3 The dispersion relation (11) is linear in leading order rather than quadratic as would be expected in vacuo . As a consequence, all excited CPs behave like a gas of free massless-like bosons with a common group veloc- ity c$_{1}$ = ℏ − $^{1}$d E ( l ) K /dK , but a variable energy determined by their CMM ℏ K . The dispersion relation (11) implies that in order for a CP to remain bound (i.e., E ( l ) K < 0) its maximum CMM wavenumber must not exceed the value |E ( l ) 0 | /c$_{1}$ ≡ K$_{0}$ since CPs with K > K$_{0}$ have E ( l ) K > 0 and thus break up [26]. Explicit expressions of relevant thermodynamic vari- ables and transport coefficients evaluated within the weak-coupling limit of the l -wave BCS theory have been derived in Refs.[6, 34, 35]. In these papers it is shown that the average behavior of most of these quantities over the cylindrical Fermi surface exhibits small variation due to the explicit realization of an l = 0 or l = 2 symme- try [6, 35]. In particular, the temperature-dependent gap equation is given by [6, 26, 34] V ( l ) 0 f l $_{k}$f l ∗ k ′ = − V$_{0}$ (8) 1 = V ( l ) 0 ∑ k | f l $_{k}$| 2 E ( l ) K − ϵ$_{k}$$_{+}$$_{K}$$_{/}$$_{2}$ − ϵ$_{k}$$_{−}$$_{K}$$_{/}$$_{2}$ . (7) Following Schrieffer [26] we assume that the angular- independent l component of the generalized BCS model interaction (4) is given by where ∑ k ′ a$_{k}$ ′ f l ∗ k ′ ≡ C ( l ) is a constant. One thus obtains a BCS-type integral relation for a CP in the eigenstate characterized by ( l, m ) a ( l ) k = C ( l ) V ( l ) 0 f l k E ( l ) K − ϵ$_{k}$$_{+}$$_{K}$$_{/}$$_{2}$ − ϵ$_{k}$$_{−}$$_{K}$$_{/}$$_{2}$ (6) with V$_{0}$ > 0 for CPs with relative momenta ( k , k $^{′}$) lying in the neighborhood of the Fermi surface and V ( l ) 0 f l $_{k}$f l ∗ k ′ = 0 otherwise. Here K$_{max}$ = √ k 2 F + k 2 D with k$_{D}$ defined in terms of the Debye energy via ℏ ω$_{D}$ ≡ ℏ $^{2}$k2 $_{D}$/ 2 m $^{∗}$. A straightforward analysis [26] reveals that (7) yields a bound state with energy E ( l ) < 0 for arbitrar- ily weak coupling so long as the potential is attractive in the region (9) in k -space. Then, a bosonic CP can form only if the tip of vector k lies within the intersection of the two spherical shells defined by (9) whose center-to- center separation is K ; fermions with wave vectors lying outside this overlap are unpairable [24]. In the quasi-2D limit the fundamental expression (7) can be evaluated by substituting the summation over k by a 2D integration. In addition, for small δ the only term in K$_{z}$ that yields a finite contribution is n = 0. By assuming a 2D cylindrical Fermi surface we obtain where β = 1 /k$_{B}$T , g $^{(0)}$( k ) = 1 for l = 0, and g $^{(2)}$( k ) = cos(2 θ ) for l = 2. The critical temperature is determined from (13) by the condition Δ ( l $^{)}$( T$_{c}$ ) = 0. In the weak- coupling limit ℏ ω$_{D}$/k$_{B}$T$_{c}$ ≫ 1 it can be calculated an- alytically and it follows that T$_{c}$ is independent of the l − state [34]: with γ ≃ 0 . 