Stable causality of Black Saturns
http://arxiv.org/abs/1102.3942
Sebastian J. Szybka

Basic definitions

The definitions presented below are based on

,,On smothness of Black Saturns''
Piotr T. Chruściel, Michał Eckstein, Sebastian J. Szybka
JHEP 1011:048,2010
http://arxiv.org/abs/1007.3668

The parameters c1, c2, q in terms of the parameters (Subscript[a, i])

c1subS=c1->(Sqrt[2]) (Sqrt[(((-a1+a3) (-a1+a4))/(-a1+a5))]);

c2subS=c2->(((Sqrt[2]) (-a2+a4) ((a1-a2) (a3-a4)+(Sqrt[(a1-a3) (-a2+a4) (a2-a5) (a3-a5)])))/(Sqrt[(a1-a4) (a2-a4) (a1-a5) (a2-a5) (a3-a5)]));

qsubS=q->(((Sqrt[2]) ((a1-a2) (a3-a4)+(Sqrt[(a1-a3) (-a2+a4) (a2-a5) (a3-a5)])) (Sqrt[(((a1-a3) (a1-a4))/(-a1+a5))]))/((a1-a4) (a3-a4)));

Parameterization with (Subscript[μ, i])

Variables which are ending with the capital "T" are used by the Init[] procedure that restores the definitions for ρ>0.

The functions (Subscript[μ, i]) form a monotonic sequence

assume={0≤ ρ≤ μ1≤μ5≤μ4≤μ3≤μ2};

Transformation from (Subscript[a, i]) to (Subscript[μ, i])

asub={a2->a1+(((-μ1+μ2) (μ1 μ2+(ρ^2)))/(2 μ1 μ2)),a3->a1+(((-μ1+μ3) (μ1 μ3+(ρ^2)))/(2 μ1 μ3)),a4->a1+(((-μ1+μ4) (μ1 μ4+(ρ^2)))/(2 μ1 μ4)),a5->a1+(((-μ1+μ5) (μ1 μ5+(ρ^2)))/(2 μ1 μ5))};

The parametersc1,c2,qintermsof(Subscript[μ, i])

qsT={q->c2 c1/(c1-c1 k2+c2 k2)}/.{k1->(a3-a1)/(a2-a1),k2->(a4-a1)/(a2-a1),k3->(a5-a1)/(a2-a1)}/.asub//FullSimplify[Together@#,assume]&

c1subSaT=FullSimplify@(c1subS/.asub);

c2subSaT=Simplify@(c2subS/.asub);

The metric functions that apear in the denominator of (Subscript[g, ψψ]) expressed in terms of (Subscript[μ, i])

ReplaceFunctions={Hx->(μ2 ((c1^2) ((μ1)^2) (((μ1-μ2))^2) μ3 (((μ2-μ4))^2) μ4 (((μ1-μ5))^2) μ5 (ρ^2) (((μ2 μ3+(ρ^2)))^2)+(((μ1-μ3))^2) (((μ2-μ4))^2) ((μ5)^2) (((μ1 μ2+(ρ^2)))^2) (((μ2 μ3+(ρ^2)))^2) (((μ1 μ4+(ρ^2)))^2)+2 c1 c2 μ1 (μ1-μ3) μ3 (μ2-μ4) μ4 (μ1-μ5) μ5 (((μ1)^2)+(ρ^2)) (((μ2)^2)+(ρ^2)) (μ2 μ3+(ρ^2)) (μ1 μ4+(ρ^2)) (μ2 μ5+(ρ^2))+(c1^2) (c2^2) ((μ1)^2) ((μ3)^2) ((μ4)^2) (((μ1-μ5))^2) (((μ1 μ2+(ρ^2)))^2) (((μ2 μ5+(ρ^2)))^2)+(c2^2) (((μ1-μ2))^2) (((μ1-μ3))^2) μ3 μ4 μ5 (ρ^2) (((μ1 μ4+(ρ^2)))^2) (((μ2 μ5+(ρ^2)))^2)))/(μ1 (((μ1-μ3))^2) (((μ2-μ4))^2) μ5 (((μ1)^2)+(ρ^2)) (((μ2)^2)+(ρ^2)) (μ1 μ3+(ρ^2)) (μ2 μ3+(ρ^2)) (((μ3)^2)+(ρ^2)) (μ1 μ4+(ρ^2)) (μ2 μ4+(ρ^2)) (((μ4)^2)+(ρ^2)) (μ2 μ5+(ρ^2)) (μ3 μ5+(ρ^2)) (((μ5)^2)+(ρ^2))),F->μ1 (((μ1-μ3))^2) (((μ2-μ4))^2) μ5 (((μ1)^2)+(ρ^2)) (((μ2)^2)+(ρ^2)) (μ1 μ3+(ρ^2)) (μ2 μ3+(ρ^2)) (((μ3)^2)+(ρ^2)) (μ1 μ4+(ρ^2)) (μ2 μ4+(ρ^2)) (((μ4)^2)+(ρ^2)) (μ2 μ5+(ρ^2)) (μ3 μ5+(ρ^2)) (((μ5)^2)+(ρ^2))};

