10 For positive ω the integral (B1) gives [21] After carrying out the angular integrals we obtain where 〈 L$_{2}$ 〉 is the angle integrated loop L$_{2}$ , which is ob- tain from (A12) via the substitution x 2 = 1 / 3. Using the transformation dpp 2 = m $^{∗}$p$_{F}$ d ( p $^{2}$/ 2 m $^{∗}$) = m $^{∗}$p$_{F}$ dξ$_{p}$, and the relation ϵ$_{p}$dϵ$_{p}$ = ξ$_{p}$dξ$_{p}$ we obtain Consider the integral (B2). First, note that the term ∝ D − 1 vanishes, since the delta function enforces ω 2 = 4 ϵ 2 $_{p}$. After dropping this term we are left with the integral This and similar integrals, which contain derivatives of the delta function, are computed via the formula To make further progress we need to separate the real and imaginary parts of the integrals. We shall first compute the imaginary parts. They are extracted with the help of the identity In the integral (B3) we again omit terms ∝ D − $^{1}$, since their prefactors are zero after integration. After inserting x 2 = 1 / 3 in the remainder we obtain ∫ f ( x ) δ $^{n}$( x − a ) dx = ( − ) $^{n}$f ( n $^{)}$( a ) . (B13) The result of integration is where P denotes the principal value and δ ( n $^{)}$( D ) is the n -th derivative of the delta function. Applying Eq. (B13) we obtain Finally, adding the three integrals (B11), (B14), and (B16) we obtain Eq. (42) of the main text. I$_{0}$ = ν ( p$_{F}$ ) − $^{1}$µ ∫ dpp 2 2 π 2 ξ$_{p}$ ϵ 3 p [ 4 ϵ 2 p ω 2 − 1 + 2 µξ$_{p}$ ϵ 2 p ] L$_{0}$, (B4) I$_{1}$ = 2 3 ν ( p$_{F}$ ) − $^{1}$µ ∫ dpp 2 2 π 2 ξ$_{p}$ ϵ 3 p [ 4 ϵ 2 p ω 2 − 1 ] L$_{1}$, (B5) I$_{2}$ = 2 ν ( p$_{F}$ ) − 1 ∫ dpp 2 2 π 2 1 ϵ$_{p}$ [ 4 ϵ 2 p ω 2 − 1 ] 〈 L$_{2}$ ( x ) 〉 , (B6) Im I$_{0}$ = − πµ ∫ ∞ Δ dϵ$_{p}$ 1 ϵ 2 p [ 4 ϵ 2 p ω 2 − 1 + 2 µξ$_{p}$ ϵ 2 p ] tanh ( ϵ$_{p}$ 2 T ) δ ( ω 2 − 4 ϵ 2 $_{p}$) = − 8 πµ 2 √$_{ω}$ 2 − 4Δ 2 sgn( ω ) ω 5 tanh ( ω 4 T ) θ ( ω − 2Δ) . (B11) I$_{0}$ = µ ∫ ∞ Δ dϵ$_{p}$ 1 ϵ 2 p [ 4 ϵ 2 p ω 2 − 1 + 2 µξ$_{p}$ ϵ 2 p ] L$_{0}$, (B7) I$_{1}$ = 2 µ 3 ∫ ∞ Δ dϵ$_{p}$ 1 ϵ 2 p [ 4 ϵ 2 p ω 2 − 1 ] L$_{1}$, (B8) I$_{2}$ = 2 ∫ ∞ Δ dϵ$_{p}$ 1 ξ$_{p}$ [ 4 ϵ 2 p ω 2 − 1 ] 〈 L$_{2}$ ( x ) 〉 . (B9) Im I$_{1}$ = 16 πµ 2 3 ∫ ∞ Δ dϵ$_{p}$ 1 ϵ 2 p ( 4 ϵ 2 p ω 2 − 1 ) √ ϵ 2 p − Δ 2 tanh ( ϵ$_{p}$ 2 T ) δ $^{(1)}$( ω 2 − 4 ϵ 2 $_{p}$) . (B12) Im I$_{1}$ = − 8 πµ 2 3 √$_{ω}$ 2 − 4Δ 2 sgn( ω ) ω 5 tanh ( ω 4 T ) θ ( ω − 2Δ) . (B14) 1 ( D + iδ ) n +1 = P D n +1 − iπ ( − 1) n n ! δ ( n $^{)}$( D ) , (B10) Im I$_{2}$ = 8 π 3 ∫ ∞ Δ dϵ$_{p}$ [ 4 ϵ 2 p ω 2 − 1 ] { µ 2 √ ϵ 2 p − Δ 2 ϵ$_{p}$ T sech 2 ( ϵ$_{p}$ 2 T ) + 2 √ ϵ 2 p − Δ $_{2}$µ$^{2}$tanh ( ϵ$_{p}$ 2 T ) +3 µ tanh ( ϵ$_{p}$ 2 T ) − µ 2 √ ϵ 2 p − Δ 2 ϵ 2 p tanh ( ϵ$_{p}$ 2 T ) } δ $^{(1)}$( ω 2 − 4 ϵ 2 $_{p}$) − 64 πµ 2 3 ∫ ∞ Δ dϵ$_{p}$ [ 4 ϵ 2 p ω 2 − 1 ] √ ϵ 2 p − Δ $^{2}$tanh ( ϵ$_{p}$ 2 T ) δ $^{(2)}$( ω 2 − 4 ϵ 2 $_{p}$) . (B15) Im I$_{2}$ = − πµ sgn( ω ) 3 T ω 5 √$_{ω}$ 2 − 4Δ 2 sech 2 ( ω 4 T ) { T [24Δ $^{2}$µ + ω $^{2}$(2 µ + 3 √ ω 2 − 4Δ $^{2}$)] sinh ( ω 2 T ) + 4 µω ( ω 2 − 4Δ $^{2}$) } θ ( ω − 2Δ) . (B16)