Dispersión de gluones NMHV 5 (Elvang y Huang, Ex 3.11)

Respuesta: Después de implementar la prescripción CSW, resulta que$\color{Red}{[1X]}$,$\color{Red}{[2X]}$, y$\color{Red}{[3X]}$aparecen en varios denominadores de los 4 diagramas on-shell mencionados, por lo que$X=1$,$X=2$o$X=3$no son opciones posibles para el espinor de referencia$|X]$. Alquiler$X$Sea una combinación lineal no trivial de$1,2,3,4,5$genera demasiados términos. Así que las opciones óptimas son$X=4$o$X=5$, que están vinculados por la simetría del diagrama en el caparazón. A continuación esbozamos el cálculo para$X=5$.

Prueba esbozada: OP ya ha notado que la expansión CSW contiene 4 diagramas:

$$\begin{align}A_{15}~:=~& A^{MHV}_3(\hat{5}^+,\hat{1}^-,-\hat{P}_{15}^-)\frac{1}{P_{15}^2}A^{MHV}_4(\hat{P}_{15}^+,\hat{2}^-,\hat{3}^-,\hat{4}^+)\cr ~=~&\frac{\langle 1\hat{P}_{15}\rangle^4}{\langle 1\hat{P}_{15}\rangle\langle \hat{P}_{15}5\rangle\langle 51\rangle}\frac{1}{P_{15}^2}\frac{\langle 23\rangle^4}{\langle 23\rangle\langle 34\rangle\langle 4\hat{P}_{15}\rangle\langle \hat{P}_{15}2\rangle}\cr ~\stackrel{\begin{array}{c}\text{CSW}\cr \text{prescript.}\end{array}}{=}&\frac{\langle 1|P_{15}|X]^3}{\langle 5|P_{15}|X]\langle 51\rangle}\frac{1}{P_{15}^2}\frac{\langle 23\rangle^3}{\langle 34\rangle\langle 4|P_{15}|X]\langle 2|P_{15}|X]}\cr ~=~&\frac{[5X]^3}{\color{Red}{[1X]}[15]}\frac{\langle 23\rangle^3}{\langle 34\rangle\langle 4|P_{15}|X]\langle 2|P_{15}|X]}\cr ~\stackrel{X=5}{=}&0,\end{align} \tag{1}$$

$$\begin{align} A_{34}~:=~&A^{MHV}_3(\hat{3}^-,\hat{4}^+,-\hat{P}_{34}^-)\frac{1}{P_{34}^2}A^{MHV}_4(\hat{P}_{34}^+,\hat{5}^+,\hat{1}^-,\hat{2}^-)\cr ~=~&\frac{\langle \hat{P}_{34}3\rangle^4}{\langle \hat{P}_{34}3\rangle\langle 34\rangle\langle 4\hat{P}_{34}\rangle}\frac{1}{P_{34}^2}\frac{\langle 12\rangle^4}{\langle 12\rangle\langle 2\hat{P}_{34}\rangle\langle \hat{P}_{34}5\rangle\langle 51\rangle}\cr ~\stackrel{\begin{array}{c}\text{CSW}\cr \text{prescript.}\end{array}}{=}&\frac{\langle 3|P_{34}|X]^3}{\langle 34\rangle\langle 4|P_{34}|X]}\frac{1}{P_{34}^2}\frac{\langle 12\rangle^3}{\langle 2|P_{34}|X]\langle 5|P_{34}|X]\langle 51\rangle}\cr ~=~&-\frac{[4X]^3}{\color{Red}{[3X]}[34]}\frac{\langle 12\rangle^3}{\langle 2|P_{34}|X]\langle 5|P_{34}|X]\langle 51\rangle}\cr ~\stackrel{X=5}{=}&-\frac{[45]^3}{[35][34]}\frac{\langle 12\rangle^2}{[15]\langle 5|P_{34}|5]\langle 51\rangle}\cr ~\stackrel{(5)}{=}&~A^{\overline{MHV}}_5 \frac{[12][23]}{[35]}\frac{\langle 12\rangle^2}{\langle 5|P_{34}|5]\langle 51\rangle}\cr ~\stackrel{(6)+(9)}{=}&-A^{\overline{MHV}}_5\frac{I}{\Delta \langle 5|P_{12}|5]} ,\end{align} \tag{2}$$

$$\begin{align} A_{12}~:=~&A^{MHV}_3(\hat{1}^-,\hat{2}^-,-\hat{P}_{12}^+)\frac{1}{P_{12}^2}A^{MHV}_4(\hat{P}_{12}^-,\hat{3}^-,\hat{4}^+,\hat{5}^+)\cr ~=~&\frac{\langle 12\rangle^4}{\langle 12\rangle\langle 2\hat{P}_{12}\rangle\langle \hat{P}_{12}1\rangle}\frac{1}{P_{12}^2}\frac{\langle \hat{P}_{12}3\rangle^4}{\langle \hat{P}_{12}3\rangle\langle 34\rangle\langle 45\rangle\langle 5\hat{P}_{12}\rangle}\cr ~\stackrel{\begin{array}{c}\text{CSW}\cr \text{prescript.}\end{array}}{=}&\frac{\langle 12\rangle^3}{\langle 2|P_{12}|X]\langle 1|P_{12}|X]}\frac{1}{P_{12}^2}\frac{\langle 3|P_{12}|X]^3}{\langle 34\rangle\langle 45\rangle\langle 5|P_{12}|X]}\cr ~=~&-\frac{1}{\color{Red}{[1X][2X]}[12]}\frac{\langle 3|P_{12}|X]^3}{\langle 34\rangle\langle 45\rangle\langle 5|P_{12}|X]}\cr ~\stackrel{X=5}{=}&\frac{1}{[15][25][12]}\frac{\langle 34\rangle^2 [45]^3}{\langle 45\rangle\langle 5|P_{12}|5]}\cr ~\stackrel{(5)}{=}~&-A^{\overline{MHV}}_5\frac{[23][34]}{[25]}\frac{\langle 34\rangle^2 }{\langle 45\rangle\langle 5|P_{12}|5]}\cr ~\stackrel{(7)+(9)}{=}&-A^{\overline{MHV}}_5\frac{II}{\Delta\langle 5|P_{12}|5]} ,\end{align} \tag{3}$$

