset \int d^{3}x \, A_ {4}' ,\\ (13) : \;& H_ {4}^{(2)}\left\lvert p \right\rangle_ {0}^{(1)} \supset \frac{g^{2}}{2\omega _ {p} }\int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \left\lvert \vec{p}_ {1,2},\vec{p}-\vec{p}_ {1}-\vec{p}_ {2} \right\rangle, \\ (14) : \;& H_ {4}^{(2)}\left\lvert p \right\rangle_ {0}^{(1)} \supset - \frac{\delta m^{2}}{2}\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1,-1},\vec{p}\right\rangle ,\\ (15) : \;& H_ {4}^{(0)}\left\lvert p \right\rangle_ {0}^{(1)} \supset - \frac{\delta m^{2}}{4\omega p^{2}}\left\lvert \vec{p} \right\rangle. \end{align*}\]

But in order to turn them into diagrammatic rules for corresponding Hilbert states. We need to inverse $(\omega _ {p}-H_ {2})$ to get components of $\left\lvert \vec{p} \right\rangle_ {2}$, eliminate the integral measures and deal with the creation operators. Note that the 1-meson states will eventually cancel out. Furthermore, we strip-off the propagators in the diagram since we are regarding them as a vertex as a whole, as shown below.

Fig 6. Diagrams for order $\mathcal{O}(g^2)$ diagrams as vertices.

As diagrammatic rules we get:

\[\begin{align*} (1) :\;& H_ {4}^{(4)} \left\lvert \vec{p} \right\rangle_ {0}^{(1)} : -\frac{g^{2}}{4(\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {1+2+3})} , \\ (2) :\;& H_ {3}^{(3)} \left\lvert \vec{p} \right\rangle_ {1}^{(2)} :\frac{3g^{2}m^{2}}{4\omega_ {p}} \frac{1}{(-\omega _ {p}+\omega_ {3}+\omega_ {p-p_ {3}})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {1+2}+\omega_ {p-p_ {3}})} , \\ (3) :\;& H_ {3}^{(3)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)}:\frac{m^{2}g^{2}}{2(\omega_ {1}+\omega_ {2}+\omega_ {1+2})(\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {4}+\omega_ {1+2}+\omega_ {3+4})}, \\ (4) :\;& H_ {3}^{(1)} \left\lvert \vec{p} \right\rangle_ {1}^{(4)} : \frac{9g^{2}m^{2}}{4\omega_ {2+3}(\omega_ {1}+\omega_ {2+3}+\omega_ {1+2+3})(\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {1+2+3})} , \\ (5) : \;& H_ {3}^{(1)}\left\lvert \vec{p} \right\rangle_ {1}^{(2)} : \frac{9g^{2}m^{2}}{4\omega_ {p}}\frac{1}{\omega_ {1+2}(-\omega _ {p} +\omega_ {p-p_ {1}-p_ {2}}+\omega_ {1+2})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {p-p_ {1}-p_ {2}})} , \\ (6) :\;& H_ {3}^{(1)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)} : \frac{3g^{2}m^{2}}{4\omega_ {p}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {1+2}+\omega_ {p-p_ {3}})} , \\ (8) :\;& H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)} : \frac{9m^{2}g^{2}}{4\omega _ {p} } \, \frac{1}{\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})(-\omega _ {p} +\omega_ {p-p_ {1}-p_ {2}}+\omega_ {1}+\omega_ {2})}, \\ (11) :\;& H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {1}^{(8)} : \frac{9m^{2}g^{2}}{8} \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} ,\\ (13): \;& H_ {4}^{(2)}\left\lvert \vec{p} \right\rangle_ {0}^{(1)} : - \frac{g^{2}}{2\omega _ {p} } \frac{1}{(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {p-p_ {1}-p_ {2}})}, \\ (14): \;& H_ {4}^{(2)}\left\lvert \vec{p} \right\rangle_ {0}^{(1)} : \frac{\delta m^{2}}{4\omega_ {1}}. \end{align*}\]

3rd Order States

Let’s start with momentum states.

$\left\lvert \vec{p} \right\rangle_ {3}^{(2)}:$

$H_ {3}^{(-3)}\left\lvert \vec{p} \right\rangle_ {2}^{(5)}$:

It includes four contributions: $H_ {3}^{(-3)}$ acting on panel (1), (2), (4) and (6) from figure 6.

