# Inverse soft construction The inverse soft construction is an [[0551 On-shell recursion relations|on-shell recursion formula]] for constructing higher point scattering amplitudes from on-shell lower point ones. Gluons, colour-ordered: $A\left[1_{+} 23 \cdots n\right]=\frac{\langle n 2\rangle}{\langle n 1\rangle\langle 12\rangle} A[\hat{2} \cdots n-1 \hat{n}]$ Gluons, colour-dressed:$A\left(1_{+}^{a_1} 2^{a_2} \cdots n^{a_n}\right)=-\sum_{i=3}^n \frac{\langle i 2\rangle}{\langle i 1\rangle\langle 12\rangle} T_i^{a_1} A\left(\hat{2}^{a_2} \cdots \hat{i}^{a_i} \cdots n^{a_n}\right)$Gravitons:$M\left(1_{+} 2 \cdots n\right)=\sum_{i=3}^n \frac{[1 i]}{\langle 1 i\rangle} \frac{\langle i 2\rangle^2}{\langle 12\rangle^2} M(\hat{2} \cdots \hat{i} \cdots n)$where the shifted momenta are $\hat{p}_2^{\alpha \dot{\alpha}}=\lambda_2^\alpha \hat{\tilde{\lambda}}_2^{\dot{\alpha}}$, $\hat{p}_n^{\alpha \dot{\alpha}}=\lambda_n^\alpha \hat{\tilde{\lambda}}_n^{\dot{\alpha}}$, and $\hat{\tilde{\lambda}}_2=\tilde{\lambda}_2+\frac{\langle n 1\rangle}{\langle n 2\rangle} \tilde{\lambda}_1, \quad \hat{\tilde{\lambda}}_n=\tilde{\lambda}_n+\frac{\langle 12\rangle}{\langle n 2\rangle} \tilde{\lambda}_1.$ ## Refs - origin: - [[2009#Arkani-Hamed, Cachazo, Cheung, Kaplan]] - gravity MHV: - [[2009#Nguyen, Spradlin, Volovich, Wen]] - YM and gravity: - [[2011#Boucher-Veronneau, Larkoski]]: derivation from [[0058 BCFW|BCFW]] - N=4 SYM: - [[2012#Nandan, Wen]] - $F^3$ and $R^3$ theories: - [[2023#Ananth, Bhave, Pandey, Pant]] - double copy and soft recursion: - [[2023#Hu, Zhou]] ## Applications - obtaining subleading [[0009 Soft theorems|soft gluon theorem]]: - [[2014#Casali]] - using soft theorems to uniquely fix all tree-level amplitudes: - [[2018#Rodina]] - all-order [[0114 Celestial OPE|celestial OPE]]: - [[2023#Ren, Schreiber, Sharma, Wang]]