5 The contribution from multiple scatterings in the electron-electron channel to the asymptotic two-particle vertex is important in particular in the calculation of the vertex corrections to the mean-field electrical conductiv- ity. The CPA vertex, as notoriously known, does not generate vertex corrections, unless we introduce odd dis- persion relations in multi-orbital models. It is only the second term in the parentheses on the right-hand side of Eq. (18), that contributes to the averaged conductivity and the we can identify from the formula for the vertex corrections to the electri- cal conductivity, Eq. (7). Inserting representation from Eq. (19) into Eq. (7) we obtain It is a general formula for the leading-order corrections where we used an abbreviated notation Representation (22) of the vertex corrections to the electrical conductivity does not explicitly contain the strength of elastic scatterings in the model. This strength is beyond the self-energy of the one-electron propaga- tors implicitly comprised in the spatial fluctuations of the two-particle bubble χ ( q ). A singular structure of the integrand in momentum representation of the vertex cor- rections to the Drude conductivity becomes transparent in Eq. (22). to the mean-field (Drude) conductivity and can be ap- plied in any dimension. Vertex ΔΓ contains the so-called Cooper pole being the image of the diffusion pole con- tained in the CPA vertex after an appropriate electron- hole transformation. The Cooper pole is responsible for the so-called weak-localization corrections in the Ander- son model of disordered electrons. 21 These corrections are here identical with those determined by Eq. (20). They are negative and diverge in spatial dimensions d ≤ 2. The mean-field corrections to the electrical conductivity from Eq. (20) can effectively be applied only in dimensions d ≥ 3 and not too close to the band edges. We can further simplify Eq. (20) in that we get rid of the local mean-field vertex γ . We utilize the Ward identity from Eq. (14) connecting the one-electron self- energy Σ and the two-particle irreducible vertex Λ. Using the stationarity equation, Eq. (10a), we easily obtain where we denoted Σ ′ + = ∂ Σ$_{+}$( ω ) /∂ω | ω $_{=0}$, G ′ + = ∂G$_{+}$ ( ω ) /∂ω | ω $_{=0}$, and Z$_{+}$ = Σ $^{′}$$_{+}$/ (Σ ′ + − 1). With these relations and the local Bethe-Salpeter equation, Eq. (13), we can rewrite Eq. (20) to To evaluate the vertex corrections to the mean-field electrical conductivity, Eq. (20), we resort to high spa- tial dimensions where we can explicitly perform the con- volutions over momenta. We are interested only in the leading-order contributions in the inverse spatial dimen- sion 1 /d . The conductivity vanishes in the mean-field limit, d = ∞ , where only local quantities survive. The actual mean-field conductivity is due to the velocity in a particular direction proportional to 1 /d and hence can be treated only asymptotically for d → ∞ . On a hyper-cubic lattice we have v$_{α}$ ( k ) = ∂ϵ ( k ) /∂k$_{α}$ = td − 1 / $^{2}$sin k$_{α}$ . In- serting this result in the mean-field conductivity, Eq. (6), we obtain the Drude conductivity on a hyper-cubic d - V. VERTEX CORRECTIONS TO THE ELECTRICAL CONDUCTIVITY FROM HIGH SPATIAL DIMENSIONS ΔΓ$_{kk}$ ′ ( q ) = γ 2 ¯ χ ( k + k ′ + q ) 1 − γ ¯ χ ( k + k ′ + q ) (19) Δ σ MF αα = e 2 2 πN 2 ∑ kk ′ { ( γ RA ) 2 ∣ G R k ∣ 2 v$_{α}$ ( k )¯ χ $^{RA}$( k + k $^{′}$) v$_{α}$ ( k $^{′}$) 1 − γ $^{RA}$¯ χ $^{RA}$( k + k $^{′}$) ∣ G R k ′ ∣ 2 −ℜ [ γ RR 2 ( G R k ) 2 v$_{α}$ ( k )¯ χ $^{RR}$( k + k $^{′}$) v$_{α}$ ( k $^{′}$) 1 − γ $^{RR}$¯ χ $^{RR}$( k + k $^{′}$) ( G R k ′ ) 2 ]} . (20) Λ$_{+}$$_{−}$ = ℑ Σ$_{+}$ G$_{+}$ = 1 χ$_{+}$$_{−}$ ( 0 ) , (21a) Δ σ MF αα = e 2 2 πN 2 ∑ kk ′    ∣ G R k ∣ $^{2}$v$_{α}$ ( k )¯ χ$_{+}$$_{−}$ ( k + k $^{′}$) v$_{α}$ ( k $^{′}$) ∣ G R k ′ ∣ 2 ( χ$_{+}$$_{−}$ ( 0 ) − |〈 G$_{+}$ 〉| 2 ) ( χ$_{+}$$_{−}$ ( 0 ) − χ$_{+}$$_{−}$ ( k + k $^{′}$)) −ℜ [ Z 2 + ( G R k ) $^{2}$v$_{α}$ ( k )¯ χ$_{++}$ ( k + k $^{′}$) v$_{α}$ ( k $^{′}$) ( G R k ′ ) 2 ( χ$_{++}$ ( 0 ) − Z$_{+}$ 〈 G$_{+}$ 〉 $^{2}$) ( χ$_{++}$ ( 0 ) − Z$_{+}$χ$_{++}$ ( k + k $^{′}$)) ]} (22) 〈 G$_{±}$ 〉 = ∫ ∞ −∞ dϵρ ( ϵ ) [ E$_{F}$ − ϵ − Σ$_{±}$ ± i 0 $^{+}$] , (23a) 〈 G 2 ± 〉 = ∞ −∞ dϵρ ( ϵ ) [ E$_{F}$ − ϵ − Σ$_{±}$ ± i 0 $^{+}$] 2 . (23b) Λ$_{++}$ = Σ ′ + G ′ + = Z$_{+}$ χ$_{++}$ ( 0 ) (21b)