The contributions $H_ {3}^{(-3)}\cdot(1)$ are shown in Figure 7 by panels (a1,a2). We have

\[\begin{align*} (a1) =& \frac{3mg^{3}}{4\sqrt{2}} \left\lvert \vec{p},0 \right\rangle \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1+2}(m+\omega_ {1}+\omega_ {2}+\omega_ {1+2})}, \\ (a2) =& \frac{9mg^{3}}{8\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \frac{1}{\omega_ {2}\omega_ {p+2}(\omega_ {1}+\omega_ {2}+\omega_ {p-1}+\omega_ {p+2})} . \end{align*}\]

The components of $H_ {3}^{(-3)}\cdot(2)$ is shown in panel (b1,b2,b3):

\[\begin{align*} (b 1) =& - \frac{27m^{3}g^{3}}{8\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1-2}} \\ & \times \frac{1}{(-\omega _ {p} +\omega_ {1}+\omega_ {p-p_ {1}})(-\omega _ {p}+2\omega_ {1} +\omega_ {2}+\omega_ {1+2}+\omega_ {p-p_ {1}})}, \\ (b 2) =& - \frac{27m^{3}g^{3}}{16\sqrt{2}\omega _ {p} ^{2}} \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {p-p_ {2}}} \\ &\times \frac{1}{(-\omega _ {p} +\omega_ {2}+\omega_ {p-p_ {2}})(\omega_ {1}+\omega_ {2}+\omega_ {p-p_ {1}}+\omega_ {p-p_ {2}})} , \\ (b 3) =& - \frac{9m^{3}g^{3}}{16\sqrt{2}\omega _ {p} }\int d^{3}x \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2,3}}{(2\pi)^{6}} \, \frac{1}{\omega_ {2}\omega_ {3}\omega_ {2+3}} \\ &\times \frac{1}{(-\omega _ {p} +\omega_ {1}+\omega_ {p-p_ {1}})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {2+3}+\omega_ {p-p_ {1}})}. \end{align*}\]

Next we calculate $H_ {3}^{(-3)}\cdot(4)$, given in panel (c1,c2,c3) in figure 6:

\[\begin{align*} (c 1) =& - \frac{81m^{3}g^{3}}{8\sqrt{2}} \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3} p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega^{2}_ {1+2}} \\ &\times \frac{1}{(\omega_ {1}+\omega_ {1+2}+\omega_ {2p_ {1}+2})(\omega_ {1}+\omega_ {1+2}+\omega_ {3}+\omega_ {2p_ {1}+2})},\\ (c 2) =& - \frac{81m^{3} g^{3}}{16\sqrt{2}} \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega _ {p} \omega^{2}_ {p-2}} \\ &\times \frac{1}{(\omega_ {1}+\omega_ {2-p}+\omega_ {1+2-p})(\omega_ {1}+\omega_ {2}+\omega _ {p} +\omega_ {1+2-p})}, \\ (c 3) =& - \frac{27m^{3}g^{3}}{16\sqrt{2}} \int d^{3}x \, \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {3}\omega_ {2+3}} \\ &\times \frac{1}{\omega_ {1+2}(\omega_ {3}+\omega_ {1+2}+\omega_ {1+2+3})(\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {1+2+3})}. \end{align*}\]

The contribution of $H_ {3}^{(-3)}\cdot(6)$, given in panel (d1,d2,d3):

\[\begin{align*} (d 1) =& - \frac{3m^{3}g^{3}\left\lvert \vec{p},0 \right\rangle}{4\sqrt{2}} \int \frac{d^{3}p_ {1,2,3}}{(2\pi)^{9}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1+2}\omega_ {p}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} \\ &\times \frac{1}{(-\omega _ {p} +\omega_ {1}+\omega_ {2}+2\omega_ {1+2}+\omega_ {p-p_ {1}-p_ {2}})},\\ (d 2) =& - \frac{9m^{3}g^{3}}{4\sqrt{2}} \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2,3}}{(2\pi)^{6}} \, \frac{1}{\omega_ {2}\omega_ {3}\omega^{2} _ {p} } \\ &\times \frac{1}{ (\omega_ {2}+\omega_ {3}+\omega_ {2+3})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {2+3}+\omega_ {p-p_ {1}})} ,\\ (d 3) =& - \frac{9m^{3}g^{3}}{4\sqrt{2}} \int \frac{d ^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2,3}}{(2\pi)^{6}} \, \frac{1}{\omega_ {2}\omega_ {3}\omega _ {p} ^{2}} \\ & \times \frac{1}{(\omega_ {1}+\omega_ {2}+\omega_ {1+2})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {3}+\omega_ {1+2}+\omega _ {p-p_ {3}} )} . \end{align*}\]

The diagrams are shown below.