The simplified formula for (Subscript[g, ψψ])=(((S1T^2)-(S2T^2))/(F Hx μ1 μ4))

psub={p1->(μ3-μ1) (μ1 μ4+(ρ^2)),
p2->(μ2-μ4)(μ1 μ2+(ρ^2)) (μ2 μ3+(ρ^2)),
p3->(μ2-μ1)(μ2 μ5+(ρ^2)),
p4->μ1 (μ5-μ1),
p5->(μ2-μ1) (μ2-μ4)(μ2 μ3+(ρ^2)),
p6->(μ1 μ2+(ρ^2)) (μ2 μ5+(ρ^2))};

A=(μ2  p1( μ5 p2 +c2 q μ1  μ3 p3)+c1  μ3 p4 (-q  (ρ^2) p5+c2 μ1 μ4 p6))/.psub//FullSimplify;

S1T=(Sqrt[μ4]) (Sqrt[μ5]) A

B= ( μ5 p1 (q μ1 p2-c2 μ4 (ρ^2) p3)+c1 μ2 μ4 p4(μ1 p5 μ5+c2 q μ3p6))/.psub//FullSimplify;

S2T=(Sqrt[μ3]) B

The Ξ funtion as defined in the article. The numerator of (Subscript[g, ψψ])is given by Ξ[[1]] Ξ[[2]].

Ξ=Evaluate[{S1+S2,S1-S2}];

Additional definitions

The substitution operators

Sc12[x_]:=Module[{t},t=Together[x/.{c1^2->(c1^2/.c1subSa),c1^4->(c1^4/.c1subSa),c1^3->(c1(c1^2/.c1subSa))}];Factor@Numerator@t/Factor@Denominator@t]

Sq[x_]:=Module[{},t=Together[x/.qs];Factor@Numerator@t/Factor@Denominator@t]

The numerical check at random values of variables

RLcheck[x_,l_]:=N[x/.(Map[#/.y_->(y->Random[])&,l])];

The check of coefficients in front of strictly positive variables

CRL[x_,l_]:=Module[{tbl},tbl=CoefficientRules[x,l]/.(xx_->yy_)->yy;Return@{Min@tbl,Max@tbl}]

The transformation to strictly positive variables (as defined by l, used before RLcheck[.,.])

toDL[x_,l_]:=x/.qs/.c2subSa/.c1subSa//.l;

The procedure that restores "ρ>0 definitions"

Init[]:=Module[{},S1=S1T;S2=S2T;c1subSa=c1subSaT;c2subSa=c2subSaT;qs=qsT;]

V1, ρ>0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of substitutions needed before the limit ρ -> 0 will be taken

sub1={};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub={μ5->d51+μ1,μ4->d45+μ5,μ3->d34+μ4,μ2->d23+μ3};

The list of striclty positive variables

nonzero={ρ,μ1,d51,d45,d34,d23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)

c1subSa=Factor@(c1subSa/.sub1)

S1=(S1T/.sub1//Factor)

S2=(S2T/.sub1//Factor)

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

We substitute c1^2. The denominator is negative

r1d=Denominator@Sc12[Δ]

Numerator@Sc12[Δ]

r1n=%[[{1,2},1;;-2]]

