where r$_{i}$ = ( x$_{i}$, y$_{i}$$_{−}$$_{1}$ ), p$_{i}$ = ( x$_{i}$, y$_{i}$ ), x$_{1}$ = y$_{0}$ = 0 (see Fig. 3) and the cumulant in the last line expansion gives is expressed in terms of averages of the exponents of the phase fields: The latter average with free energy, Eq. (6) is well known, where tilde over the product sign excludes i = j . In what follows we sum up the most singular terms in expansion (16). Our solution is based on the physical idea that at the mass shell the horizontal segments in the staircase diagram in Fig. 3(a) are parametrically longer then the vertical ones. In other words, the mass shell singularity comes from the integration region y$_{i}$ ≪ x$_{j}$ . Accordingly, we introduce new variables ξ$_{i}$ , and argue that the important domain of integration is ξ$_{i}$ ≪ 1. We expand the cumulant (17) in powers of ξ$_{i}$ s and retain the lowest power term to get the most singular contribution. It can be shown by inspection that this In this section we turn to the analysis of the behavior of the Green function at the “shadow side” of the pocket, The sum of contributions (12), (13), (24) and (26) yields the final result Eq. (7). where The contributions of the orders n ≥ 3 give For n ≥ 3 the integral in Eq. (23) is We notice that for n = 2 , 3 the integral in Eq. (23) di- verges at the upper limit. In this case the expansion in Eq. (21) is not justified. These values of n have to be treated separately, (see App. A for details). For n = 2 the most singular part is given in Eq. (13) and for n = 3 we obtain, (see App. A 2). where We substitute Eqs. (20) and (21) in Eq. (16), and per- form integrations over x$_{n}$ . We now analyze the remaining integrations over ξ$_{i}$ s, C (2 n ) ≈ a 2 dn ∏ i =1 ( y$_{i}$ − y$_{i}$$_{−}$$_{1}$ ) − 2 d × [ [ x 2 n + ( y$_{n}$$_{−}$$_{1}$ − y$_{0}$ ) $^{2}$] $^{d}$[ x 2 n + ( y$_{n}$ − y$_{1}$ ) $^{2}$] d [ x 2 n + ( y$_{n}$ − y$_{0}$ ) $^{2}$] $^{d}$[ x 2 n + ( y$_{n}$$_{−}$$_{1}$ − y$_{1}$ ) $^{2}$] d − 1 ] ≈ − 2 da 2 $^{dn}$( x$_{n}$ ) $^{n}$ξ$_{1}$ ξ$_{n}$ n ∏ i =1 ξ − 2 d i . (21) C (2 n $^{)}$( r$_{1}$ , . . . , r$_{n}$ ; p$_{1}$ , . . . , p$_{n}$ ) = δ$_{1}$$_{,n}$ − n − 1 ∑ i =1 δ$_{1}$$_{,i}$δ$_{i}$$_{+1}$$_{,n}$ + . . . + ( − 1) $^{n}$δ$_{1}$$_{,}$$_{1}$ · · · δ$_{n,n}$ (17) δ$_{i,i}$$_{+}$$_{l}$ = 〈 e i φ ( r$_{i}$ )+ ... + i φ ( r$_{i}$$_{+}$$_{l}$ $^{)}$e − i φ ( p$_{i}$ ) − ... − i φ ( p$_{i}$$_{+}$$_{l}$ ) 〉 . (18) Σ (2 n ) =2 d ( − i) 2 n $^{+1}$( Ja $^{d}$) 2 $^{n}$Γ[ n (2 − 2 d ) − 1] ( n − 2)!( − i( ω − k$_{y}$ )) n (2 − 2 d ) − 1 I$_{n}$ ( α ) , (22) I$_{n}$ ( α )= ∫ ∞ 0 n ∏ i =1 dξ$_{i}$ ξ 1 − 2 d ξ − 2 d 2 · · · ξ − 2 d n − $_{1}$ξ 1 − 2 d n ( α + ξ$_{1}$ + ξ$_{2}$ + . . . + ξ$_{n}$ ) n (2 − 2 d ) − 1 . (23) 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 . (24) I$_{n}$ ( α ) = (1 − 2 d ) $^{2}$α3 − $_{n}$Γ$^{n}$(1 − 2 d )Γ( n − 3) Γ[ n (2 − 2 d ) − 1] . (25) ∑ n ≥ 3 Σ (2 n ) = 2 dα − i( Ja $^{d}$) $^{4}$Γ$^{2}$(2 − 2 d ) ( − i( ω − k$_{y}$ )) 3 − 4 d log(1 + x ) 2 d − i( Ja $^{d}$) $^{6}$Γ(1 − 2 d )Γ $^{2}$(2 − 2 d ) ( − i( ω − k$_{y}$ )) 5 − 6 d [1 − log(1 + x )] , (26) x = α − $^{1}$( Ja $^{d}$) $^{2}$Γ(1 − 2 d ) ( − i( ω − k$_{y}$ )) (2 − 2 d ) . (27) δ$_{1}$$_{,n}$ = a 2 dn ( n − 2) { ˜ ∏ n i,j =1 | r$_{i}$ − r$_{j}$ | ˜ ∏ n i,j =1 | p$_{i}$ − p$_{j}$ | ∏ n i,j =1 | r$_{i}$ − p$_{j}$ | } 2 d , (19) FIG. 3: (color online) Graphical representation of the self- energy correction of the order 2 n in coupling constant, Eq. (16). Solid vertical and horizontal lines represent seg- ments of a real space electron trajectory for (a) particle close to the mass shell, ω ≈ k$_{x}$ , (b) particle close to the shadow mass shell, ω ≈ k$_{y}$ . Incoming (blue) and outgoing (red) ar- rowed skew dashed lines represent exponential factors e i φ ( r $_{i}$) and e − i φ ( p $_{j}$) respectively. 4 Y n x 2 p X ... 1 p 1 r 2 r n p n y Y X ... 1 p n p n y (a) (b) n x 1 r 2 r n r y$_{n}$ − y$_{n}$$_{−}$$_{1}$ = ξ$_{n}$x$_{n}$, . . . , y$_{1}$ − y$_{0}$ = ξ$_{1}$x$_{n}$ (20) IV. GREEN FUNCTION AT THE SHADOW MASS SHELL, ω ∼ k$_{y}$ .