The Electro-weak theory

In 1961, S. L. Glashow [3] proved that the weak and electromagnetic interactions are not separated, but are two aspects of the same force: the electro-weak inter-action. The theory of the electro-weak interactions is a Gauge theory based on a symmetry groupSU(2)L⊗ SU(1)^Y. The weak hypercharge Y , the third compo-nent of the weak isospinI and the electric charge Q are related by the Gell Mann - Nishima relation:

Q = I3+Y

2. (1.2)

By requiring that the Lagrangian of the electro-weak interaction is invariant under the Gauge transformation SUL(2) ⊗ SU^Y(1) and substituting the expression of the standard derivative with the covariant derivative:

Dµ= ∂µ+ ig1Y Bµ+ ig2

τi

2W_µⁱ, (1.3)

(where ~τ are the Pauli matrices and g1, g2 are the coupling constant of the in-teraction ), four vector bosons are introduced: W_µⁱ withi = 1, 2, 3 and Bµ. The Standard Model Lagrangian can be written as sum of four independent terms:

L = LF + LG+ LH+ LY, (1.4)

whereLF andLGdescribe respectively the kinetic term and the gauge interaction of fermions and bosons, whereas LH and LY describe the mass generation of bosons and fermions by the introduction of Higgs scalar boson, in addition to the kinetic term and interaction of the Higgs particles.

The termL_F = i ¯ψD_µψ is related to massless fermionic particles fields and to the interactions with gauge fields; the term

LG = −1

4W_µνⁱ W_µνⁱ −1

4BµνBµν, (1.5)

with

W_µνⁱ = ∂νW_µⁱ− ∂^µW_νⁱ− g²ǫ^ijkW_µⁱW_ν^k (1.6)

and

Bµν = ∂νBµ− ∂^µBν (1.7)

contains the kinetic term of gauge fieldsW and B and the self-interaction of fields^→

−→

W due to the fact that the group SU(2)_weakis non abelian.

As explained in the following section, the mass eigenstates of the fieldW are:^→ W_µ^± = 1

2(W_µ¹∓ Wµ²), (1.8)

whereas a combination of neutral bosons describes the photon Aµ and the Zµ

boson:

Aµ= Bµcos θW + W_µ³sin θW (1.9a) Zµ= −B^µsin θW + W_µ³cos θW. (1.9b) Theθ_W parameter is the weak mixing angle and experimentally we have that:

sin θW ≈ 0.231,

furthermore, the coupling constantsg1andg2are related withθW by the formula:

g1sin θW = g2cos θW = e. (1.10) Neglecting the self-interactions terms, the gauge term can be written as

LG = −1

4FµνF^µν− 1

2FW µνF_W^µν− 1

4ZµνZ^µν, (1.11) where Fµν is the electromagnetic field tensor, FW µν is the weak charged field tensor andZ_µνis the weak neutral field tensor given by expression similar to (1.6) and (1.7).

The Lagrangian described so far does not contain mass terms and, conse-quently, bosons and fermions are massless. This is because the presence of direct mass terms would destroy the invariance of the theory under the transformation SUL(2) ⊗ SU^Y(1). To generate the bosons and fermions mass “inside” the theory and to be, therefore, consistent with experimental evidence, it is necessary to in-troduce a new scalar field and apply the Higgs mechanism [4], to generate boson masses, and the Yukawa potential, to generate fermion masses.

1.2.1 The spontaneous symmetry breaking mechanism

To generate the particles mass without destroying the invariance under the gauge transformation, it is possible to use a spontaneous (i.e. “implicit”¹) breaking of the symmetry

1in this case, spontaneous symmetry breaking means that the Lagrangian is symmetric under a certain transformation, but the solutions of equation of motion are not.

This is done by introducing a complex scalar field that self-interact with a phenomenological potential:

V (φ) = µ²φ^†φ + λ(φ^†φ)², (1.12) where:

φ = φ1+ iφ2 (1.13)

and the parameters are chosen in such a way that the origin is a local maximum:

µ² < 0, λ > 0. (1.14)

For simplicity, we consider first the breaking of the Abelian gauge groupSU(1).