The numerator contains negative factor times some nontrivial w

w=%%[[{1,2},-1]]

In the article we cancel - 1 in the numerator with - 1 in the denominator. Now, we check w at a random point

RLcheck[toDL[w,dsub],nonzero]

Exponent[w,c1]

We solve w == 0 for c1

c1pm={Solve[w[[1]]==0,c1],Solve[w[[2]]==0,c1]};

(c1/.c1pm[[1]])+(c1/.c1pm[[2]])//Factor

Both polynomials w[[1]], w[[2]] give the same solution. We would like to check if this value of c1 could coincide with the true value of c1

{U}=Together[(c1^2/.c1pm[[1]])-(c1^2/.c1subSa)]

Factor@Numerator@U

This is (OverHat[U])

hU=%[[-1]]

and the factor in front of it

r2n=%%[[1;;-2]]

Factor@Denominator@U

This is (OverTilde[U])

tU=%[[-1]]

and the factor in front of it

r2d=%%[[1;;-2]]

Now, we write  (OverHat[U]) and  (OverTilde[U]) in terms of strictly positive variables and print the smallest and the biggest coefficient

r3n=CRL[toDL[hU,dsub],nonzero]

r3d=CRL[toDL[tU,dsub],nonzero]

Finally, we check at a random point that Δ is really negative

RLcheck[toDL[Δ,dsub],nonzero]

Results

The coefficient behind w is

r1n/r1d//FullSimplify

The coefficient before (OverHat[U]) is

r2n

The coefficient before (OverTilde[U]) is

r2d

The range of coefficients in  (OverHat[U]) is

r3n

The range of coefficients in  (OverTilde[U]) is

r3d

V2, ρ=0, (Subscript[μ, 1])>0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of substitutions needed before the limit ρ -> 0 will be taken

sub1={};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub={μ5->d51+μ1,μ4->d45+μ5,μ3->d34+μ4,μ2->d23+μ3};

The list of striclty positive variables

nonzero={μ1,d51,d45,d34,d23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)/.ρ->0

c1subSa=Factor@(c1subSa/.sub1)/.ρ->0

c2subSa=Factor@(c2subSa/.sub1)/.ρ->0;

S1=(S1T/.sub1//Factor)/.ρ->0

S2=(S2T/.sub1//Factor)/.ρ->0

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate the discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

We substitute c1^2. The denominator is negative

r1d=Denominator@Sc12[Δ]

Numerator@Sc12[Δ]

r1n=%[[{1,2},1;;-2]]

The numerator contains negative factor times some nontrivial w

w=%%[[{1,2},-1]]

In the article we cancel - 1 in the numerator with - 1 in the denominator. Now, we check w at a random point

RLcheck[toDL[w,dsub],nonzero]

Exponent[w,c1]

We solve w == 0 for c1

c1pm={Solve[w[[1]]==0,c1],Solve[w[[2]]==0,c1]};

(c1/.c1pm[[1]])+(c1/.c1pm[[2]])//Factor

Both polynomials w[[1]], w[[2]] give the same solution. We would like to check if this value of c1 could coincide with the true value of c1

{U}=Together[(c1^2/.c1pm[[1]])-(c1^2/.c1subSa)]

Factor@Numerator@U

This is (OverHat[U])

hU=%[[-1]]

and the factor in front of it

r2n=%%[[1;;-2]]

Factor@Denominator@U

This is (OverTilde[U])

tU=%[[-1]]

and the factor in front of it

r2d=%%[[1;;-2]]

Now, we write  (OverHat[U]) and  (OverTilde[U]) in terms of strictly positive variables and print the smallest and the biggest coefficient

r3n=CRL[toDL[hU,dsub],nonzero]

r3d=CRL[toDL[tU,dsub],nonzero]

Finally, we check at a random point that Δ is really negative

RLcheck[toDL[Δ,dsub],nonzero]

Results

The coefficient behind w is

r1n/r1d//FullSimplify

The coefficient before (OverHat[U]) is

r2n

The coefficient before (OverTilde[U]) is

r2d

The range of coefficients in  (OverHat[U]) is

r3n

The range of coefficients in  (OverTilde[U]) is

r3d

V3, ρ=(Subscript[μ, 1])=0, (Subscript[μ, 5])>0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of substitutions needed before the limit ρ -> 0 will be taken

sub1={μ1->ρ^2 hμ1};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub=Join[{μ5->d51+μ1,μ4->d45+μ5,μ3->d34+μ4,μ2->d23+μ3},sub1,{ρ->0}];

