Relic Abundance and The Spin Independent Cross Section

3.7 Relic Abundance and The Spin Independent

Here, we approximate that the incoming momenta are zero, and using the conservation of energy, then we can simply perform the integral p_φ− to ob-tain:

h^σvi =n^EQ_{σL −}2Z 1 (2π)³

d³pσ_L

2EσL

1 (2π)³

d³pσ_L

2EσL

1 (2π)³

d³p_φ⁺ (2E_φ⁺)²

1 (2π)³ (2π)⁴¹

2δ(Eσ_L−^Eφ⁺)G_φσ² exp(−^EσL/T)exp(−^EσL/T). Then usingd³p_φ⁺ =4πp²

φ⁺dp_φ⁺ =4πp_φ⁺E_φ⁺dE_φ⁺ =4πq E²

φ⁺ −^m2_DM^Eφ⁺dE_φ⁺, we arrive at

h^σvi = n^EQ_{σL −}2Z 1 (2π)³

d³pσ_L

2EσL

1 (2π)³

d³pσ_L

2EσL

1 (2π)⁶

4πq E²

φ⁺ −^m2_DM^Eφ⁺dE_φ⁺ (2E_φ⁺)²

(2π)⁴¹

2δ(Eσ_L−^Eφ⁺)G_φσ² exp(−^EσL/T)exp(−^EσL/T)

= n^EQσ_L

₋2Z 1 (2π)³

d³pσ_L

2Eσ_L

1 (2π)³

d³pσ_L

2Eσ_L

1 (2π)⁶²

4π q

E²_σL−^m2_DM^EσL

(2Eσ_L)²

(2π)⁴¹

2G²_φσexp(−^2Eσ_L/T).

SubstitutingE_σL =^qp²_σL +m²_σL =^q0+m²_σL =m_σL, we obtain the thermally averaged cross section forσLσL →^φ+φ−:

h^σ(σLσL;φ⁺φ⁻)vi =n^EQσ_L

₋2Z d³pσ_L

(2π)³ d³pσ_L

(2π)³ 1 2Eσ_L

exp(−^2EσL/T) ^G

2 φσ

32πm_σ²

L

× r

1−^mDM mσ_L

= ^G

2 φσ

32πm_σ²

L

r

1−^mDM mσ_L

. (3.77)

First diagram in Fig.3.8 correspond to φ⁺φ⁻ → ^W+W−. From this dia-gram, we obtain:

(−^iGφh) ⁱ

p²_h−^m2_h(−^igνh) ^d

4p_h

(2π)⁴(2π)⁴e^µ_Weµ,Wδ⁴(p_φ⁺ +p_φ−−^ph)(2π)⁴δ⁴(ph−^pW⁺ −^pW⁻)

−→ −^iGφhgν_he_W^µ e_µ,W

(p_φ⁺ +p_φ−)²−^m2_h(2π)⁴δ⁴(p_φ⁺ +p_φ− −^pW⁺ −^pW⁻),

Mφ⁺φ⁻→W^± = h φ⁻

φ⁺ W⁺

W⁻ (a)

Mφ⁺φ⁻→Z⁰Z⁰ = h φ⁻

φ⁺ Z⁰

Z⁰ (b)

Mφ⁺φ⁻→tt = h φ⁻

φ⁺ t

t (c)

M^φ+φ−→hh = φ⁻

φ⁺ h

h

+ h

φ⁻

φ⁺ h

h

+ φ

φ⁺ h

h φ⁻

(d)

FIGURE 3.8: Annihilation of Dark Matterφinto the Standard Model particles

so that the amplitude is given by

M= ^Gφhgνhe

µ Weµ,W

(p_φ⁺ +p_φ−)²−^m2_h or

M = ^2GφhMWe

µ Weµ,W

(2m_DM)²−^m2_h ^{, (3.78)} where,g =2M_W² /ν_h and p_φ⁺ +p_φ− = 2mDM. Because we want to calculate thermal average cross section, we should have squared the amplitude above:

polarization

∑

|M|² = ^4G

2 φhM_W⁴ (2m_DM)²−^m2_h²

g^µν− ^p

µp^ν m²_W

!

g_µν− ^pµpν m²_W

!

= ^4G

2 φhM_W⁴ (2m_DM)²−^m2_h²

4− ^m

2 W

m²_W − ^m

2 W

m²_W + (p·^p)² m⁴_W

!

= ^4G

φh2 M_W⁴ (2m_DM)²−^m2_h²

2+ ¹ m⁴_W

s 2

2

−^2s 2

m²_W+m⁴_W !