In order to have the Lagrangian invariant for a phase transformation like:

φ → e^iα(x)φ, (1.15)

it is necessary to replace the standard derivative with covariant derivative:

Dµ= ∂µ− ieA^µ, (1.16)

introducing the gauge fieldAµthat transforms according to:

Aµ → A^µ+ 1

e ∂µα. (1.17)

Then, the gauge-invariant Lagrangian is given by:

L = (∂^µ− ieA^µ)φ^∗(∂µ− ieA^µ)φ − µ²φ^∗φ − λ²(φ^∗φ)²− 1

4FµνF^µν. (1.18) The potentialV (φ) has a minimum in the points of space (φ₁,φ₂) belonging to a circle with radiusv given by:

v² = φ²₁+ φ²₂ with v² = −µ²

λ . (1.19)

Around a minimum energy point (φ1 = v, φ2 = 0), we can write φ in terms of two real fields (η, ξ) defined by:

φ(x) = 1

√2[v + η(x) + iξ(x)] . (1.20) By substituting (1.20) into (1.18), the last equation becomes:

L = 1

2(∂µξ)²+ 1

2(∂µη)²− v²λη²+1

2e²v²AµAµ

− evA^µ∂^µξ −1

4FµνF^µν + interaction terms. (1.21)

The equation (1.21) describes the dynamics of a massless boson ξ, a massive scalar boson η and a massive vector boson Aµ. The Lagrangian (1.21) has one degree of freedom more than the Lagrangian (1.18). Because a change of coordi-nates cannot change the number of degrees of freedom, we deduce that equation (1.21) contains an unphysical field not representing a real particle. It is possible to choice a specific gauge transformation by which the unphysical field disappears from the Lagrangian. Indeed, by writing:

φ = 1

In this particular case, θ(x) is chosen such that h is real. Therefore we have the Lagrangian: in which we get two massive particles, the vectorial boson A_µ and the scalar h (the Higgs boson) and no off-diagonal terms, like the termevAµ∂^µξ of (1.21).

For the case of the breaking ofSU(2) group symmetry, we start by considering a Lagrangian defined as:

L = (∂^µφ)^†(∂^µφ) − µ²φ^†φ − λ² φ^†φ
2

, (1.25)

whereφ is a complex scalar SU(2) doublet.

φ =φ_α

In order to makeL invariant under the local gauge transformation defined by:

φ −→ φ^′ = e^{i αa(x)τa/2}φ, (1.27) it is necessary to use in equation (1.25) instead of the standard derivative the co-variant derivative:

Dµ = ∂µ+ igτ_a

2 W_µ^a. (1.28)

In this case, three gauge fieldsW_µ(x) (with a = 1, 2, 3) are introduced. Under the Therefore the gauge invariant Lagrangian corresponding to equation (1.28) is:

L = Ifµ² > 0, the equation (1.31) describes a physical system of four scalar particles φ_i interacting with three massless gauge bosons W_µ^a. If µ² < 0 and λ > 0, the potentialV (φ) of (1.32) has a minimum at the points satisfying the conditions:

φ^†φ = 1

2(φ²₁+ φ²₂+ φ²₃+ φ²₄) = −µ²

2λ. (1.34)

We can expandφ(x) in a neighbourhood of a chosen minimum:

φ1 = φ2 = φ4 = 0 φ²₃ = −µ²

λ ≡ v². (1.35)

Therefore, by expandingφ(x) in the neighbourhood of the selected vacuum state:

φ₀ =r 1

into the Lagrangian (1.31), we obtain that the only scalar field surviving is the Higgs fieldh(x). Indeed, if we write φ(x) as

φ(x) = e^{iτ ·θ(x)/v}

withθ1, θ2, θ3andh real fields the exponential term drop out from the Lagrangian.

By substitutingφ0 (defined in (1.36)) into the Lagrangian, we obtains:

and the mass of vector boson is given by M = ¹₂gv. Therefore, the Lagrangian describes three massive gauge fields and one massive scalarh.