Finding root of function with numerical integration

Question

I have two function, $\alpha20$ and $\alpha22$, which are given in terms of some integrals. They depend on some parameters. I fix most of them except $(T,\mu)$. I want to find the roots of these function (equivalently finding $T$) for various parameter $\mu$. I used Findroot but no success specially when I am working with $\alpha22$. These are my functions:

α20[(A_)?NumberQ, (μ_)?NumberQ, (T_)?NumberQ, (ω_)?
   NumberQ, (lMin_)?NumberQ, (lMax_)?NumberQ, (Nf_)?NumberQ, (Nc_)?
   NumberQ, 
     (b_)?NumberQ, (Λ_)?NumberQ, (r_)?
   NumberQ] := Λ^2/(2*b) - 
  A^2*((Nf*Nc)/(8*Pi^2))*
   NIntegrate[1/τ^2, {τ, Λ^(-2), Infinity}, 
         Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
    AccuracyGoal -> 10] + 
     A^2*((Nf*Nc)/(4*Pi^2))*
   Sum[NIntegrate[
     p*(BesselJ[ℓ, p*r]^2 + BesselJ[ℓ + 1, p*r]^2)*
             (1/(E^((Sqrt[
                 k^2 + p^2] - μ - ω*(ℓ + 1/2))/
              T)*((1 + 
               E^(-((Sqrt[
                    k^2 + p^2] - μ - ω*(ℓ + 1/2))/
                    T)))*Sqrt[k^2 + p^2])) + 
                
        1/(E^((Sqrt[k^2 + p^2] + μ + ω*(ℓ + 1/2))/
              T)*((1 + 
               E^(-((Sqrt[
                    k^2 + p^2] + μ + ω*(ℓ + 1/2))/
                    T)))*Sqrt[k^2 + p^2]))), 
           {p, 0, Infinity}, {k, 0, Infinity}, 
     Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
     AccuracyGoal -> 10], 
         {ℓ, lMin, lMax, 1}]

and

α22[(A_)?NumberQ, (μ_)?NumberQ, (T_)?NumberQ, (ω_)?
    NumberQ, (lMin_)?NumberQ, (lMax_)?NumberQ, (Nf_)?NumberQ, (Nc_)?
    NumberQ, 
       (b_)?NumberQ, (Λ_)?NumberQ, (r_)?NumberQ] := 
  A^2*((Nf*Nc)/(2*Pi^(5/2)))*
    NIntegrate[(1/τ^(1/2))*((3 - 2*k^2*τ)/E^(k^2*τ)), 
           {k, 0.000001, Infinity}, {τ, Λ^(-2), 
      Infinity}, 
     Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
     AccuracyGoal -> 10] - 
       A^2*((Nf*Nc)/(4*Pi^2))*T*
    Sum[NIntegrate[
      p*(BesselJ[ℓ, p*r]^2 + 
         BesselJ[ℓ + 1, p*r]^2)*
               ((3*E^((3*Sqrt[k^2 + p^2])/T)*p^2*T^2 + 
           6*E^((3*μ)/T)*p^2*T^2 + 
           3*E^((3*(Sqrt[k^2 + p^2] + 
                   2*(μ + ω*(ℓ + 1/2))))/T)*p^2*
                       T^2 + 
           9*E^((3*Sqrt[k^2 + p^2] + 
                 2*(μ + ω*(ℓ + 1/2)))/T)*p^2*
            T*(2*Sqrt[k^2 + p^2] + 3*T) + 
                     
           9*E^((3*Sqrt[k^2 + p^2] + 
                 4*(μ + ω*(ℓ + 1/2)))/T)*p^2*
            T*(2*Sqrt[k^2 + p^2] + 3*T) - 
                     
           E^((5*Sqrt[k^2 + p^2] + 
                 2*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 - 3*p^2*T*(Sqrt[k^2 + p^2] + T)) - 
                     
           6*E^((4*Sqrt[k^2 + p^2] + 
                 3*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 - 3*p^2*T*(Sqrt[k^2 + p^2] + T)) - 
                     
           E^((5*Sqrt[k^2 + p^2] + 
                 4*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 - 3*p^2*T*(Sqrt[k^2 + p^2] + T)) + 
                     
           E^((4*Sqrt[
                   k^2 + p^2] + (μ + ω*(ℓ + 
                    1/2)))/T)*(k^4 + k^2*p^2 + 
              3*p^2*T*(Sqrt[k^2 + p^2] + 2*T)) + 
                     
           6*E^((2*Sqrt[k^2 + p^2] + 
                 3*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 + 3*p^2*T*(Sqrt[k^2 + p^2] + 2*T)) + 
                     
           E^((4*Sqrt[k^2 + p^2] + 
                 5*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 + 3*p^2*T*(Sqrt[k^2 + p^2] + 2*T)) - 
                     
           
           E^((2*Sqrt[
                   k^2 + p^2] + (μ + ω*(ℓ + 
                    1/2)))/T)*(k^4 + k^2*p^2 - 
              3*p^2*T*(Sqrt[k^2 + p^2] + 4*T)) - 
                     
           E^((2*Sqrt[k^2 + p^2] + 
                 5*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 - 3*p^2*T*(Sqrt[k^2 + p^2] + 4*T)) + 
                     
           E^((Sqrt[k^2 + p^2] + 
                 2*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 + 3*p^2*T*(Sqrt[k^2 + p^2] + 5*T)) + 
                     
           E^((Sqrt[k^2 + p^2] + 
                 4*(μ + ω*(ℓ + 1/2)))/T)*(k^4 + 
              k^2*p^2 + 3*p^2*T*(Sqrt[k^2 + p^2] + 5*T)))/
                  ((E^(Sqrt[k^2 + p^2]/T) + 
              E^((μ + ω*(ℓ + 1/2))/T))^3*(1 + 
              E^((Sqrt[
                    k^2 + p^2] + μ + ω*(ℓ + 1/2))/
                 T))^3*(k^2 + p^2)^(5/2)*
                     T^2)), {p, 0.000001, Infinity}, {k, 0.000001, Infinity}, 
      Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
      AccuracyGoal -> 10], 
           {ℓ, lMin, lMax, 1}]; 

I use Findroot to find $T$ for different values of $\mu$. But the code run for a long period of time and I do not receive any results (specially for $\alpha22$).This is my sample code

Block[{A = N[1], lMin = -2, lMax = 2, Nf = 2, 
  Nc = 3, Λ = 0.86, r = 0.1, 
  b = 11, ω = 0.05}, 
   list = {}; 
 Table[roots = 
   FindRoot[α22[A, μ, T, ω, lMin, lMax, Nf, Nc, 
     b, Λ, r], {T, 0.05}, Method -> "Newton"]; 
roots2 = 
   FindRoot[α20[A, μ, T, ω, lMin, lMax, Nf, Nc, 
     b, Λ, r], {T, 0.05}, Method -> "Newton"]; 
       AppendTo[
   list, {roots[[1, 2]],roots2[[1, 2]], μ, ω}]; , {μ, 
   0.36, 0.4, 0.005}]]

Is there a way to find roots of these function? I also want the code run a bit faster! Any help is appreciated.