577 · · · Euler’s constant. In the zero- temperature limit Δ ( l ) 0 ≡ Δ ( l $^{)}$( T = 0) the energy inte- gration in (13) leads to the gap relation [34] 1 = N$_{0}$ V$_{0}$ ∫ ℏ ω D 0 dϵ$_{k}$ 〈 tanh ( 1 $_{2}$β √ ϵ 2 $_{k}$+ Δ ( l ) $^{2}$| g ( l $^{)}$( k ) | 2 ) √ ϵ 2 $_{k}$+ Δ ( l ) $^{2}$| g ( l $^{)}$( k ) | 2 〉 F (13) k$_{B}$T$_{c}$ = 2 e γ π ℏ ω$_{D}$ exp( − 1 / N$_{0}$ V$_{0}$ ) ≃ 1 . 13 ℏ ω$_{D}$ exp( − 1 / N$_{0}$ V$_{0}$ ) (14) k$_{F}$ < | k + K / 2 | , | k − K / 2 | < K$_{max}$ (9) On the other hand, the average superfluid density ρ$_{s}$ ( T ) ≡ λ 2 $_{ab}$(0) /λ 2 $_{ab}$( T ) exhibits a more pronounced an- where c$_{1}$ ≡ 2 v$_{F}$ /π in 2D and E ( l ) 0 is the binding energy of the CP ground state ( ℏ K = 0) [27] E ( l ) K ≃ E ( l ) 0 + c$_{1}$ ℏ K + O ( K $^{2}$) (11) E ( l ) 0 = − 2 ℏ ω$_{D}$/ [exp(2 / N$_{0}$ V ( l ) 0 ) − 1] . (12) 2Δ (0) 0 k$_{B}$T$_{c}$ ≃ 3 . 53 2Δ (2) 0 k$_{B}$T$_{c}$ ≃ 4 . 28 . (16) where Γ ( l ) = exp[ − 〈 | g ( l $^{)}$|$^{2}$ln | g ( l $^{)}$| 〉 $_{F}$]. For l = 0, Γ (0) = 1, while for l = 2, Γ (2) = 2 exp( − 1 / 2) ≃ 1 . 213, so that combining (14) and (15) we are led to the gap-to- T$_{c}$ ratios ∣$^{|}$$^{E}$$^{(}$$^{l}$$^{)}$ |E K | + v$_{F}$ ℏ K cos θ F where k$_{1}$ = k$_{F}$ +( K/ 2) cos θ , k$_{2}$ = k$_{F}$ + k$_{D}$ − ( K/ 2) cos θ , and the approximate equality in the second row holds up to terms of order ( k$_{D}$/k$_{F}$ ) 2 ≡ Θ$_{D}$ /T$_{F}$ . The angular brackets denote an average over a 2D cylindrical Fermi surface < ... >$_{F}$ → (1 / 2 π ) ∫ 2 π 0 dθ . The Fermi average then gives the energy spectrum of excited CPs [26]: 1 = V$_{0}$ (2 π ) 2 ∫ 2 π 0 dθ ∫ k$_{2}$ k$_{1}$ kdk |E ( l ) K | + ϵ$_{k}$$_{+}$$_{K}$$_{/}$$_{2}$ + ϵ$_{k}$$_{−}$$_{K}$$_{/}$$_{2}$ (10) ≃ m $^{∗}$V$_{0}$ 4 π 〈 ln ∣ K | + 2 ℏ ω$_{D}$ − v$_{F}$ ℏ K cos θ |E ( l ) K | + v$_{F}$ ℏ K cos θ ∣ 〉 F Δ ( l ) 0 = 2Γ ( l ) ℏ ω$_{D}$ exp( − 1 / N$_{0}$ V ( l ) 0 ) (15) where a$_{k}$ = a$_{k}$Y m l (Ω$_{k}$). Eq.(5) can be now rewritten as For l = 0 one recovers the standard BCS result [25] and the somewhat higher value for the l = 2 d -wave case. We note that the quantity Δ ( l ) 0 / Γ ( l ) has the same functional dependence as the zero-temperature gap of the BCS the- ory [25]. Considering that measurements of the energy gap for any given cuprate show some scatter about a cen- tral value Δ exp 0 [38] in the following we shall assume that Δ exp 0 ≃ Δ (2) 0 / Γ (2) ≃ Δ (0) 0 ≡ Δ$_{0}$.