= ^4G

φh2 M_W⁴ (2m_DM)²−^m2_h²

2+ ¹ m⁴_W

4m⁴_DM−^4m2DMm²_W+m⁴_W

!

= ^4G

φh2 M_W⁴ (2m_DM)²−^m2_h²

3−^4m

2DM

m_W² +4m⁴_DM m⁴_W

!

, (3.79)

where √

s is the center of mass energy. Now, for the thermal average cross section, we have the formula

h^σvi=n^EQ_{φ −}2Z 1 (2π)³

d³p_φ⁺ 2E_φ⁺

1 (2π)³

d³p_φ−

2E_φ−

1 (2π)³

d³p_W⁺ 2E_W⁺

1 (2π)³

d³p_W−

2E_W−

(2π)⁴δ⁴(p_φ⁺+p_φ− −^pW⁺ −^pW⁻)|M|^2exp−^Eφ⁺/T exp

−^Eφ⁻/T .

Substituting eq.(3.79) to the equation above, we obtain h^σvi=n^EQ_{φ −}2Z 1

(2π)³ d³p_φ⁺

2E_φ⁺ 1 (2π)³

d³p_φ−

2E_φ−

1 (2π)³

d³p_W⁺ 2E_W⁺

1 (2π)³

d³p_W−

2E_W−

(2π)⁴δ(E_φ⁺ +E_φ−−^EW⁺ −^EW⁻)δ³(−^pφ⁺−^pφ⁻ +p_W⁺ +p_W−) 4G_φh² M⁴_W

(2m_DM)²−^m2_h²

3−^4m

2 DM

m²_W +4m⁴_DM m⁴_W

! exp

−^Eφ⁺/T

exp

−^Eφ⁻/T .

Then settingE_φ⁺ = E_φ−, E_W⁺ = E_W−, and assuming that the incoming mo-menta are equal to zero, we find

h^σvi=n^EQ_{φ −}2Z d³p_φ⁺ (2π)³

d³p_φ−

(2π)^3exp

−^Eφ⁺/T exp

−^Eφ⁻/T 1

4E²_φ+

1 (2π)⁶²

d³p_W⁺ 2E_W⁺

d³p_W−

2E_W⁺

4G²_φhM_W⁴ (2m_DM)²−^m2_h²

3−^4m

2DM

m²_W +4m⁴_DM m⁴_W

!

(2π)⁴¹

2δ(E_φ⁺ −^EW⁺)δ³(p_W⁺ +p_W−)

=n^EQ_{φ −}2Z d³p_φ⁺ (2π)³

d³p_φ−

(2π)^3exp

−^Eφ⁺/T exp

−^Eφ⁻/T d³p_W⁺

32E²_φ+E²_W+

1 (2π)²

4G²_φhM_W⁴ (2mDM)²−^m2_h²

3−^4m

2DM

m²_W +4m⁴_DM m⁴_W

!

δ(E_φ⁺ −^EW⁺).

Noting thatd³p_W⁺ =4πp²_W+dp_W⁺ =4πp_W⁺E_W⁺dE_W⁺andp_W⁺ =^qE²_W+ +m²_W+ = E_W⁺

r

1+ ^m_E^2W+

W+, we arrive at

h^σ(φ⁺φ⁻;W⁺W⁻)vi =n^EQ_{φ −}2Z d³p_φ⁺ (2π)³

d³p_φ−

(2π)^{3 exp}

−^Eφ⁺/T exp

−^Eφ⁻/T

4π

r

1+ ^m_E^2W+

W+

E²_W+dE_W⁺ 32E²

φ⁺

E_W² +

1

(2π)^2π

4G_φh² M_W⁴ (2m_DM)²−^m2_h² 3−^4m

2 DM

m²_W +4m⁴_DM m⁴_W

!

δ(E_φ⁺ −^EW⁺)

=

n^EQ_{φ −}2

((((((((((((((((((((((( Z d³p_φ⁺

(2π)³ d³p_φ−

(2π)^{3 exp}

−^Eφ+ T

exp

−^Eφ− T

1 32πE²

φ⁺

4G²_φhM_W⁴ (2m_DM)²−^m2_h²

3−^4m

2DM

m²_W +4m⁴_DM m⁴_W

!

× s

1+ ^m

2 W⁺

E_φ⁺

= ¹

32πm²_DM

4G_φh² M_W⁴ (2mDM)²−^m2_h²

3−^4m

2 DM

m_W² +4m⁴_DM m⁴_W

!

