Masses of Fermion and Coupling to Higgs

In section 2.1.2 we discuss the mass term of fermion in the form of −m ¯ψLψR and m ¯ψRψLare not gauge invariant. With introduction of the Higgs boson, we can construct combinations of fermion and Higgs field, so that the Lagrangian is invariant under gauge transformation.

We start with Lepton, since the right-handed neutrino is absences in SM, the only combination will be:

L^e_Yukawa = −^hy^e¯eRϕ^†L_L+ y^e∗¯L_Lϕe_Rⁱ (2.43) The second term are just h.c. (Hermition conjugate) of the first term, which makes the Lagrangian Hermition. Since fermion field is dimension 3/2 and Higgs field is dimension 1, then y^e must be a dimensionless constant. In general, y^e could be complex constant, but we can just absorbe the phase by rephrasing the fermion field. In the unitarity gaue, we can rewrite the Lagrangian:

L^e_Yukawa = −y^e

√2[(v + h)(¯eRe_L+ ¯eLe_R)] = −y^ev

√2¯ee − y^e

√2h¯ee (2.44) The first term is mass of the electron:

m_e= y^ev

√2 (2.45)

and the second term is interaction of electron with Higgs, which the coupling is Yukawa coupling of e:

h¯ee : λe = y^e

√2 = m_e

v (2.46)

We can see the coupling is proportional to the mass of electron. Similar deduction can applies to mu and tau lepton. The Yukawa couplings are free parameter, so the masses of fermion is not predicted in the SM.

We only discussed one generation of leptons, but we know in the SM we have three generations of leptons, the SU(2)L doublet/singlet are transforming in the same way for different generation, so in general we can write down the Lagrangian:

L^l_Yukawa = −

2.2. THE SM HIGGS MECHANISM 15 Now the couplings y_ij^e are complex numbers that form a 3 × 3 matrix, contains totally 9 complex couplings. Inserting the Higgs field in unitarity gauge:

L^l_Yukawa = − v√+ h

In the above representation we write the leptons in generation space, and we want to find the leptons in mass eigenstate, which means we will diagonalize the mass matrix.

We know a matrix can be diagonalize to a real diagonal matrix through the biunitary transformation:

V_R^l†Y^lV_L^l = M^l

v , with M^l_ij = m^l_iδ_ij = v

√2y_i^lδ_ij (i, j = e, µ, τ) (2.49)

V_L^l and V_R^l are two appropriate 3 × 3 unitary matrices^††. On the other hand, the lepton array in mass eigenstate have become:

ℓ^′_L= V_L^l†ℓ_L =

Now we can rewrite the Lagrangian in eq.(2.48) in mass eigenstate:

L^l_Yukawa =(v + h)¯ℓ^′L of the charged lepton can only be obtained through experiment.

Moving on to the quark sector. As we can see in the lepton case, the Higgs doublet is only interacting with the lower component in the fermion doublet, so we introduce the conjugate doublet which transforms in the same way as the doublet under SU(2)

††recall that for a unitary matrix U⁻¹ = U^†

16 CHAPTER 2. BRIEF REVIEW OF THE STANDARD MODEL

With this doublet, we can write down another Lagrangian term:

L^q_Yukawa = − Now the couplings y^uij and yij^d are complex numbers that form two 3 × 3 matrices, containing totally 18 complex couplings. Let’s discuss the quark masses by inserting the ϕfield with (0, v/√

2)^T and rewrite the Lagrangian in the form of matrix in generation basis:

then we define the quark mass matrices as:

Y_ij^u = v

√2y_ij^u, Y_ij^d= v

√2y_ij^d (2.55)

Follow the procedure we had discussed in lepton sector, we diagnolize the mass matrices:

U_R^†Y^uU_L= M^u

Where UL,R and DL,R are proper unitarity matrices to diagonal mass matrices

u^′_L,R= U^l†L,Ru_L,R=

When we expand the kinetic Lagrangian of fermion, we will have a term relate to charged current:

J_L^+µ ∝ ¯uLγ^µd_L = ¯u^′LU_Lγ^µD_L^†d^′_L= ¯u^′Lγ^µV^†d^′_L (2.58)

2.2. THE SM HIGGS MECHANISM 17 Where UL^†D_L≡ V is the Cabibbo–Kobayashi–Maskawa(CKM) matrix[10][11], which is unitary matrix. In general, CKM is not identity matrix, hence there are mixing between different generation of up-type quark and down-type quark, then we have generation-changing charge current. For the right-handed quark, since they are SU(2)L

singlet, means the up-type and down-type right-handed quark are not tie together by charge current, thus the UR and DR have no physical consequence in the SM. On the other hand, the fermion bilinears enter the neutral current interactions (γ and Z mediator) can be writing down as:

J_L^0µ ∝ ¯uLγ^µu_L = ¯u^′_LU_Lγ^µU_L^†u^′_L= ¯u^′_Lγ^µu^′_L (2.59) The neutral current are automatically diagonal in generation basis, so the photon and Z boson couplings are the same in each generation, same argument can apply to down-type quark. As we see the SM do not have flavor changing neutral currents(FCNC) at tree level in SM, and the FCNC induced in 1-loop by W boson are far too small to detect in current technology.[12]

