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Higher-Order Gravitational Lensing Reconstruction using Feynman Diagrams

We develop a method for calculating the correlation structure of the Cosmic Microwave Background (CMB) using Feynman diagrams, when the CMB has been modified by gravitational lensing, Faraday rotation, patchy reionization, or other distorting effects. This method is used to calculate the bias of the Hu-Okamoto quadratic estimator in reconstructing the lensing power spectrum up to O(ϕ^4) in the lensing potential $ϕ$. We consider both the diagonal noise TTTT, EBEB, etc. and, for the first time, the off-diagonal noise TTTE, TBEB, etc. The previously noted large O(ϕ^4) term in the second order noise is identified to come from a particular class of diagrams. It can be significantly reduced by a reorganization of the $ϕ$ expansion. These improved estimators have almost no bias for the off-diagonal case involving only one $B$ component of the CMB, such as EEEB.