Figure 7. The diagrams of $H_3^{(-3)}\left\lvert \vec{p} \right\rangle_ {2}^{(5)}$.

$H_ {4}^{(0)}\left\lvert \vec{p} \right\rangle_ {1}^{(2)}:$

The diagrams are shown below.

Figure 8. Diagrams for $H_ {4}^{(0)}\left\lvert \vec{p} \right\rangle_ {1}^{(2)}$.

Their contributions correspondingly are

\[\begin{align*} (e 1) =& \frac{9mg^{3}}{8\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {p-2}(\omega _ {p} -\omega_ {2}-\omega_ {p-p_ {2}})} , \\ (e 2) =& -\frac{\delta m^{2} 3mg}{2\sqrt{2}\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{\left\lvert \vec{p}_ {1}, \vec{p}-\vec{p}_ {1} \right\rangle}{\omega_ {1}(-\omega _ {p} +\omega_ {1}+\omega_ {p-p_ {1}})}, \\ (e 3) =& \frac{3A_ {4}'mg}{2\sqrt{2}\omega _ {p} }\int d^{3}x \, \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{\left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle}{-\omega _ {p} +\omega_ {1}+\omega_ {p-p_ {1}}}. \end{align*}\]

$H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {2}^{(3)}:$

It includes three parts: $H_ {3}^{(-1)}$ acting on panel (5,8,11,13,14) in figure 6. The figure are shown below.

First of all, $H_ {3}^{(-1)}$ acting on panel (5). They correspond to panel (f1) and (f2) in figure 9 below.

Figure 9. The diagrams for $H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {2}^{(3)}$.

Their contributions are

\[\begin{align*} (f 1) =& - \frac{27g^{3}m^{3}}{8\sqrt{2}\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1-2}} \\ & \times \frac{1}{(-\omega _ {p} +\omega_ {1}+\omega_ {p-p_ {1}})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {1-2})}, \\ (f 2) =& -\frac{27m^{3}g^{3}}{4\sqrt2\omega _ {p}} \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {1+2}\omega_ {p-p_ {1}-p_ {2}}} \\ &\times \frac{1}{(-\omega _ {p} +\omega_ {p-p_ {1}-p_ {1}}+\omega_ {1+2})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {p-p_ {1}-p_ {2}})}, \end{align*}\]

Then we move on to $H_ {3}^{(-1)}$ acting on panel (8), given by panel (g1) and (g2) in figure 9. The contributions are

\[\begin{align*} (g 1) =& - \frac{27m^{3}g^{3}}{8\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}}\left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \, \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1}\omega_ {2}\omega_ {1-2}} \\ & \times \frac{1}{(\omega_ {1}+\omega_ {2}+\omega_ {1-2})(-\omega _ {p} +\omega_ {p-p_ {1}}+\omega_ {2}+\omega_ {1-2})}, \\ (g 2) =& - \frac{27m^{3}g^{3}}{4\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {1+2}\omega_ {p-p_ {1}-p_ {2}}} \\ & \times \frac{1}{(\omega_ {1}+\omega_ {2}+\omega_ {1+2})(-\omega _ {p} +\omega_ {1}+\omega_ {2}+\omega_ {p-p_ {1}-p_ {2}})}. \end{align*}\]

Then $H_ {3}^{(-1)}$ acting on panel (11),given by panel (h1,h2):

\[\begin{align*} (h 1) =& - \frac{27m^{3}g^{3}}{16\sqrt{2}} \left\lvert \vec{p},0 \right\rangle \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{1}{\omega_ {1}^{3}\omega_ {2}\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})}, \\ (h 2) =& - \frac{27m^{3}g^{3}}{8\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle \frac{1}{\omega_ {1}^{2}\omega_ {2}\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} . \end{align*}\]

It looks weird that we have a $\left\lvert \vec{p}=0 \right\rangle$ state… it just pops out from the vacuum. Anyway, let’s move on.