Chapter 3

Brife Review of Dark Matter

Many observations indicate the existence of DM. Historically the first evidence was observed by Fritz Zwicky in 1937[13], he measured mass of the Coma cluster, and found it’s not compatible with the galaxies velocity within the cluster, the mass of visible matter^∗ cannot sustain the orbits of galaxies. However, his work was overlooked for many years. Next breakthrough was provided by Vera Rubin et al.[14], they measured the rotation curve of 21 Sc galaxies, and found the rotation velocity grow and approximate to a constant as the radius grow. In contrast, the velocity at radius r related the visible matter is described by the Newtonian gravity v = (GM/r)^1/2 was propotional to r^−1/2, which indicates it must have more mass than we can observe, and those matters do not emit or absorb light that we can measure. They only interact through gravity. Later, we have more experiment providing the evidence of exsistence of DM from scale as small as dwarf galaxies to as large as Cosmic Microwave Background (CMB), the detail will not within the scope of this thesis.

There are few basic properties that we know about DM:

• From "not" observe DM interact with light, hence in most of models, the DM are electric neutral.

• DM is "almost" collisionless

• DM obey the gravitational interaction like atomic matter.

• DM must be stable, otherwise it will decay in the early universe.

There are many different types of DM candidate with different mass and different strength of interactions as shown in Fig.3.1, but we will only focus on the Weakly

∗Although at that time he did not know there has a halo of hot gas in the Coma cluster.

20 CHAPTER 3. BRIFE REVIEW OF DARK MATTER

Fig. 3.1 The mass range in this figure are only approximate. Figure is taken from [1].

Interacting Massive Particle (WIMP) scenario, which the DM mass are in GeV to TeV range with typical cross sections at weak scale.

3.1 Weakly Interacting Massive Particle and Relic Abundance

Consider the DM χ interacts with SM particle X through a process:

¯χχ ↔ ¯XX (3.1)

In the early universe, when the temperature was high enough, particles were in thermal equilibrium, which means the process of creation and annihilation of particles were equally efficient. The temperature of universe is decreasing as it is exanding. When the temperature was lower than mχ, the creation of particles were supressed. However, the annihilation continued, so the number of density was decreased as the temperature dropped. At some point, the expansion rate of the universe are high enough so that annihilation processes were ineffective, thus the relic abundance of DM are almost remain the constant till today. The period are often addressed as DM freeze-out or decoupling.

Follow the Kolb and Turner’s method[15] in deduction of relic density (RD). The evolution of number density n of DM can be described by Boltzmann equation:

dn

dt = −3Hn − ⟨σAv⟩(n²− n²_EQ) (3.2) We assume the DM can only create and annihilate like in eq.3.1, and nχ = nχ¯ = n.

Here, H = ˙a/a is Hubble parameter that describe the expansion rate of the universe, a is scale factor of the universe and ⟨σAv⟩ is the thermally averge DM annihilation

3.1. WEAKLY INTERACTING MASSIVE PARTICLE AND RELIC ABUNDANCE21 cross section times the relative DM velocity. The Big Bang Nucleosynthesis (BBN) happened when the temperature T ≃ O(1) MeV, the heavy particle like DM were created before BBN when it is radiation dominated period[16].

There are three terms on the right-hand side of eq.3.2, each stand for the expansion of the universe, annihilation of DM and creantion of DM. The Boltzmann equation can further represent in another form:

x equilibrium annihilation rate. Noted the conservation of entropy per comoving volume S = sa³ = constant, thus the number of particles per comoving volume Y ∝ na³ become constant when dY/dx = 0. Such case will lead to ΓA≪H. The numerical solution of eq.3.3were showed in Fig.3.2. As the cross section increase, DM can stay in equilibrium longer, which will decoupling in later time with smaller RD.

As we discussed before, the universe is at high temperature in the early stage, thus the interaction rate between particles is faster than the expansion of the universe. As the temperature decreases ΓA is decreasing faster than H, and when the annihilation rate drops below the expansion rate of universe, DM will decouple from thermal bath:

ΓA(Tf o) = ⟨σAv⟩_{T =T}

f on_EQ(Tf o)≃H(Tf o) (3.4) We can get freeze-out temperature Tf o by solving numerically the Boltzmann equation:

x_{f o}= m_χ and g∗ is the number of external degrees of freedom at freeze-out(in the SM, g∗ ≈120 for T ≈1 TeV and g∗ ≈65 for T ≈1 GeV).

If the mχ is at GeV to TeV scale, and cross section at weak-scale, freeze-out happened at xf o≈20 − 30, resulting DM density today as:

Ωχh² ≈0.1^x_{f o}

22 CHAPTER 3. BRIFE REVIEW OF DARK MATTER

Fig. 3.2 Evolution of DM number density per comoving volume through the thermal freeze-out, where the solid line show the equilibrium value of comoving number density in different time, and dashed line show the actual relic density for the different value of σAv. Figure is taken from [2].

Thus we can see that for cross section on the order of 10⁻²⁶cm³/s, will give Ωh² ≈0.1.

The current measurment by PLANCK gave Ωh² = 0.120 ± 0.001[17]. The WIMP scenario can have RD in the correct order, which sometimes so-called "WIMP miracle".