The list of striclty positive variables

nonzero={hμ1,d51,d45,d34,d23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)/.ρ->0

c1subSa=Factor@(c1subSa/.sub1)/.ρ->0

c2subSa=c2->Limit[Factor@(c2/.c2subSa/.sub1),ρ->0,Direction->-1]

S1=(S1T/.sub1//Factor)/ρ^4/.ρ->0

S2=(S2T/.sub1//Factor)/ρ^4/.ρ->0

The denominator in terms of strictly positive variables (there is a factor ρ^8 in front of the denominator)

μ1 μ4 Hx F/.ReplaceFunctions/.sub1//Factor;

Short@%

powρ=8;

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate the discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

Δ//Factor

We substitute c1^2. The denominator is positive

r1d=Denominator@Sc12[Δ]

Numerator@Sc12[Δ]

r1n=%[[{1,2},1;;-2]]

The numerator contains negative factor times some nontrivial w

w=%%[[{1,2},-1]]

Now, we check w at a random point

RLcheck[toDL[w,dsub],nonzero]

We solve w == 0 for c1

c1pm={Solve[w[[1]]==0,c1],Solve[w[[2]]==0,c1]};

(c1/.c1pm[[1]])+(c1/.c1pm[[2]])//Factor

Both polynomials w[[1]], w[[2]] give the same solution. We would like to check if this value of c1 could coincide with the true value of c1

{U}=Together[(c1^2/.c1pm[[1]])-(c1^2/.c1subSa)]

Factor@Numerator@U

This is (OverHat[U])

hU=%;

and the factor in front of it

r2n=1

Factor@Denominator@U

This is (OverTilde[U])

tU=%[[-1]]

and the factor in front of it

r2d=%%[[1;;-2]]

Now, we write  (OverHat[U]) and  (OverTilde[U]) in terms of strictly positive variables and print the smallest and the biggest coefficient

r3n=CRL[toDL[hU,dsub],nonzero]

r3d=CRL[toDL[tU,dsub],nonzero]

Finally, we check at a random point that Δ is really negative

RLcheck[toDL[Δ,dsub],nonzero]

Results

The factor in front of the numerator and the denominator of (Subscript[g, ψψ])

powρ

The coefficient behind w is

r1n/r1d//FullSimplify

The coefficient before (OverHat[U]) is

r2n

The coefficient before (OverTilde[U]) is

r2d

The range of coefficients in  (OverHat[U]) is

r3n

The range of coefficients in  (OverTilde[U]) is

r3d

A part of the black ring event horizon, ρ =(Subscript[μ, 1])= (Subscript[μ, 5])=0, (Subscript[μ, 4])>0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of substitutions needed before the limit ρ -> 0 will be taken

sub1={μ1->ρ^2 hμ1,μ5->ρ^2 hμ5};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub=Join[{hμ5->hd51+hμ1,μ4->d45+μ5,μ3->d34+μ4,μ2->d23+μ3},sub1,{ρ->0}];

The list of striclty positive variables

nonzero={hμ1,hd51,d45,d34,d23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)/.ρ->0

c1subSa=Factor@(c1subSa/.sub1)/.ρ->0

c2subSa=c2->Limit[Factor@(c2/.c2subSa/.sub1),ρ->0,Direction->-1]

S1=(PowerExpand[S1T/.sub1]//Factor)/ρ^7/.ρ->0

S2=(PowerExpand[S2T/.sub1]//Factor)/ρ^7/.ρ->0

There is a factor ρ^14 in front of the denominator

μ1 μ4 Hx F/.ReplaceFunctions/.sub1//Factor;

Short[%,2]

powρ=14;

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate the discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

Δ//Factor

We substitute c1^2. The denominator is positive

r1d=Denominator@Sc12[Δ]

Numerator@Sc12[Δ]

r1n=%[[{1,2},1;;-2]]

