where sech $^{2}$( x ) ≡ 1 − tanh ( x ) $^{2}$. On substituting Eqs. (A7) and (A9) in Eq. (A1) we obtain factor where The terms that are odd in x drop out on integration in symmetrical limits. Similarly, We further use the relation Next we expand these functions in powers of small pa- rameter y to order O ( y $^{2}$) to write Finally, where the coefficients of the expansion (A9) can be writ- ten using the shorthand expressions D ≡ ω 2 − 4 ϵ 2 p and ζ ≡ ϵ$_{p}$/ 2 T as Here we evaluate the integrals (39)–(41), which are given as Appendix B: Evaluating phase-space integrals 9 A ( q ) = ∫ d $^{3}$p (2 π ) $_{3}$a ( p , q , ω ) L ( p , q , ω ) , (A1) B ( q ) = Δ ω ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L ( p , q , ω ) , (A2) C ( q ) = − ∫ d $^{3}$p (2 π ) $_{3}$c ( p , q , ω ) L ( p , q , ω ) − ∫ d $^{3}$p (2 π ) 3 1 2 ϵ$_{p}$ [1 − 2 f ( ϵ$_{p}$ )] , (A3) A ( q ) = ∫ d $^{3}$p (2 π ) 3 [ 2Δ 2 ϵ$_{p}$ L$_{0}$ + 2Δ 2 ϵ$_{p}$ L$_{2}$y 2 − 2Δ 2 µ ξ$_{p}$ ϵ 3 p L$_{1}$ x 2 y 2 + Δ 2 µ ξ$_{p}$ ϵ 5 p (6 µξ$_{p}$x 2 − ϵ 2 $_{p}$) L$_{0}$ y 2 ] = A $_{0}$( q ) + A $_{2}$( q ) y 2 + O ( y $^{4}$) . (A13) a ( p , q , ω ) = ϵ$_{p}$ + ϵ$_{p}$$_{+}$$_{q}$ 2 ( 1 − ξ$_{p}$ξ$_{p}$$_{+}$$_{q}$ ϵ$_{p}$ϵ$_{p}$$_{+}$$_{q}$ + Δ 2 ϵ$_{p}$ϵ$_{p}$$_{+}$$_{q}$ ) , (A4) C ( q ) = − ∫ d $^{3}$p (2 π ) 3 [ 2 ϵ$_{p}$L$_{0}$ + 2 µξ$_{p}$ ϵ$_{p}$ L$_{1}$ x 2 y 2 + µ ξ$_{p}$ ϵ$_{p}$ L$_{0}$ y 2 + 2 ϵ$_{p}$L$_{2}$y 2 ] − ∫ d $^{3}$p (2 π ) 3 1 2 ϵ$_{p}$ [1 − 2 f ( ϵ$_{p}$ )] . (A14) c ( p , q , ω ) = ϵ$_{p}$ + ϵ$_{p}$$_{+}$$_{q}$ 2 ( 1 + ξ$_{p}$ξ$_{p}$$_{+}$$_{q}$ ϵ$_{p}$ϵ$_{p}$$_{+}$$_{q}$ + Δ 2 ϵ$_{p}$ϵ$_{p}$$_{+}$$_{q}$ ) , (A5) L ( p , q , ω ) = 1 − f ( ϵ$_{p}$ ) − f ( ϵ$_{p}$$_{+}$$_{q}$ ) ( ω + iδ ) 2 − ( ϵ$_{p}$ + ϵ$_{p}$$_{+}$$_{q}$ ) $_{2}$. (A6) − ∫ d $^{3}$p (2 π ) $_{3}$2 ϵ$_{p}$L$_{0}$ − ∫ d $^{3}$p (2 π ) 3 1 2 ϵ$_{p}$ [1 − 2 f ( ϵ$_{p}$ )] = − ω 2 2 ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L$_{0}$ (A15) C ( q ) = − ω 2 2 ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L$_{0}$ − ∫ d $^{3}$p (2 π ) 3 [ 2 µξ$_{p}$ ϵ$_{p}$ L$_{1}$ x 2 y 2 + µ ξ$_{p}$ ϵ$_{p}$ L$_{0}$ y 2 + 2 ϵ$_{p}$L$_{2}$y 2 ] = C $_{0}$( q ) + C $_{2}$( q ) y $^{2}$. (A16) a ( p , q , ω ) = 2Δ 2 ϵ$_{p}$ − 2Δ 2 µ ξ$_{p}$ ϵ 3 p x y + Δ 2 µ ξ$_{p}$ ϵ 5 p (6 µξ$_{p}$x 2 − ϵ 2 $_{p}$) y $^{2}$, (A7) c ( p , q , ω ) = 2 ϵ$_{p}$ + 2 µξ$_{p}$ ϵ$_{p}$ x y + µ ξ$_{p}$ ϵ$_{p}$ y $^{2}$, (A8) L ( p , q , ω ) = L$_{0}$ + L$_{1}$xy + L$_{2}$y $^{2}$, (A9) B ( q ) = Δ ω ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ ( L$_{0}$ + L$_{2}$ y $^{2}$) = B $_{0}$( q ) + B $_{2}$( q ) y $^{2}$, (A17) and A $_{0}$/ C 0 = − 4Δ $^{2}$/ω$^{2}$, and B $_{0}$/ C 0 = − 2Δ /ω . L$_{0}$ = tanh ζ D + iδ , (A10) L$_{1}$ = 8 µ ξ$_{p}$ tanh ζ ( D + iδ ) 2 + µ ξ$_{p}$ 2 ϵ$_{p}$ T sech 2 ζ ( D + iδ ) , (A11) L$_{2}$ = 4 µ $^{2}$x2 ξ$_{p}$ $^{2}$sech2 ζ ϵ$_{p}$ T ( D + iδ ) 2 + µ 2 x $^{2}$sech 2 ζ 2 ϵ$_{p}$ T ( D + iδ ) + µ ξ$_{p}$ sech 2 ζ 4 ϵ$_{p}$ T ( D + iδ ) − µ 2 x 2 ξ$_{p}$ $^{2}$sech2 ζ 2 ϵ$_{p}$ 3 T ( D + iδ ) + 64 µ $^{2}$x2 ξ$_{p}$ 2 tanh ζ ( D + iδ ) 3 + 8 µ 2 x 2 tanh ζ ( D + iδ ) 2 + 4 µ ξ$_{p}$ tanh ζ ( D + iδ ) 2 − 4 µ 2 x 2 ξ$_{p}$ 2 tanh ζ ϵ$_{p}$ $^{2}$( D + iδ ) 2 − µ 2 x 2 ξ$_{p}$ 2 tanh ζ sech $^{2}$ζ 2 ϵ$_{p}$ 2 T 2 ( D + iδ ) , (A12) I$_{0}$ = ν ( p$_{F}$ ) − $^{1}$µ ∫ d $^{3}$p (2 π ) 3 ξ$_{p}$ ϵ 3 p [ 4 ϵ 2 p ω 2 − 1 + 6 µξ$_{p}$ ϵ 2 p x 2 ] L$_{0}$, (B1) I$_{1}$ = 2 ν ( p$_{F}$ ) − $^{1}$µ ∫ d $^{3}$p (2 π ) 3 ξ$_{p}$ ϵ 3 p [ 4 ϵ 2 p ω 2 − 1 ] L$_{1}$x $^{2}$, (B2) I$_{2}$ = 2 ν ( p$_{F}$ ) − 1 ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ [ 4 ϵ 2 p ω 2 − 1 ] L$_{2}$ ( x ) . (B3)