$$\begin{align} A_{23}~:=~&A^{MHV}_3(\hat{2}^-,\hat{3}^-,-\hat{P}_{23}^+)\frac{1}{P_{23}^2}A^{MHV}_4(\hat{P}_{23}^-,\hat{4}^+,\hat{5}^+,\hat{1}^-)\cr ~=~&\frac{\langle 23\rangle^4}{\langle 23\rangle\langle 3\hat{P}_{23}\rangle\langle \hat{P}_{23}2\rangle}\frac{1}{P_{23}^2}\frac{\langle 1\hat{P}_{23}\rangle^4}{\langle 1\hat{P}_{23}\rangle\langle \hat{P}_{23}4\rangle\langle 45\rangle\langle 51\rangle}\cr ~\stackrel{\begin{array}{c}\text{CSW}\cr \text{prescript.}\end{array}}{=}&\frac{\langle 23\rangle^3}{\langle 3|P_{23}|X]\langle 2|P_{23}|X]}\frac{1}{P_{23}^2}\frac{\langle 1|P_{23}|X]^3}{\langle 4|P_{23}|X]\langle 45\rangle\langle 51\rangle}\cr ~=~&-\frac{1}{\color{Red}{[2X][3X]}[23]}\frac{\langle 1|P_{23}|X]^3}{\langle 4|P_{23}|X]\langle 45\rangle\langle 51\rangle}\cr ~\stackrel{X=5}{=}&\frac{1}{[25][35][23]}\frac{\langle 14\rangle^2[45]^3}{[15]\langle 45\rangle\langle 51\rangle}\cr ~\stackrel{(5)}{=}~&-A^{\overline{MHV}}_5\frac{[12][34]}{[25][35]}\frac{\langle 14\rangle^2}{\langle 45\rangle\langle 51\rangle}\cr ~\stackrel{(8)+(9)}{=}&-A^{\overline{MHV}}_5\frac{III}{\Delta} .\end{align} \tag{4}$$

Arriba, anticipándonos al resultado, introdujimos la siguiente notación de línea corta:

$$A^{\overline{MHV}}_5~:=~\frac{[45]^4}{[51][12][23][34][45]}~=~\frac{[45]^3}{[51][12][23][34]}, \tag{5}$$

$$\begin{align}I~:=~&P_{12}^2\langle 12\rangle[23][25]\langle 45\rangle\cr ~=~&P_{12}^2\langle 12\rangle[23]([23]\langle 34\rangle-[12]\langle 14\rangle)\cr ~=~&P_{12}^2\langle 12\rangle[23]^2\langle 34\rangle-P_{12}^4[23]\langle 14\rangle ,\end{align} \tag{6}$$

$$\begin{align}II~:=~&P_{34}^2\langle 34\rangle[23][35]\langle 51\rangle\cr ~=~&P_{34}^2\langle 34\rangle[23]([34]\langle 14\rangle-[23]\langle 12\rangle)\cr ~=~&P_{34}^4[23]\langle 14\rangle -P_{34}^2[23]^2\langle 34\rangle\langle 12\rangle,\end{align} \tag{7}$$

$$III~:=~[12][34]\langle 14\rangle^2,\tag{8}$$

$$\begin{align}-\Delta~:=~&-[25]\langle 45\rangle[35]\langle 51\rangle\cr ~=~&-([23]\langle 34\rangle-[12]\langle 14\rangle)([34]\langle 14\rangle-[23]\langle 12\rangle)\cr ~=~&[23]^2\langle 34\rangle\langle 12\rangle -(P_{12}^2+P_{34}^2)[23]\langle 14\rangle +[12][34]\langle 14\rangle^2\cr ~\stackrel{(6)+(7)+(8)}{=}&\frac{I+II}{P_{12}^2-P_{34}^2}+III .\end{align} \tag{9}$$

A continuación, tenga en cuenta que la conservación del momento y la condición sin masa implican

$$\begin{align}\langle 12\rangle[12]-\langle 34\rangle[34] ~=~&P_{12}^2-P_{34}^2~=~P_{345}^2-P_{34}^2\cr ~=~&2 P_{34}\cdot P_5 \cr ~=~&-\langle 5|P_{34}|5]\cr ~=~&\langle 5|P_{12}|5].\end{align} \tag{10}$$

En total, calculamos

$$\begin{align}A^{NMHV}_{5}(\hat{1}^-,\hat{2}^-,\hat{3}^-,\hat{4}^+,\hat{5}^+) ~\stackrel{\text{CSW}}{=}&A_{15}+A_{34}+A_{12}+A_{23}\cr ~\stackrel{(1)+(2)+(3)+(4)+(9)+(10)}{=}&A^{\overline{MHV}}_5, \end{align} \tag{11}$$

que es el resultado buscado.$\Box$