Electroweak Interactions

Imposing a local SU(2)L×U(1)Y symmetry leads to a theory of fermions in SU(2)L dou-blets of left-handed particles, QL,i=1,2,3 for the quarks and LL,i=1,2,3 for the leptons, and in SU(2)L singlets of right-handed particles, u_R,i=1,2,3, d_R,i=1,2,3 and e_R,i=1,2,3 for up-type quarks, down-type quarks and charged leptons, respectively⁵. The SU(2)L doublets have a U(1)Y hypercharge Y = −1, the SU(2)L singlets have a U(1)Y hypercharge Y = −2. A spontaneous breaking of SU(2)L ×U(1)Y → U(1)EM is considered, with QEM = I + ¹₂Y, where I is the weak isospin, the charge of SU(2)L. This requires to extend the theory by a scalar SU(2) doublet, φ, with hypercharge Y = 1. The four generators, three for SU(2) and one for U(1) result in four gauge bosons, W_a,i=1,2,3^µ and B^µ, all without hypercharges.

Two coupling constants determine the strengths of the interactions, g for SU(2)L and g⁰ for U(1)Y.

The kinetic part of the Lagrangian for fermions and vector bosons takes the following form:

Lkin = −1

4W^iµνW_µνⁱ − 1

4B^µνBµν (1.5)

+iQγ¯ ^µDµQ+iuγ¯ ^µDµu+idγ¯ ^µDµd+iLγ¯ ^µDµL+i¯eγ^µDµe,

with

Dµ=∂µ−igW_µ^aT^a− i

2g⁰Bµ. (1.6)

The generators of SU(2) are given byT^a= ¹₂σ^a, where σ^a denote the Pauli matrices.

4Furthermore, techniques such as QCD sum rules relate hadronic parameters like masses, couplings or magnetic moments, to characteristics of the QCD vacuum, i.e. quark and gluon condensates. Also the so-called quark-hadron duality allows to describe observed reactions either as interactions between partons or of hadronic resonances.

5Note that right-handed neutrinos are not included here.

Chapter 1. The Standard Model 25

Spontaneous Symmetry Breaking

The model, as described up to now, does not allow the gauge bosons to have mass. However, W andZ bosons, mediating the weak interactions, are measured to be massive. In order to account for that, a scalarSU(2)doublet is introduced. The part of the Lagrangian governing the scalar kinematics and its potential reads:

L^φ= (Dµφ)^†(D^µφ)−µ²φ^†φ−λ(φ^†φ)², (1.7) where λ is dimensionless, real and needs to be positive to make the potential be bounded from below, andµ² has mass dimension two and is assumed to be negative. In this case, the potential has a local maximum at the origin and degenerate minima on a circle around it, satisfying φ^†φ = −µ²/(2λ) and v² = −µ²/λ can be defined. By an SU(2) gauge transfor-mation, a particular vacuum expectation value (vev) of φ can be chosen: φ0 = ^√1₂(⁰_v). This presents the spontaneous symmetry breaking (SSB): one specific solution that minimises the potential is chosen. When considering small excitations around the vacuum state, φ reads:

φSSB(x) = 1

√2

v+h(x)0

. (1.8)

The scalar field h(x) is the only one remaining from the original four fields of φ after the SSB. It is identified with the Higgs boson. Substituting φ with its vev, the kinetic part of the scalar Lagrangian written in (1.7) can be expressed as:

(Dµφ)^†(D^µφ)⊃ |(gW_µ^aT^a+g⁰BµY)·(_v⁰)|² (1.9)

= v²

4 g²(W_µ¹W^1µ+W_µ²W^2µ) + (gW_µ³−g⁰Bµ)(gW^3µ−g⁰B^µ)

. (1.10)

The first part can be written as m²_WW⁺W⁻ with W_µ^± = ^√1₂(W_µ¹∓iW_µ²) and hence leads to the mass terms for the charged W bosons:

m²_W =v²g²/4. (1.11)

The second term of Eq. 1.10 leads to the mass of the neutral Z boson withZµ =W_µ³cosθW− BµsinθW, whereθW is the so-called Weinberg angle with tanθ =g⁰/g:

m²_Z =v²(g² +g⁰²)/4. (1.12)

The neutral counterpart of the Z boson, the photon (A), remains massless:

Aµ=W_µ³sinθ+BµcosθW. (1.13)

26 Chapter 1. The Standard Model The mass eigenstates Z and A are formed by W³ and B as:

Z γ

!

= cosθW −sinθW

sinθW cosθW

! W³ B

!

(1.14)

A real, massive scalar degree of freedom, the Higgs boson, whose mass is given bym²_h = 2λv results from the SSB. It has been measured to bemH = 125.09±0.24 GeV [7].

Fermion masses

Without SSB, it is not possible to write down mass terms for the fermions. They arechiral: left-handed and right-handed particles behave differently, since only left-handed particles are charged under SU(2)L. This excludes a Dirac mass term in which left- and right-handed particles ought to be combined and so would breakSU(2)L. Furthermore, a Majorana mass term is excluded, since the fermions are charged and hence cannot be their own antiparticles.