The numerator contains negative factor times some nontrivial w

w=%%[[{1,2},-1]]

Now, we check w at a random point

RLcheck[toDL[w,dsub],nonzero]

There is no c1 in w

Exponent[w,c1]

We write w in terms of strictly positive variables and print the smallest and the biggest coefficient

wrange=CRL[toDL[w[[1]],dsub],nonzero]

CRL[toDL[w[[2]],dsub],nonzero]

Hence, w is positive and Δ is negative

Results

The factor in front of the numerator and the denominator of (Subscript[g, ψψ])

powρ

The coefficient behind w is

r1n/r1d//FullSimplify

The range of the coefficients in w

wrange

V4, ρ=(Subscript[μ, 1])=(Subscript[μ, 5])=(Subscript[μ, 4])=0, (Subscript[μ, 3])>0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of the substitutions needed before the limit ρ -> 0 will be taken

sub1={μ1->ρ^2 hμ1,μ5->ρ^2 hμ5,μ4->ρ^2 hμ4};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub=Join[{hμ5->hd51+hμ1,hμ4->hd45+hμ5,μ3->d34+μ4,μ2->d23+μ3},sub1,{ρ->0}];

The list of striclty positive variables

nonzero={hμ1,hd51,hd45,d34,d23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)/.ρ->0

c1subSa=Factor@(c1subSa/.sub1)/.ρ->0

c2subSa=c2->Limit[Factor@(c2/.c2subSa/.sub1),ρ->0,Direction->-1]

S1=(PowerExpand[S1T/.sub1]//Factor)/ρ^8/.ρ->0

S2=(PowerExpand[S2T/.sub1]//Factor)/ρ^8/.ρ->0

There is a factor ρ^16 in front of the denominator of (Subscript[g, ψψ])

μ1 μ4 Hx F/.ReplaceFunctions/.sub1//Factor;

Short@%

powρ=16;

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate the discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

Δ//Factor

We substitute c1^2. The denominator is negative

r1d=Denominator@Sc12[Δ]

Numerator@Sc12[Δ]

r1n=%[[{1,2},1;;-2]]

One numerator contains a negative factor times some nontrivial w[[1]], the second numerator contains a positive factor times some nontrivial w[[2]]

w=%%[[{1,2},-1]]

Now, we check w at a random point

RLcheck[toDL[w,dsub],nonzero]

We solve w == 0 for c1

c1pm={Solve[w[[1]]==0,c1],Solve[w[[2]]==0,c1]};

(c1/.c1pm[[1]])+(c1/.c1pm[[2]])//Factor

Both polynomials w[[1]], w[[2]] give the same solution. We would like to check if this value of c1 could coincide with the true value of c1

{U}=Together[(c1^2/.c1pm[[1]])-(c1^2/.c1subSa)]

Factor@Numerator@U

This is (OverHat[U])

hU=%[[-1]];

and the factor in front of it

r2n=%%[[1;;-2]]

Factor@Denominator@U

This is (OverTilde[U])

tU=1

and the factor in front of it

r2d=%%

Now, we write  (OverHat[U]) and  (OverTilde[U]) in terms of strictly positive variables and print the smallest and the biggest coefficient

r3n=CRL[toDL[hU,dsub],nonzero]

r3d=CRL[toDL[tU,dsub],nonzero]

Finally, we check at a random point that Δ is really negative

RLcheck[toDL[Δ,dsub],nonzero]

Results

The factor in front of the numerator and the denominator of (Subscript[g, ψψ])

powρ

The coefficient behind w is

r1n/r1d//FullSimplify

The coefficient before (OverHat[U]) is

r2n

The coefficient in the denominator of U is

r2d

The range of coefficients in  (OverHat[U]) is

r3n

The range of coefficients in  (OverTilde[U]) is

r3d

A part of the spherical black hole event horizon, ρ=(Subscript[μ, 1])=(Subscript[μ, 5])=(Subscript[μ, 4])=(Subscript[μ, 3])=0, (Subscript[μ, 2])>0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of the substitutions needed before the limit ρ -> 0 will be taken

sub1={μ1->ρ^2 hμ1,μ5->ρ^2 hμ5,μ4->ρ^2 hμ4,μ3->ρ^2 hμ3};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub=Join[{hμ5->hd51+hμ1,hμ4->hd45+hμ5,hμ3->hd34+hμ4,μ2->d23+μ3},sub1,{ρ->0}];

