0.5 1 1.5 2 0.5 1 1.5 2 0 2 4 6 k$_{y}$ A$_{ω}$$_{=0}$ ( k ) k$_{x}$ (a) 0.5 1 1.5 2 0.5 1 1.5 2 0 2 4 6 k$_{y}$ A$_{ω}$$_{=0}$ ( k ) k$_{x}$ (b) At the shadow mass shell, ω ∼ k$_{y}$ we get Our analytical results, are presented graphycally in Fig. 2. The area of validity of these expressions is con- trolled by the energy scale Equation (7) is valid for | k$_{y}$ | > T$_{K}$/v . Equation (9) is valid for | k$_{x}$ | > T$_{K}$/v . The remaining part of the paper contains a deriva- tion of the results, Eqs. (7), (9). Following the approach of [3] we develop a perturbation theory in the coupling constant J . Though this perturbation theory is free of ultra-violet singularities it contains infra-red singularities at ω ∼ k$_{x}$$_{(}$$_{y}$$_{)}$ which we sum up. In this section we derive the result (7). It is useful to consider the self-energy Σ$_{ω}$( k ) defined in the standard way by the Dyson equation As is shown below, the self-energy is a regular function of frequency at the mass shell, and only weakly logarith- mically non-analityc in the coupling constant. The J 2 contribution to the self-energy is We notice that in the limit of infinite phase stiffness, d = 0, Eq. (12) reproduces the mean field spectrum, Eq. (4), as expected. Note that the self energy Eq. (12) is regular at the mass shell. This, however, is not the case for higher order contribution. In fact, the fourth order contribution has a weak logarithmic singularity (see App. A 1), where The aforementioned analyticity of the self-energy at the mass shell is restored once the leading on-shell singulari- ties in all orders in J 2 are summed up. The reminder of the present section is devoted to this task. we write the self-energy at the order J 2 n with n ≥ 3 in the form (see Fig.3) Using the expressions for the bare (retarded) Green functions FIG. 2: (color online) The spectral density at the Fermi sur- face, A$_{ω}$$_{=0}$ ( k ), as given by Eqs. (7) and (9) at k$_{y}$ > k$_{x}$ and k$_{x}$ > k$_{y}$ respectively. These two graphs are separated by the area where the presented derivation ceases to be valid. The parameter d is (a) d = 0 . 08, and (b) d = 0 . 2. 3 G = J $^{2}$e i $^{πd}$Γ(1 − 2 d ) ( ω − k$_{x}$ ) 2 × [ ω − k$_{y}$ + i J $^{2}$a2 $^{d}$Γ(1 − 2 d ) ( − i( ω − k$_{x}$ )) 1 − 2 d ] − 1+2 d . (9) T$_{K}$ = J ( Ja ) d/ (1 − d $^{)}$. (10) G$_{ω}$ ( k ) = [ ω − k$_{x}$ − Σ$_{ω}$( k )] − 1 . (11) Σ (2) ω ( k ) = − i( Ja $^{d}$) $^{2}$Γ(1 − 2 d ) ( − i( ω − k$_{y}$ )) 1 − 2 d . (12) Σ (4) = 2 d − i( Ja $^{d}$) $^{4}$Γ$^{2}$(2 − 2 d ) ( − i( ω − k$_{y}$ )) 3 − 4 d α log α , (13) α = ω + i0 − k$_{x}$ ω + i0 − k$_{y}$ . (14) Σ (2 n ) ω ( k ) =i( − i J ) 2 n ∫ ∞ 0 dx$_{n}$ e i ( ω − k$_{x}$ ) x$_{n}$ ∫ ∞ 0 dy$_{n}$ e i ( ω − k$_{y}$ ) y$_{n}$ × n − 1 ∏ i =2 ∫ x$_{i}$$_{+1}$ 0 dx$_{i}$ n − 1 ∏ i =1 ∫ y$_{i}$$_{+1}$ 0 dy$_{i}$ × C (2 n $^{)}$( r$_{1}$ , . . . , r$_{n}$ ; p$_{1}$ , . . . , p$_{n}$ ) , (16) i G (0) $_{1(2)}$( ω, r ) = θ ( r$_{x}$$_{(}$$_{y}$$_{)}$ ) δ ( r$_{y}$$_{(}$$_{x}$$_{)}$ ) e i ωr$_{x}$$_{(}$$_{y}$$_{)}$ . (15) III. BEHAVIOR AT THE MASS SHELL: ω ∼ k$_{x}$