Electroweak interaction

Formulation

Mathematically, the unification is accomplished under an SU(2) × U(1) gauge group. The corresponding gauge bosons are the three W bosons of weak isospin from SU(2) (W₁, W₂, and W₃), and the B boson of weak hypercharge from U(1), respectively, all of which are massless.

In the Standard Model, the
W±
and
Z0
bosons, and the photon, are produced by the spontaneous symmetry breaking of the electroweak symmetry from SU(2) × U(1)_Y to U(1)_em, caused by the Higgs mechanism (see also Higgs boson). U(1)_Y and U(1)_em are different copies of U(1); the generator of U(1)_em is given by Q = Y/2 + I₃, where Y is the generator of U(1)_Y (called the weak hypercharge), and I₃ is one of the SU(2) generators (a component of weak isospin).

The spontaneous symmetry breaking makes the W₃ and B bosons coalesce into two different bosons – the
Z0
boson, and the photon (γ),

( γ Z 0 ) = ( cos ⁡ θ W sin ⁡ θ W − sin ⁡ θ W cos ⁡ θ W ) ( B W 3 )

Where θ_W is the weak mixing angle. The axes representing the particles have essentially just been rotated, in the (W₃, B) plane, by the angle θ_W. This also introduces a mismatch between the mass of the
Z0
and the mass of the
W±
particles (denoted as M_Z and M_W, respectively),

M Z = M W cos ⁡ θ W .

The W₁ and W₂ bosons, in turn, combine to give massive charged bosons

W ± = 1 2 ( W 1 ∓ i W 2 ) .

The distinction between electromagnetism and the weak force arises because there is a (nontrivial) linear combination of Y and I₃ that vanishes for the Higgs boson (it is an eigenstate of both Y and I₃, so the coefficients may be taken as −I₃ and Y): U(1)_em is defined to be the group generated by this linear combination, and is unbroken because it does not interact with the Higgs.

Before electroweak symmetry breaking

The Lagrangian for the electroweak interactions is divided into four parts before electroweak symmetry breaking

L E W = L g + L f + L h + L y .

The L g term describes the interaction between the three W particles and the B particle.

L g = − 1 4 W a μ ν W μ ν a − 1 4 B μ ν B μ ν ,

where W a μ ν ( a = 1 , 2 , 3 ) and B μ ν are the field strength tensors for the weak isospin and weak hypercharge fields.

L f is the kinetic term for the Standard Model fermions. The interaction of the gauge bosons and the fermions are through the gauge covariant derivative.

L f = Q ¯ i i D / Q i + u ¯ i i D / u i + d ¯ i i D / d i + L ¯ i i D / L i + e ¯ i i D / e i ,

where the subscript i runs over the three generations of fermions, Q , u , and d are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields, and L and e are the left-handed doublet and right-handed singlet electron fields.

The h term describes the Higgs field F.

L h = | D μ h | 2 − λ ( | h | 2 − v 2 2 ) 2

The y term gives the Yukawa interaction that generates the fermion masses after the Higgs acquires a vacuum expectation value.

L y = − y u i j ϵ a b h b † Q ¯ i a u j c − y d i j h Q ¯ i d j c − y e i j h L ¯ i e j c + h . c .

After electroweak symmetry breaking

The Lagrangian reorganizes itself after the Higgs boson acquires a vacuum expectation value. Due to its complexity, this Lagrangian is best described by breaking it up into several parts as follows.

L E W = L K + L N + L C + L H + L H V + L W W V + L W W V V + L Y

The kinetic term L K contains all the quadratic terms of the Lagrangian, which include the dynamic terms (the partial derivatives) and the mass terms (conspicuously absent from the Lagrangian before symmetry breaking)

L K = ∑ f f ¯ ( i ∂ / − m f ) f − 1 4 A μ ν A μ ν − 1 2 W μ ν + W − μ ν + m W 2 W μ + W − μ − 1 4 Z μ ν Z μ ν + 1 2 m Z 2 Z μ Z μ + 1 2 ( ∂ μ H ) ( ∂ μ H ) − 1 2 m H 2 H 2

where the sum runs over all the fermions of the theory (quarks and leptons), and the fields A μ ν , Z μ ν , W μ ν − , and W μ ν + ≡ ( W μ ν − ) † are given as

X μ ν = ∂ μ X ν − ∂ ν X μ + g f a b c X μ b X ν c , (replace X by the relevant field, and f^abc with the structure constants for the gauge group).

The neutral current L N and charged current L C components of the Lagrangian contain the interactions between the fermions and gauge bosons.

L N = e J μ e m A μ + g cos ⁡ θ W ( J μ 3 − sin 2 ⁡ θ W J μ e m ) Z μ ,

where the electromagnetic current J μ e m and the neutral weak current J μ 3 are

J μ e m = ∑ f q f f ¯ γ μ f ,

and

J μ 3 = ∑ f I f 3 f ¯ γ μ 1 − γ 5 2 f

q f and I f 3 are the fermions' electric charges and weak isospin.

The charged current part of the Lagrangian is given by

L C = − g 2 [ u ¯ i γ μ 1 − γ 5 2 M i j C K M d j + ν ¯ i γ μ 1 − γ 5 2 e i ] W μ + + h . c .

L H contains the Higgs three-point and four-point self interaction terms.

L H = − g m H 2 4 m W H 3 − g 2 m H 2 32 m W 2 H 4

L H V contains the Higgs interactions with gauge vector bosons.

L H V = ( g m W H + g 2 4 H 2 ) ( W μ + W − μ + 1 2 cos 2 ⁡ θ W Z μ Z μ )

L W W V contains the gauge three-point self interactions.

L W W V = − i g [ ( W μ ν + W − μ − W + μ W μ ν − ) ( A ν sin ⁡ θ W − Z ν cos ⁡ θ W ) + W ν − W μ + ( A μ ν sin ⁡ θ W − Z μ ν cos ⁡ θ W ) ]

L W W V V contains the gauge four-point self interactions

L W W V V = − g 2 4 { [ 2 W μ + W − μ + ( A μ sin ⁡ θ W − Z μ cos ⁡ θ W ) 2 ] 2 − [ W μ + W ν − + W ν + W μ − + ( A μ sin ⁡ θ W − Z μ cos ⁡ θ W ) ( A ν sin ⁡ θ W − Z ν cos ⁡ θ W ) ] 2 }

and L Y contains the Yukawa interactions between the fermions and the Higgs field.

L Y = − ∑ f g m f 2 m W f ¯ f H

Note the 1 − γ 5 2 factors in the weak couplings: these factors project out the left handed components of the spinor fields. This is why electroweak theory (after symmetry breaking) is commonly said to be a chiral theory.