The list of striclty positive variables

nonzero={hμ1,hd51,hd45,hd34,d23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)/.ρ->0

c1subSa=Factor@(c1subSa/.sub1)/.ρ->0

c2subSa=c2->Limit[Factor@(c2/.c2subSa/.sub1),ρ->0,Direction->-1]

S1=(PowerExpand[S1T/.sub1]//Factor)/ρ^12/.ρ->0

S2=(PowerExpand[S2T/.sub1]//Factor)/ρ^12/.ρ->0

There is a factor (ρ^24) in front of the denominator of (Subscript[g, ψψ])

μ1 μ4 Hx F/.ReplaceFunctions/.sub1//Factor;

Short@%

powρ=24;

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate the discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

Δ//Factor

We substitute c1^2. The denominator is negative

r1d=Denominator@Sc12[Δ]

Numerator@Sc12[Δ]

The numerator is stroctly positive

r1n=%[[{1,2}]]

Finally, we check at a random point that Δ is really negative

RLcheck[toDL[Δ,dsub],nonzero]

Results

The factor in front of the numerator and the denominator of (Subscript[g, ψψ])

powρ

The determinant Δ is

r1n/r1d//FullSimplify

(Subscript[V, ψ]), ρ=(Subscript[μ, 1])=(Subscript[μ, 5])=(Subscript[μ, 4])=(Subscript[μ, 3])=(Subscript[μ, 2])=0

Calculations

We restore "ρ>0 definitions"

Init[]

The list of the substitutions needed before the limit ρ -> 0 will be taken

sub1={μ1->ρ^2 hμ1,μ5->ρ^2 hμ5,μ4->ρ^2 hμ4,μ3->ρ^2 hμ3,μ2->ρ^2 hμ2};

The transformation from (Subscript[μ, i]) to strictly positive variables

dsub=Join[{hμ5->hd51+hμ1,hμ4->hd45+hμ5,hμ3->hd34+hμ4,hμ2->hd23+hμ3},sub1,{ρ->0}];

The list of striclty positive variables

nonzero={hμ1,hd51,hd45,hd34,hd23};

The functions in terms of strictly positive variables

qs=Factor@(qsT/.sub1)/.ρ->0

c1subSa=Factor@(c1subSa/.sub1)/.ρ->0

c2subSa=c2->Limit[Factor@(c2/.c2subSa/.sub1),ρ->0,Direction->-1]

S1=(PowerExpand[S1T/.sub1]//Factor)/ρ^16/.ρ->0

S2=(PowerExpand[S2T/.sub1]//Factor)/ρ^15/.ρ->0

There is a factor (ρ^30) in front of the denominator of (Subscript[g, ψψ])

μ1 μ4 Hx F/.ReplaceFunctions/.sub1//Factor;

Short@%

powρ=30;

It turns out that the numerator behaves like  ρ^32 (at least)

Sc12@Sq[S2]

S1=(PowerExpand[S1T/.sub1]//Factor)/ρ^16/.ρ->0

S2=CoefficientList[(PowerExpand[S2T/.sub1]),ρ][[17]]

pownρ=2 16;

Ξ-{S1+S2,S1-S2}

We check Ξ at a random point

RLcheck[toDL[Ξ ,dsub],nonzero]

Now, we would like to verify if Ξ could vanish. We substitute q

Sq@Ξ

and calculate the discriminant in respect to c2

Δ=Discriminant[Numerator[%],c2]

Exponent[Δ,c1]

Δ//Factor

We substitute c1^2. The denominator is negative

r1d=Denominator@Sc12[Δ]

The numerator is positive

Numerator@Sc12[Δ]

r1n=%

Finally, we check at a random point that Δ is really negative

RLcheck[toDL[Δ,dsub],nonzero]

Results

The factor in front of the denominator of (Subscript[g, ψψ])

powρ

The factor front in numerator of (Subscript[g, ψψ])

pownρ

The discriminant Δ is

r1n/r1d//FullSimplify