DensityPlot, RegionPlot, and ContourPlot Runtime with NMinValue

Question

I need to make a DensityPlot, a RegionPlot, and a ContourPlot of a function MinΔBG that I've defined in the code below, but my problem is:

the DensityPlot of the function shows after 15 minutes; however the computation runtime of the RegionPlot and ContourPlot of the same function are above 30 minutes in my laptop, and I stoped the processes after that time. Is there a better way to write the code below and so to optimize the runetime of the previous plots?

Here is my code:

SMValues := {g -> 0.64, g1 -> 0.34, yt -> 1.0, mh -> 125.18, vh -> 246.22}

ΔBGH[Λ_,mr_,λHR_] := Max[{(Λ^2 Abs[(3 g^2)/2+(9 g1^2)/2+(6 mh^2)/vh^2-12 yt^2-(3 mh^4)/(vh^2 (mh^2+Λ^2))+2 λHR-(2 mr^2 λHR)/(mr^2+Λ^2)])/(16 mh^2 π^2),(mr^2 Abs[λHR (-(Λ^2/(mr^2+Λ^2))+Log[1+Λ^2/mr^2])])/(8 mh^2 π^2),(λHR Abs[Λ^2-mr^2 Log[1+Λ^2/mr^2]])/(8 mh^2 π^2)}]
ΔBGR[Λ_,mr_,λHR_,λR_] := Max[{(Λ^2 Abs[(Λ^2 λHR)/(mh^2+Λ^2)+(6-(3 mr^2)/(mr^2+Λ^2)) λR])/(8 mr^2 π^2),(3 Abs[λR (-(Λ^2/(mr^2+Λ^2))+Log[1+Λ^2/mr^2])])/(8 π^2),(λHR Abs[Λ^2-mh^2 Log[1+Λ^2/mh^2]])/(8 mr^2 π^2),(3 λR Abs[-2 Λ^2+mr^2 Log[1+Λ^2/mr^2]])/(8 mr^2 π^2)}]

ΔBGHR[Λ_,mr_,λHR_,λR_] := Max[{ΔBGH[Λ,mr,λHR], ΔBGR[Λ,mr,λHR,λR]}]

MinΔBG[Λ_,λR_] := NMinValue[ΔBGHR[Λ,mr,λHR,λR], {mr,λHR} ∈ Rectangle[{50,-3},{65,-1.5}]]

DensityPlot[
  Evaluate[MinΔBG[Λ*10^3,λR]/.SMValues],{Λ,1,2},{λR,0,1},
  PlotLegends->BarLegend[Automatic],
  FrameLabel->{Style["Λ [TeV]",12,Bold], Style["λR",12,Bold]},
  Mesh->19,ColorFunction->"StarryNightColors"];

RegionPlot[Evaluate[MinΔBG[Λ*10^3,λR]/.SMList]<=10,{Λ,1,2},{λR,0,1}];

ContourPlot[Evaluate[MinΔBG[Λ*10^3,λR]/.SMList]==10,{Λ,1,2},{λR,0,1}];

Incidentally, when I evaluate the function MinΔBG[Λ,λR] for virtually any par of values (like in the following example), it yields the following message (error?):

MinΔBG[10^3, 0.4]/.SMValues
NMinValue::nnum: The function value Max[8.41802,-(36387.5/mh^2),616.062/mh^2,3.54321 Abs[2.39573 -2.93213*10^6/(1000000+mh^2)],(62500 Abs[-5.84337+(3 g^2)/2+(9 g1^2)/2+(6 mh^2)/vh^2-(3 mh^4)/((1000000+mh^2) vh^2)-12 yt^2])/(mh^2 π^2),-0.0000103891 Abs[1000000-mh^2 Log[1+1000000 Power[<<2>>]]]] is not a number at {mr,λHR} = {59.787,-2.93213}.
NMinValue::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations.
7.11126

which I'm not sure it could be generating an increse in the running time of the plots.

I am working on Mathematica 12.0. My laptop is an HP Pavilion with intel i7 and 16 GB of ram.

Answer 1

This is a partial solution only.

g = 0.64; g1 = 0.34; yt = 1.0; mh = 125.18; vh = 246.22;

(* unchanged *)
ΔBGH[Λ_, mr_, λHR_] := Max[{(Λ^2 Abs[(3 g^2)/2 + (9 g1^2)/2 + (6 mh^2)/vh^2 - 12 yt^2 - (3 mh^4)/(vh^2 (mh^2 + Λ^2)) + 2 λHR - (2 mr^2 λHR)/(mr^2 + Λ^2)])/(16 mh^2 π^2), (mr^2 Abs[λHR (-(Λ^2/(mr^2 + Λ^2)) + Log[1 + Λ^2/mr^2])])/(8 mh^2 π^2), (λHR Abs[Λ^2 - mr^2 Log[1 + Λ^2/mr^2]])/(8 mh^2 π^2)}]
ΔBGR[Λ_, mr_, λHR_, λR_] := Max[{(Λ^2 Abs[(Λ^2 λHR)/(mh^2 + Λ^2) + (6 - (3 mr^2)/(mr^2 + Λ^2)) λR])/(8 mr^2 π^2), (3 Abs[λR (-(Λ^2/(mr^2 + Λ^2)) + Log[1 + Λ^2/mr^2])])/(8 π^2), (λHR Abs[Λ^2 - mh^2 Log[1 + Λ^2/mh^2]])/(8 mr^2 π^2), (3 λR Abs[-2 Λ^2 + mr^2 Log[1 + Λ^2/mr^2]])/(8 mr^2 π^2)}]
ΔBGHR[Λ_, mr_, λHR_, λR_] := Max[{ΔBGH[Λ, mr, λHR], ΔBGR[Λ, mr, λHR, λR]}]

(* added pattern `?NumericQ` to function arguments, 
   and changed the way the constraints are written *)
MinΔBG[Λ_?NumericQ, λR_?NumericQ] := NMinValue[{
   ΔBGHR[Λ, mr, λHR, λR],
   mr > 50 && mr < 65 && λHR > -3 && λHR < -1.5}, {mr, λHR}]

Calling MinΔBG with the example values now yields a number:

MinΔBG[10^3, 0.4]
  (* 7.11126 *)

DensityPlot and CountourPlot work (I used a small number for PlotPoints to make it faster).

DensityPlot[MinΔBG[Λ*10^3, λR], {Λ, 1, 2}, {λR, 0, 1}, 
 PlotLegends -> BarLegend[Automatic], FrameLabel -> {Style["Λ [TeV]", 12, Bold], 
   Style["λR", 12, Bold]}, Mesh -> 19, ColorFunction -> "StarryNightColors", PlotPoints -> 4]

ContourPlot[MinΔBG[Λ*10^3, λR] == 10, {Λ, 1, 2}, {λR, 0, 1}, PlotPoints -> 4]

I can't figure out what's wrong with RegionPlot - I'm getting errors like Break::nofwd: No enclosing For, While, or Do found for Break[].