However, a Yukawa interaction between fermions and the complex scalar field φ can be written down in the following way:

L^{Y uk} =Y_ij^dQ¯LiφdRj +Y_ij^uQ¯Liφu˜ Rj+Y_ij^lL¯LiφeRj +h.c. . (1.15) The matrices Yij contain the different Yukawa coupling strengths. Note that the up-type quarks couple to φ˜ = −iσ2φ∗. After spontaneous symmetry breaking, i.e. the transition φ→ (⁰_v) (and φ˜→ (^v₀)) the above Yukawa terms have the form of a Dirac mass term with the fermion mass given bymf =vyij/√

2, wherev denotes the Higgs vev andyij the relevant Yukawa coupling. Since the Standard Model does not contain right-handed neutrinos, mass terms for neutrinos are not possible and hence they remain massless and degenerate.

For leptons, the interaction basis can always be made consistent with the mass basis, such that Y_ij^l is diagonal. This is not the case for the quarks: generally, no interaction basis can be found that is also a mass basis for both up- and down-type quarks, i.e. that diagonalises both Y_ij^d and Y_ij^u at the same time. Hence, the mass eigenstates of the quarks generally do not coincide with the flavour eigenstates which take part in the electroweak interaction.

This leads to a mixing of flavour states to form the mass eigenstates that is described by a unitary matrix, called CKM matrix after Cabbibo, Kobayashi and Maskawa. Its complex phase gives rise to CP-violating processes within the Standard Model.

Chapter 1. The Standard Model 27

Electromagnetic interactions

Electromagnetic interactions are vectorial, parity conserving, diagonal and universal:

LEM =eqfψ¯fiγµA^µψfi, (1.16) where e = gsinθW is the electromagnetic coupling, qf is the electromagnetic charge of the (left- or right-handed) fermionψf and A^µ denotes the photon.

Neutral Currents

Neutral weak interactions, mediated by theZ boson, are chiral, i.e. they distinguish between left- and right-handed particles and hence violate parity. They are diagonal and universal since fermions are in the same representation as for U(1)EM.

L^{N C} = g cos(θW)

I³1−γ⁵

2 −qfsin²(θW)

ψ¯fiγµZ^µψfi. (1.17)

As above, qf denotes the electromagnetic charge and ψfi stands for any fermion. I³ is the third component of the weak isospin. The projection operator (1−γ⁵)/2 selects only the left-handed components of ψfi. This means, the neutral current consists of a purely left-handed part, proportional to I³ and a part treating left- and right-handed particles equally, proportional toqf.

Charged Currents

Charged weak interactions are mediated by the W bosons. Since they arise purely from SU(2)L, charged currents involve only left-handed fermions and are hence maximally parity-violating. As long as neutrinos are treated massless and degenerate, the lepton interactions are diagonal and universal, while the above-mentioned CKM matrix describes the mixing in the case of quarks, where the charged currents are neither universal nor diagonal.

L^CC =− g

√2

ψ¯L,u_iγ^µMCKMψL,d_i + ¯ψL,ν_iγ^µψL,l_i

W⁺+h.c.. (1.18) Here, ψL,d_i and ψL,u_i stand for down- and up-type quarks, respectively, whileψL,d_i and ψν,d_i

indicate charged leptons and neutrinos.

28 Chapter 1. The Standard Model

Vector Boson Interactions

In addition, the Lagrangian contains the following terms, covering three- and four-point interactions between the vector gauge bosons:

L^{V V V} =−ig[(W_µν⁺W^−µ−W^+µW_µν⁻)(A^νsinθW −Z^νcosθW)

+W_ν⁻W_µ⁺(A^µνsinθW −Z^µνcosθW)] (1.19)

LV V V V =−g² 4

(2W_µ⁺W^−µ+ (AµsinθW −ZµcosθW)²)²

−(W_µ⁺W_ν⁻+W_ν⁺W_µ⁻+ (AµsinθW −ZµcosθW)(AνsinθW −ZνcosθW))²

(1.20)

Higgs Interactions

The Yukawa couplings between the Higgs and the fermions, as introduced above, are pro-portional to the particle masses, heavier particles couple stronger to the Higgs. The Yukawa couplings are diagonal.

Recall that after SSB the Higgs field φ takes the following form: φSSB(x) = ^√1_{2 v+h(x)}⁰ . Dimensionless couplings such ashhhh and hhV V involving only h(x)but not v, arise in the Lagrangian. Trilinear couplings likehhhand hV V vertices are proportional tov. The cubic and quartic Higgs self-interactions are given by:

L^H ⊃+λv

2 H³+λ

4H⁴. (1.21)

The interactions between the Higgs and the vector bosons reads:

L^HV ⊃

g²v

2H+g²

4H² W_µ⁺W^−µ+ 1 2 cos²θW

ZµZ^µ

. (1.22)

Couplings between the Higgs and the photon are not allowed since the Higgs is not charged electromagnetically and the photon is massless and does not have a Yukawa interaction with the Higgs.