where r$_{1}$ = (0 , 0), r$_{2}$ = ( x$_{2}$, y$_{1}$ ), p$_{1}$ = (0 , y$_{1}$ ), and p$_{2}$ = ( x$_{2}$, y$_{2}$ ) (see Fig. 3). Equation (21) gives We expect the main contribution to come from the re- gion y$_{1}$, y$_{2}$ − y$_{1}$ ≪ x$_{2}$ it is convenient to introduce new variables, y$_{2}$ = ξx$_{2}$ , y$_{1}$ = ηξx$_{2}$ . To isolate the leading sin- gularity in Eq. (A1) we expand the cumulant in Eq. (A2) for ξ ≲ 1 This expansion holds for any d < 1 / 2. The integral in Eq. (A4) is not convergent at the upper limit. This is an artifact of the approximation (A3) which is not justified for ξ ≳ 1. To overcome this we differenti- ate equation (A1) twice with respect to the parameter α defined by Eq. (14), The integrations in Eq. (A5) are easily done with the result Integrating Eq. (A6) back we finally get with A , B constants. In the last equation the most sin- gular term is presented in Eq. (13). In this case the general expression (16) with n = 3 reduces to In the fourth order the general expression (16) takes the form FIG. 4: False color plot representing the spectral density at the Fermi surface, A$_{ω}$$_{=0}$ ( k ), as given by Eqs. (7) and (9) at k$_{y}$ > k$_{x}$ and k$_{x}$ > k$_{y}$ respectively. The region k$_{x}$, k$_{y}$ < T$_{K}$ where the results are inapplicable is not shown. The param- eter d is (a) d = 0 . 04, and (b) d = 0 . 13. Dashed line shows the mean field Fermi surface as given by Eq. (4). Σ (4) ω ( k ) =i J 4 ∫ ∞ 0 dx$_{2}$ e i ( ω − k$_{x}$ ) x$_{2}$ ∫ ∞ 0 dy$_{2}$ e i ( ω − k$_{y}$ ) y$_{2}$ ∫ y$_{2}$ 0 dy$_{1}$C $^{(4)}$( r$_{1}$ , r$_{2}$ ; p$_{1}$ , p$_{2}$ ) , (A1) Σ (6) ω ( k ) =i( − i Ja $^{d}$) 6 ∫ ∞ 0 dx$_{3}$ e i ( ω − k$_{x}$ ) x$_{3}$ ∫ ∞ 0 dy$_{3}$ e i ( ω − k$_{y}$ ) y$_{3}$ × ∫ x$_{3}$ 0 dx$_{2}$ ∫ y$_{3}$ 0 dy$_{2}$ ∫ y$_{2}$ 0 dy$_{1}$ × ABC − B − C + 1 y 2 d 1 ( y$_{2}$ − y$_{1}$ ) 2 $^{d}$( y$_{3}$ − y$_{2}$ ) 2 d , (A8) Σ (4) = − 2 d iΓ $^{2}$(2 − 2 d )( Ja $^{d}$) $^{4}$( − i( ω − k$_{y}$ )) 4 d − 3 × ( α log α + Aα + B ) (A7) ∂ $^{2}$Σ(4) ∂α 2 = − 2 d i α − $^{1}$Γ$^{2}$(2 − 2 d ) ( Ja $^{d}$) 4 ( − i( ω − k$_{y}$ )) 3 − 4 d . (A6) ∂ $^{2}$Σ(4) ∂α 2 = − 2 d iΓ(5 − 4 d )( Ja $^{d}$) $^{4}$( − i( ω − k$_{y}$ )) 4 d − 3 × ∫ ∞ 0 dξ ∫ 1 0 dη ξ 3 − 4 $^{d}$η 1 − 2 $^{d}$(1 − η ) 1 − 2 d ( ξ + α ) 5 − 4 d . (A5) Σ (4) = − 2 d i( Ja $^{d}$) $^{4}$Γ(3 − 4 d ) × ∫ ∞ 0 dξ ∫ 1 0 dη ξ 3 − 4 $^{d}$η 1 − 2 $^{d}$(1 − η ) 1 − 2 d [ − i (( ω − k$_{y}$ ) ξ + ( ω − k$_{x}$ ))] 3 − 4 d . (A4) C (4) = a 4 d y 2 d 1 ( y$_{2}$ − y$_{1}$ ) 2 d × [ ( y$_{2}$ − y$_{1}$ ) 2 + x 2 ) d ( y 2 1 + x 2 ) d ( x 2 + y 2 $_{2}$) $^{d}$x 2 d − 1 ] . (A2) 6 C (4) ≈ − 2 da 4 $^{d}$ξ 2 − 4 $^{d}$η 1 − 2 $^{d}$(1 − η ) 1 − 2 d . (A3) 2. Order J 6 1. Fourth order contributions k$_{x}$ k$^{y}$ (a) 0 0.5 1 1.5 2 0.5 1 1.5 2 k$_{x}$ k$^{y}$ (b) 0 0.5 1 1.5 2 0.5 1 1.5 2