× s

1+ ^m

2 W⁺

m_DM, (3.80)

whereE²

φ⁺ = p²

φ⁺ +m²

φ⁺ =0+m²

φ⁺.

In the caseh^σvi^forφ+φ− → ZZ, we have the same result except for a 1/2 (the statistical factor for the identical particles) andm_W is replaced withm_Z,

h^σ(φ⁺φ⁻;Z⁰Z⁰)vi= ¹ 32πm²_DM

2G²_φhM⁴_Z

4m²_DM−^m2_h^{2 3}−^4m

2DM

m²_Z +4m⁴_DM m⁴_Z

!

s

1+ ^m

2Z

mDM. (3.81)

For φ⁺φ⁻ → hh, we have to compute the last three diagrams in Fig.3.8.

In this case, the squared amplitude has a form

|M|²= ¹

2G_φh 1+24 λH

4m_DM−^m2_h m²_W

g² +8G_φh 1

−^2mDM+m²_h m²_W

g²

!2

, (3.82)

and from this result we obtain h^σ(φ⁺φ⁻;hh)vi = ^Gφh

64πm²_DM(1−^m2h/m²_DM)^1/2

× ¹+24 λ_H 4mDM−^m2_h

m²_W

g² +8G_φh 1

−^2mDM+m²_h m²_W

g²

!2

. (3.83) Forφ⁺φ⁻ →tt, with the same pattern of calculation, we obtain

h^σ(φ⁺φ⁻;tt)vi= ^Gφh

64πm²_DM(1−^m2h/m²_DM)^1/2

× ^{24 λH}

4mDM−^m2_h^m

2t(m²_DM−^m2t)

!

(3.84)

MσLσL→W^± = h σL

σL W⁺

W⁻ (a)

M^σLσL→Z⁰Z⁰ = h σL

σL Z⁰

Z⁰ (b)

M^σLσL→tt= h σL

σL t

t (c)

M^σLσL→hh= σL

σL h

h

+ h

σL

σL h

h

+ σL

σL h

σL h (d)

FIGURE3.9: The diagrams contributing the annihilation cross section ofσ_Linto the Standard Model particles

M^σL→W⁺W⁻ =

h

h σL

W⁺

W⁻ (a)

M^σL→Z⁰Z⁰ =

h

h σL

Z⁰

Z⁰ (b)

M^σL→tt=

h

h σL

t

t (c)

FIGURE3.10: The diagrams contributing the decay width ofσ_L

The thermally averaged annihilation cross section and decay width forσ can be calculated in a similar way. The corresponding diagram are shown in Fig.3.9. Summing up all the results we obtain:

h^σ(σLσL;φ⁺φ⁻)vi = ^G

2 φσ

32πm²_σ(1−^m2DM/m²_σ)^1/2, (3.85) h^σ(φLφL; SM)vi = ¹

32πm²_DM

∑

I=W,Z,t,h

(1−^m2I/m²_DM)^1/2

×^aI(G_φh,mDM), (3.86)

h^σ(σLσL; SM)vi = ¹

32πm²_L

∑

I=W,Z,t,h

(1−^m2I/m²_L)^1/2

×^aI(G_σh,mL), (3.87)

h^γ(σL)i = ¹

16πmL

∑

I=W,Z,t

(1−^4m2I/m²_L)^1/2

×^aI(G^¯σh,mL/2) + ^G¯σh 32πmL

(1−^4m2h/m²_L)^1/2

×

1+24λH∆h(mL/2)^mL/2 g²

, (3.88)

where

a_W(Z)(G,m) =4(2)G²∆²_h(m)m_W(Z) 3+4 m⁴

m⁴_W(Z) −^{4 m2} m²_W(Z)

! , at(G,m) =24G²∆²_h(m)m²_t(m²−^m2t),

a_h(G,m) =¹

2G² 1+24λH∆h(m)^m

2 W

g² +8G∆^t_h(m)^m

2 W

g²

!2

. (3.89)

Hereg=0.65 is theSU(2)L gauge coupling constant, and

∆h(m) = (4m²−^m2h)⁻¹(∆h(m)^t = (−^2m2+m²_h)⁻¹) (3.90) is the Higgs propagator in thes(t)−^channel.

