6 multiply both sides of Eq. (38) by d αβ and obtain, f A q $_{⊥}$( x, k$_{⊥}$ ) ≈ A π Δ$_{2}$$_{F}$ ∫ d $^{2}$ℓ$_{⊥}$ ( ⃗ k$_{⊥}$ · ⃗ ℓ$_{⊥}$ ) ⃗ k 2 ⊥ e − ( ⃗ k$_{⊥}$ − ⃗ ℓ$_{⊥}$ ) $^{2}$/ Δ$_{2}$ F f N q $_{⊥}$( x, ℓ$_{⊥}$ ) (39) = A π Δ$_{2}$$_{F}$ ( 1 + Δ$_{2}$$_{F}$ 2 ⃗ k 2 ⊥ ⃗ k$_{⊥}$ · ⃗ ∂$_{k}$$_{⊥}$ ) ∫ d $^{2}$ℓ$_{⊥}$ e − ( ⃗ k$_{⊥}$ − ⃗ ℓ$_{⊥}$ ) $^{2}$/ Δ$_{2}$ F f N q $_{⊥}$( x, ℓ$_{⊥}$ ) . (40) Given the twist-2 TMD quark distribution function f N q ( x, k$_{⊥}$ ) and the twist-3 quark distribution f N q $_{⊥}$( x, ℓ$_{⊥}$ ) in a nucleon and using the above convolution, Eqs.(3) and (40), one can then calculate the nuclear dependence of the azimuthal asymmetry 〈 cos φ 〉 and 〈〈 cos φ 〉〉 . In general, if the convoluted twist-3 TMD quark distribution is a decreasing function of the transverse momentum in the region of interest, the second term involving a derivative in the above equation will be negative. Therefore, one would expect the azimuthal asymmetry to decrease because of the multiple scattering in nuclei. To illustrate the nuclear dependence of the azimuthal asymmetry qualitatively, we consider an ansatz of the Gaussian distributions in k$_{⊥}$ for both the twist-2 and 3 TMD quark distributions, The corresponding TMD distributions in nuclei are, One can then calculate the the azimuthal asymmetry for SIDIS off both nucleon and nuclear targets The nuclear modification factor for the azimuthal asymmetry is then, In the special case when α = β , we have 〈 cos φ 〉 eN = − 2(2 − y $_{)}$√$_{1}$ − y 1 + (1 − y ) 2 α β | ⃗ k$_{⊥}$ | Q x$_{B}$f N q $_{⊥}$( x$_{B}$ ) f N q ( x$_{B}$ ) exp { − α − β αβ k 2 ⊥ } , (45) f A q $_{⊥}$( x, k$_{⊥}$ ) ≈ Aβ π ( β + Δ$_{2}$$_{F}$ ) $_{2}$f N q $_{⊥}$( x ) e − k 2 $_{⊥}$/ ( β +Δ$_{2}$ F $^{)}$. (44) 〈 cos φ 〉 eA = − 2(2 − y $_{)}$√$_{1}$ − y 1 + (1 − y ) 2 β ( α + Δ$_{2}$$_{F}$ ) ( β + Δ$_{2}$$_{F}$ ) 2 | ⃗ k$_{⊥}$ | Q x$_{B}$f N q $_{⊥}$( x ) f N q ( x ) exp { − α − β ( α + Δ$_{2}$$_{F}$ )( β + Δ$_{2}$$_{F}$ ) k 2 ⊥ } . (46) 〈 cos φ 〉 eA cos φ eN = β $^{2}$( α + Δ$_{2}$$_{F}$ ) α ( β + Δ$_{2}$$_{F}$ ) 2 exp { ( α − β )Δ$_{2}$$_{F}$ ( α + β + Δ$_{2}$$_{F}$ ) αβ ( α + Δ$_{2}$$_{F}$ )( β + Δ$_{2}$$_{F}$ ) ⃗ k 2 ⊥ } . (47) 〈 cos φ 〉 eN = − 2(2 − y $_{)}$√$_{1}$ − y 1 + (1 − y ) 2 | ⃗ k$_{⊥}$ | Q x$_{B}$f N q $_{⊥}$( x$_{B}$ ) f N q ( x$_{B}$ ) , (48) 〈 cos φ 〉 eA = − 2(2 − y $_{)}$√$_{1}$ − y 1 + (1 − y ) 2 β β + Δ$_{2}$$_{F}$ | ⃗ k$_{⊥}$ | Q x$_{B}$f N q $_{⊥}$( x ) f N q ( x ) , (49) 〈 cos ϕ 〉 eA cos ϕ eN = β β + Δ$_{2}$$_{F}$ . (50) f A q ( x, k$_{⊥}$ ) ≈ A π ( α + Δ$_{2}$$_{F}$ ) f N q ( x ) e − k 2 $_{⊥}$/ ( α +Δ$_{2}$ F $^{)}$, (43) f N q $_{⊥}$( x, k$_{⊥}$ ) = 1 πβ f N q $_{⊥}$( x ) e − k 2 $^{$_{⊥}$/β}$. (42) f N q ( x, k$_{⊥}$ ) = 1 πα f N q ( x ) e − k 2 $^{$_{⊥}$/α}$, (41)