$H_ {3}^{(-1)}$ acting on panel (13),given by panel (i1):

\[(i 1) = \frac{9mg^{3}}{4\sqrt{2}\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {1-2}(-\omega _ {p} +\omega_ {2}+\omega_ {1-2}+\omega_ {p-p_ {1}})} ,\]

and $H_ {3}^{(-1)}$ acting on panel (13) represented by panel (j1,j2), whose contributions are

\[\begin{align*} (j 1) =& - \frac{3\delta m^{2}m g}{32\sqrt{2}} \left\lvert \vec{p},0 \right\rangle \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{1}{\omega^{3}_ {1}}, \\ (j 2) =& - \frac{3mg\delta m^{2}}{4\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{\left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle}{\omega_ {1}^{2}}. \end{align*}\]

$H_ {4}^{(-2)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)}:$

This contribution is relatively simple, including one contribution only: $H_ {4}^{(-2)}$ acting on panel 3 in Figure 4. The diagrams are given in Figure 10, with contributions

\[\begin{align*} k 1=& \frac{3mg^{3}\left\lvert \vec{p},0 \right\rangle}{4\sqrt{2}} \int \frac{d^{3}p_ {1,2}}{(2\pi)^{6}} \, \frac{1}{\omega_ {1} \omega_ {2} \omega_ {3}(\omega_ {1} + \omega_ {2} + \omega_ {1+2})} ,\\ k 2=& \frac{9 mg^{3}}{4\sqrt{2}\omega _ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \left\lvert \vec{p},\vec{p}-\vec{p}_ {1} \right\rangle \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {1+2}(\omega_ {1}+\omega_ {2}+\omega_ {1+2})} ,\\ \ell 1=& - \frac{3mg \delta m^{2} }{4\sqrt{2}} \left\lvert \vec{p},0 \right\rangle \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1}^{2}(2\omega_ {1}+m)} ,\\ \ell 2=& - \frac{3mg \delta m^{2}}{4\sqrt{2}\omega^{2}_ {p} } \int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{\left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle}{\omega_ {1}+\omega _ {p} +\omega_ {p-1}} . \end{align*}\]
Fig. 10 Diagrams for $H_ {4}^{(-2)}\left\lvert \vec{p} \right\rangle_ {1}^{(4)}$.

Reducible Diagrams

The blob stands for all divergent diagrams. These three are the divergent contributions to $\left\lvert \vec{p} \right\rangle_ {3}^{(2)}$, in the sense that they can be amputated by cutting an external leg.

Now we can sum up the previous results and organize diagrams according to their topology.

For the reducible diagrams we have

\[\begin{align*} H_ {4}^{(0)} \left\lvert \vec{p} \right\rangle_ {1}^{(2)} =& - \frac{3m\delta m^{2}g}{2\sqrt{2}\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \, \frac{\left\lvert \vec{p}_ {1}, \vec{p}-\vec{p}_ {1} \right\rangle_ {0}}{\omega_ {1}(-\omega _ {p} +\omega_ {1}+\omega_ {p+p_ {1}})},\\ H_ {3}^{(-1)}\left\lvert \vec{p} \right\rangle_ {1}^{(3)} =& - \frac{27m^{3}g^{3}}{8\sqrt{2}\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi^{3})} \, \frac{\left\lvert \vec{p}_ {1},\vec{p}-\vec{p}_ {1} \right\rangle_ {0}}{\omega_ {1}(-\omega _ {p} +\omega_ {1}+\omega_ { p-p_ {1}})} \\ &\times \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {1+2}\omega_ {2}(-\omega _ {p} +\omega_ {1+2}+\omega_ {2}+\omega_ {p-p_ {1}})}, \\ H_ {3}^{(-3)}\left\lvert \vec{p} \right\rangle_ {2}^{(5)} &= - \frac{27m^{3}g^{3}}{8\sqrt{2}\omega _ {p} }\int \frac{d^{3}p_ {1}}{(2\pi)^{3}} \frac{\left\lvert \vec{p}_ {1}, \vec{p}-\vec{p}_ {1} \right\rangle}{\omega_ {1}(-\omega _ {p} +\omega_ {1}+\omega_ {p-p_ {1}})} \\ &\times \int \frac{d^{3}p_ {2}}{(2\pi)^{3}} \, \frac{1}{\omega_ {2}\omega_ {1+2}(-\omega _ {p} +2\omega_ {1}+\omega_ {2}+\omega_ {1+2}+\omega_ {p-p_ {1}})} \end{align*}\]