Issues

* Include all parameters as function arguments.
* NSum[], which is a combination symbolic-numerical solver, evaluates the summand symbolically.

Alternative 1

L=200;​​Ω=1;​​σ=5;​​λ=0.1;​​​​ClearAll[x,t];​​t[τ_,a_]:=1/aSinh[aτ]​​x[τ_,a_]:=1/a(Cosh[aτ]-1)​​​​(*Ensuremisnumerictoavoidsymbolicevaluations*)​​ClearAll[η0,η1,int0,int1];​​int0[a_?NumericQ,m_?NumericQ]:=NIntegrate[Exp[-τ^2/(2σ^2)]Exp[IΩτ+(2IPi)/L(Abs[m]t[τ,a]-mx[τ,a])]1/Sqrt[4PiAbs[m]],{τ,-100,100}];​​η0[a_]:=λNSum[int0[a,m],{m,1,100}]​​​​int1[a_?NumericQ,m_?NumericQ]:=NIntegrate[Exp[-τ^2/(2σ^2)]Exp[IΩτ-(2IPi)/L(Abs[m]t[τ,a]-mx[τ,a])]1/Sqrt[4PiAbs[m]],{τ,-100,100}];​​η1[a_]:=λNSum[int1[a,m],{m,1,100}]​​​​F[a_]:=-(η0[a]^2/(1+η0[a]^2))+1/(1+η1[a]^2)​​​​η0[0.21]
Out[]=
-1.85339×
-7
10
+5.52228×
-6
10


Alternative 2

Use Total[] and Table[] to carry out a purely numerical sum.
L=200;​​Ω=1;​​σ=5;​​λ=0.1;​​​​ClearAll[x,t];​​t[τ_,a_]:=1/aSinh[aτ]​​x[τ_,a_]:=1/a(Cosh[aτ]-1)​​​​(*Ensuremisnumerictoavoidsymbolicevaluations*)​​ClearAll[η0,η1];​​η0[a_]:=λTotal@Table[NIntegrate[Exp[-τ^2/(2σ^2)]Exp[IΩτ+(2IPi)/L(Abs[m]t[τ,a]-mx[τ,a])]1/Sqrt[4PiAbs[m]],{τ,-100,100}],{m,1,100}];​​​​η1[a_]:=λTotal@Table[NIntegrate[Exp[-τ^2/(2σ^2)]Exp[IΩτ-(2IPi)/L(Abs[m]t[τ,a]-mx[τ,a])]1/Sqrt[4PiAbs[m]],{τ,-100,100}],{m,1,100}]​​​​F[a_]:=-(η0[a]^2/(1+η0[a]^2))+1/(1+η1[a]^2)​​​​η0[0.21]
Out[]=
-1.85304×
-7
10
+5.5223×
-6
10