Y_ϕ

Y_ϕ

Y_σ Y_σ

0 10 20 30 40 50μm/T

10^-23 10^-19 10^-15 10^-11 Y_i

FIGURE3.11:Y_φ, ¯Y_φ,Y_σL, and ¯Y_σLas a function ofx=µ_m/T

The plot of the coupled Boltzmann equation is displayed in Fig.3.11, where we use:

m_DM =500 GeV, mσ_L =550 GeV, h^σ(σ_Lσ_L;φ⁺φ⁻)vi =5.2×^{10−6GeV−2} h^σ(σLσL; SM)vi =10⁻¹¹GeV⁻², h^σ(φ⁺φ⁻; SM)vi =10⁻¹¹GeV⁻²

h^γ(σL)i =10⁻⁹GeV.

In this graph, we see thatYφ (orYσ_L) and ¯Yφ (or ¯Yσ_L) have the same value at the beginning, but thenYφ (orYσ_L) separate from the equilibrium values asx increases. The Dark Matter particle number over entropy becomes constant (Freeze out), while σ−^L decreases with the speed slower than the equilib-rium value.

Relic abundances forσLandφ^±can be obtained fromY_σL;∞ ≡^YσL(x=∞) and Y_φ;∞ ≡ ^Yφ(x = ∞) in the coupled Boltzmann equations (eq.(3.75) and

eq.(3.76)):

Ωσ_L,φh² = ^{gσL,φmDMYσL,φ;∞S0}

ρc/h² , (3.91)

where gσL,φ is the degree of freedom of σL,φ, s0 = 2890 cm⁻³ is the en-tropy density at present, andρc/h²=1.05×^{10−5GeV/cm3}is the critical en-ergy density over the dimensionless Hubble constant at present [18]. When h^γ(σL)i is sufficiently large, Yσ_L may be approximated by its equilibrium value ¯Yσ_L, then the {} in the right-hand side of eq.(3.76) can be rewritten as

"

1

2h^σ(φ⁺φ⁻; SM)vi+¹

4h^σ(σLσL;φ⁺φ⁻)vi ^m

3 σ_L

m³_DM exp 2xm²_DM−^m2_L mDMmL

!#

× ^YφYφ−^Y¯φY¯φ

. (3.92)

From this equation, ifmL ^mDM, then the second term is neglected or sup-pressed. SomL must be nearly equal tomDM to make it effectively increases the annihilation rate of the Dark matter. In the Dark matter case, relic abun-dance must be at certain value, which is ΩDM ' 0.12, thus if the second term in the above equation suppressed, we must setλHSi big, to fulfill this value. However, if the second term is efficient, we can loosen the value of λ_HSi which is useful when we consider the spin-independent elastic cross section.

In order to compare this model result with the WIMP DM direct-detection search experiment [20–22], evaluation of the spin-independent elastic cross section of the nucleonσSI is done in [19]. Here, the spin-independent elastic cross section of the nucleon calculation is given by [23]:

σSI = ¹

4π rˆG_φhm²_N m_DMm²_h

!2

mDM

mN+m_DM 2

, (3.93)

where mN ' 940 MeV is the nucleon mass and ˆr ∼ 0.3 originated from the nucleonic matrix element [24–26]. Here, the parameter space where v_h = 246 GeV, m_h = 125 GeV, and ΩDMh² ' 0.12 are satisfied. From all of this data, eq.(3.91) can be written as a constraint with form:

mDMY_φ;∞ =4.35986×^{10−10GeV (3.94)}

FIGURE3.12: The Spin-Independent cross section in the case of U(2)(red),U(1)×^U0(1)×^Z(2)(blue), andU(1)×^U0(1)(pink)

[19]

and eq.(3.93), which is the main equation to produce Fig.(3.12), can be written as:

σ_SI(m_DM) =7.6345×¹⁰⁻¹¹

G_φh 0.94+m_DM

2

. (3.95)

There are several constraints to obtainλs values other than those mentioned above. First is from the assumption thatλs must be positive. The second is from eq.(3.19):

