Problem 2.1

a)

(* π is protected in Mathematica *)

V= 1/2 μ^2(ρ^2 + πp^2) +1/4 λ(ρ^2 + πp^2)^2

vevs={ρ-> v Cos[θ],πp-> v Sin[θ]}

Eq1=D[V,ρ]/. vevs //Simplify

Eq2=D[V,πp]/. vevs //Simplify

Sol=Part[Solve[Eq1==0,λ],1]

NewVac=Flatten[{Sol,{ρp->0,πpp->0}}]

b)

subs={ρ-> v Cos[θ]+ρp,πp-> v Sin[θ]+πpp}

VNew= V/. subs

M11=D[D[VNew,ρp],ρp]/. NewVac //Simplify

M12=D[D[VNew,ρp],πpp]/. NewVac //Simplify

M21=D[D[VNew,πpp],ρp]/. NewVac //Simplify

M22=D[D[VNew,πpp],πpp]/. NewVac //Simplify

MassMat={{M11,M12},{M21,M22}}

c)

Eigenvalues[MassMat]

Note:  Take x={ρp,πpp} and Notice M_12=M_21

V= 1/2 x_i M_ij x_j =
1/2 M_11 x_1^2 + M_12 x_1 x_2 +1/2 M_22 x_2^2

Problem 3.1

e)

MatMass=1/2v^2{{g^2,-g gp},{-g gp, gp^2}}

Eigenvalues[MatMass]

g)

A={Sin[θw],Cos[θw]};
Z={Cos[θw],-Sin[θw]};

MatMass.A/. gp->g Tan[θw]//Simplify

EigenZ=MatMass.Z/. gp->g Tan[θw]//Simplify

MZ2=1/2 g^2 v^2/Cos[θw]^2;

EigenZ/MZ2

Problem 4.1

a)

DPhi1={I g/Sqrt[2] Wp/Sqrt[2] v1, (-I g/2 W3 + I gp/2 B) v1};

DPhi2={I g/Sqrt[2] Wp/Sqrt[2] v2, (-I g/2 W3 + I gp/2 B) v2};

DPhi1Adj={-I g/Sqrt[2] Wm/Sqrt[2] v1, (I g/2 W3 - I gp/2 B) v1};

DPhi2Adj={-I g/Sqrt[2] Wm/Sqrt[2] v2, (I g/2 W3 - I gp/2 B) v2};

LMass= DPhi1Adj. DPhi1 +DPhi2Adj. DPhi2//Simplify

Problem 4.4

Clear[DPhi1,DPhi2,DPhi1Adj,DPhi2Adj];
SubAZ={W3->Sin[θw] A + Cos[θw] Z,B-> Cos[θw] A -Sin[θw] Z};

DPhi1={I g/Sqrt[2] Wp(v1+rho1+I eta1)/Sqrt[2] , (-I g/2 W3 + I gp/2 B) (v1+rho1+I eta1)/Sqrt[2] };
DPhi2={I g/Sqrt[2] Wp(v2+rho2+I eta2)/Sqrt[2] , (-I g/2 W3 + I gp/2 B) (v2+rho2+I eta2)/Sqrt[2] };
DPhi1Adj={-I g/Sqrt[2] Wm (v1+rho1+I eta1)/Sqrt[2] , (I g/2 W3 - I gp/2 B) (v1+rho1-I eta1)/Sqrt[2] };
DPhi2Adj={-I g/Sqrt[2] Wm (v2+rho2+I eta2)/Sqrt[2] , (I g/2 W3 - I gp/2 B) (v2+rho2-I eta2)/Sqrt[2] };

DPhi1 /. SubAZ /. gp-> g Tan[θw]//Simplify

Lag= DPhi1Adj. DPhi1 +DPhi2Adj. DPhi2//Simplify

Lag2=Lag /.SubAZ//Expand

b)

Coefficient[Lag2,Wp Wm rho1]

Coefficient[Lag2,Wp Wm rho2]

Coefficient[Lag2,Z^2 rho1]/. gp-> g Tan[θw]//Simplify

Coefficient[Lag2,A]/. gp-> g Tan[θw]

c)

vevs={v1,v2};

rhos={Cos[α] h - Sin[α] H,Sin[α] h + Cos[α] H};

vevrhos=vevs.rhos /. {v1-> v Cos[β],v2-> v Sin[β]}//Simplify