where self energy correction, and We write It is apparent form Eq. (A12) that the leading singularity originates from the first term, ( A − 1) BC ≈ A − 1 ≈ − 2 d ( y$_{1}$ − 0)( y$_{3}$ − y$_{2}$ ). Introducing new variables as in the Sec. III and integrating over x$_{i}$ s we obtain Here again, the integrals are divergent on the upper limit. Similarly to the previous section we differentiate it once with respect to the variable α introduced in Eq. (14) in order to isolate the leading logarithmic singularity, The remaining integrals are easily evaluated. The sub- sequent integration over α restores the singularity in the We stress that contrary to the mass shell singularities discussed in App. A, where it was important to compute the self-energy, at the shadow mass shell it is enough to consider the Green function itself. We notice that the singularity at ω ∼ k$_{y}$ comes from the region of small ξ . Therefore we keep only the first term in the square brackets in Eq. (B1). After performing remaining integrations over ξ$_{i}$ s we obtain In this appendix we evaluate the singular contributions to the Green function at the shadow mass shell, ω ∼ k$_{y}$ to fourth order in the coupling constant. We start with the expression (A1) introduced in App. A 1. In contrast to the discussion in App. A 1 we anticipate the singular- ity at ω = k$_{y}$ to come from the region y$_{2}$ ≫ x$_{2}$ , and introduce new variables accordingly, x$_{2}$ = ξy$_{2}$ , y$_{1}$ = ηy$_{2}$ . Performing integration over y$_{2}$ we obtain where C is an integration constant. The singular part of Eq. (A15) is given by Eq. (24). Σ (6) =2 d − i( Ja $^{d}$) $^{6}$Γ(1 − 2 d )Γ $^{2}$(2 − 2 d ) ( − i( ω − k$_{y}$ )) 5 − 6 d × (log α + C ) , (A15) Σ (4) =i( Ja $^{d}$) 4 ∫ ∞ 0 dξ ∫ 1 0 dη Γ(3 − 4 d ) η − 2 $^{d}$(1 − η ) − 2 d [( − i(( ω − k$_{y}$ ) + ( ω − k$_{x}$ ) ξ )] 3 − 4 d × [ | (1 − η ) 2 + ξ $^{2}$| $^{d}$| η 2 + ξ $^{2}$| d | 1 + ξ $^{2}$| $^{d}$| ξ | 2 d − 1 ] . (B1) Σ (4) = i( Ja $^{d}$) $^{4}$Γ(1 − 2 d )Γ(2 − 2 d ) α 2 − 2 d ( − i( ω − k$_{x}$ )) 3 − 4 d . (B2) ∂ Σ (6) ∂α = − 2 d i α − 1 ( Ja $^{d}$) $^{6}$Γ(6 − 6 d ) ( − i( ω − k$_{y}$ )) 5 − 6 d × ∫ ∞ 0 3 ∏ i =1 dξ$_{i}$ ξ 1 − 2 d ξ − 2 d 2 ξ 1 − 2 d 3 (1 + ξ$_{1}$ + ξ$_{2}$ + ξ$_{3}$ ) 6 − 6 d . (A14) Σ (6) = 2 d i ( Ja $^{d}$) $^{6}$Γ(5 − 6 d ) ( − i( ω − k$_{y}$ )) 5 − 6 d × ∫ ∞ 0 3 ∏ i =1 dξ$_{i}$ ξ 1 − 2 d ξ − 2 d 2 ξ 1 − 2 d 3 ( α + ξ$_{1}$ + ξ$_{2}$ + ξ$_{3}$ ) 5 − 6 d . (A13) ABC − B − C +1 = ( A − 1) BC +( B − 1)( C − 1) . (A12) A = ( x 2 3 + y 2 $_{2}$) $^{d}$| x 2 3 + ( y$_{3}$ − y$_{1}$ ) $^{2}$| d ( x 2 3 + ( y$_{2}$ − y$_{1}$ ) $^{2}$) $^{d}$( x 2 3 + y 2 $_{3}$) d , (A9) B = [( x$_{3}$ − x$_{1}$ ) 2 + ( y$_{2}$ − y$_{1}$ ) $^{2}$] d ( x$_{3}$ − x$_{1}$ ) 2 d × [( x$_{3}$ − x$_{1}$ ) 2 + ( y$_{3}$ − y$_{2}$ ) $^{2}$] d [( y$_{3}$ − y$_{1}$ ) 2 + ( x$_{3}$ − x$_{1}$ ) $^{2}$] d , (A10) C = | x 2 1 + y 2 $_{1}$| $^{d}$| x 2 1 + ( y$_{2}$ − y$_{1}$ ) $^{2}$| d | x$_{1}$ | 2 $^{d}$| y 2 + x 2 $_{1}$| d . (A11) Appendix B: Leading singularities at the shadow mass shell, ω ∼ k$_{y}$ to the fourth order. 7 [1] Ar. Abanov, A. V. Chubukov and J. Schmalian, Adv. Phys. 52 , 119 (2003) and references therein. [2] S. Sachdev, M. A. Metlitski, Y. Qi and C. Xu, arXiv: 0907.3732 and references therein. [3] M. Khodas and A. M. Tsvelik, arXiv:0910.3967. [4] N. Doiron-Leyraud et.al. , Nature (London) 447 , 565 (2007). [5] E. A. Yelland et.al. , Phys. Rev. Lett. 100 , 047003 (2008). [6] S. Sachdev, arXiv:0907.0008.