1

2 +lnh^M2_i0i Λ²_H >0

h^M2i0i>Λ²_He⁻¹². (3.96) The third is the constraint data where it used the values of m_h = 125 GeV and v_h = 246 GeV, where these values are bounding of the value of λ1, λ2, λ12, λ⁰₁₂, λHS1, λHS2, λH, and ΛH. The fourth is the value of h^fii ^and h^Mi0i are not allowed to be negative. In this calculation, the value ofλH is set around 0.145−^{0.153, λ}HSi is around 0.05−^{0.2, and λ}12,λ⁰₁₂ are around 0.01−4. The result of this model in U(1)×^U0(1) symmetry is given in Fig.3.12 with the pink points. We can see from eq.(3.95) thatσ_SI depend on theG_φhwhich is proportional toλ_HSi, from this relation we have to setλ_HSi not too big to make σSI overcome the constraint from the direct detection experiment. However, we know that relic abundance of the Dark matter de-pend onh^σ(φ⁺φ⁻; SM)iwhich also proportional toλHSi. To fulfill the value that given in eq.(3.94),λHSimust be big. However, thanks to the second term in the eq.(3.92), as long as this second term is efficient, the mass ofσL is nearly proportional to Dark matter mass, we can loosen the value ofh^σ(φ⁺φ⁻; SM)i which mean loosen the value ofλHSi.

In this figure, the green and yellow bands denote the 1σ and 2σ bands of XENON1T [27], respectively. The black dashed, solid, and dotted line inside of the green bands stand for the current upper bound from the direct detection experiments of LUX [28], XENON1T [27], and PandaX-II [29]. Here, theU(2) flavor symmetry which shown by the red dot, is almost excluded because it does not have much points below the constraint that given by the experiments. Enlarging the model with the permutation symmetry,U(1)× U⁰(1)×^Z2, which requires λ1 = λ2 and λHS₁ = λHS₂ inLeff from eq.(3.3) also shown. This model enlargement also has a good result likeU(1)×^U0(1) symmetry because of it still far from the given constraint. Single-scalar model of dark matter is also included for comparison that shown in the brown point.

Chapter 4

Concluding Remarks

We have considered a scale-invariant extension of the Standard model with introducing the hidden sector described by anSU(2) gauge theory with the scalar fieldsS^a_i (a =1· · ·^Nc,i =1· · ·^Nf) in the fundamental representation ofSU(Nc) and U(1)^Nf. Here we restrict ourself to the minimal model, i.e., N_f =2 and set Nc =6. We assume that the origin of the electroweak scale is a scalar-bilinear condensationh^Si^†S_ji which forms due to theSU(Nc) gauge interaction. From this scaleless Lagrangian, we have shown that mass can be generated from massless Lagrangian, where scale is generated by quantum effect within scaleless effective theory (eq.(3.3)) of the hidden sector. This scale characterizes the origin of the scales of both the hidden sector and the electroweak sector.

By using mean field approximation and auxiliary method, we investigate the vacuum structure of the effective LagrangianLeff. From this discussion, by setting some parameters we obtain the term of the Higgs mass squared m²_h0with the value 135GeV. After considering one loop contribution from the standard model sector,δm²_h, we obtain the Higgs mass 125.8 GeV in eq.(3.36).

As a byproduct of this model, three new particlesφ,σL, andσHintroduced with assumptionmσH > mσL > mφ where the lightest particle (the complex scalar particle) considered as a dark matter candidate. Mass of the Dark mat-ter particle has been calculated by calculating its inverse propagator and set it equal to zero, this inverse propagator also used to calculate the wave func-tion renormalizafunc-tion. Relic abundance and the interacfunc-tion of the dark matter particles with the Standard model particles were evaluated where the effec-tive interaction of dark matter and the other particles obtained by setting the external momenta equal to zero.

To evaluate relic abundance, we encounter the coupled Boltzmann equa-tion that describes the evoluequa-tion of the number density ofσL andφparticles over entropy density (eq.(3.76) and eq.(3.75)). By solving this coupled Boltz-mann equation, we obtain Fig.3.11 which tells us about the evolution of the number of particles over entropy density (Y_i) over x = µm/T. From this graph, we see that in the smallx,YσL(Yφ) coincides with its equilibrium state Y¯σL( ¯Yφ), and after some value, they parted away with its equilibrium state.

Yφ becomes constant after some value of x, on the other hand YσL still goes down but with a slower speed compared with its equilibrium state. After some value, we can make an approach and assumeY_σL ≈ ^Y¯σL and from this we can obtain the approximation expression that given by eq.(3.92). In this equation, the second term will have an effective role if the mass ofσ_L is big-ger but nearly equal to dark matter massm_DM, because its much larger than dark matter mass it will be suppressed and do not have a contribution. Higgs portal coupling, λHSi, which introduced in this model, not only useful as a bridge between the standard model sector and the hidden sector but also important in the dark matter case. This λHSi has a double role in the relic abundance calculation in eq.(3.91) which is loosened its value by the exis-tence of second term in eq.(3.92) as long as this term is effective, and in the calculation of the spin independent in eq.(3.93).

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