Weak Force

The Weak Interacting force is involved in β decay, n → p+e⁻ + ¯ν_e and in the decay processes of many meson particles, such asπ^± →µ^±+νµ, which is seen as beam halo background in the experiments.

Weak interactions were postulated to explain the long lifetimes of some particles such as the muon (τ ≈ 2.2 × 10⁻⁶ sec) and the neutron (τ ≈ 960 sec)[72]. The evolution of the theory of Weak interactions is an interesting example of the en-twined nature of theory and experimental results. Initially, Weak interactions were formulated by E. Fermi as a four fermion interactions theory, described diagram-matically in figure 6.2 with the amplitude given in equation 6.3 [73].

M=−G_F

√2[¯p(x)γλn(x)][¯e(x)γ^λνe(x)] (6.3) However, it was subsequently noted in experiments involvingKaonsthat the decay ofK⁺can result in two different final states each with opposite parities. In the case of K⁺ →π⁺π⁻, Parity = -1, whereas for K⁺ →π⁺π⁺π⁻, Parity = +1. This led to the suggestion that Weak interactions do not conserve parity, as EM and Strong

n ν_e

p e⁻

Figure 6.2: The vector formulation by Fermi postulated that all the interaction takes place the same space-time point.

interactions do [74]. Experimental evidence quickly confirmed this suggestion [75], leading on to the theory of a vector minus axial vector form (V − A) of the charged Weak current; one which violates Parity maximally. As a result, charged Weak interactions only couple to left (right) handed fermions (anti-fermions). The V −A formulation of a charged Weak current described the phenomenology of Weak interactions very well. But it did not include neutral currents which were eventually discovered in 1973 at the Gargamelle bubble chamber at CERN [76].

Additionally, theV −Atheory was non-renormalisable and also violated unitarity.

A new theory had to be found which incorporated all the good properties ofV −A theory but was also renormalisable, described neutral currents and the relative weakness of the Weak force.

The final amendment to the Weak theory is known as the Intermediate Vector Boson (IVB) theory. This theory assumes that the Weak currents are mediated by massive vector bosons (spin = 1). The charged Weak currents involved, for example, in β decay, are conveyed by the W⁺ and W⁻ bosons. The interactions involved in the interaction ν_µe⁻ → ν_µe⁻ seen in Gargamelle can only happen via the neutral weak current, shown in figure 6.3. The boson responsible for neutral current weak interactions, the Z⁰ boson, was discovered at CERN in 1983 at the UA1 and UA2 [77] experiments shortly after the discovery of theW^± bosons.

The confirmation of the existence of theW^± andZ⁰bosons as predicted by theory, solidified the understanding of Weak interactions. As well as predicting the three Weak vector bosons, the theory included the photon γ of EM and unified the U(1)_em theory of QED with the SU(2) theory of weak interactions into a single SU(2)_L×U(1)_Y theory of ElectroWeak interactions.

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e⁻ νµ

e⁻ νµ

Figure 6.3: The interaction seen in Gargamelle (CERN) can only proceed via the Weak neutral current. Strong interactions are excluded as all particles involved are leptons. The presence of the neutrinos excludes EM interactions.

Finally, no charge transfer takes place, excluding the involvement ofW^±bosons.

Electroweak Unification

The Standard Model of Glashow, Weinberg and Salam unifies the Electromagnetic force (with its U(1) symmetry), and the Weak force (SU(2) symmetry) to form an SU(2)_L×U(1)_Y symmetry group. Here, the subscript Lrefers to the charged weak coupling to left handed fermions. The subscript Y refers to the more fundamental quantum number of the weak hypercharge. The electronic charge Q is a linear combination of this hypercharge and the conserved quantity of the z component of the weak isospin, T_z.

Q=T_z+ 1

2Y (6.4)

BothT_z and Y are conserved currents and therefore, electronic charge is also con-served. This is different to the U(1)em symmetry group where Q was conserved directly as a consequence of imposing global gauge invariance. The SU(2)_L sym-metry group has 3 generators, W_1,2,3^µ while the U(1)_Y has one, B^µ.

The Lagrangian for the EW theory of the standard model can be written as:

L_SM =L_f +L_G+L_SBS+L_{Y W} (6.5) whereL_f is the fermion Lagrangian andL_G is the Lagrangian of the gauge fields.

L_f = Σ_ff iγD¯ _µf

L_G =−1

4W_µνⁱ W_i^µν− 1

4B_µνⁱ B^µν_i +...

where;

W_µνⁱ =δµW_νⁱ−δνW_µⁱ +g^ijkW_µ^jW_ν^k

B_µν =δ_muB_ν −δ_νB_µ

The last two terms of equation 6.5 are the Spontaneous Symmetry Breaking La-grangian, required to introduce masses to the gauge bosons in a gauge invariant way, and the Yukawa Lagrangian to give masses to the fermions. The physical gauge bosons, W_µ^±,Z_µ and A_µ are formed from these electroweak eigenstates by:

W_µ^±= 1

√2(W_µ¹∓iW_µ²)

Z_µ=cosθ_wW_µ³−sinθ_wB_µ A_µ=sinθ_wW_µ³+cosθ_wB_µ

where θ_w is the weak mixing angle of the neutral components, Z_µ and A_µ. This forms the W^±, Z and photon of the weak and electromagnetic forces. However, none of these bosons have mass as expected of the short range weak force. The Higgs Mechanism is the simplest method of introducing mass terms without spoil-ing the gauge invariance of the local symmetry.

Higgs Mechanism

A method of bestowing mass to the W^± and Z bosons while leaving the U(1)em

photon massless needs to be done in an way which still leaves the Lagrangian in-variant. The simplest way to do this is to introduce a complex scalar doublet field:

Φ = Φ⁺

Φ⁰

= 1

√2

Φ⁺₁ +iΦ⁺₂ Φ⁰₁ +iΦ⁰₂

(6.6)

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Figure 6.4: The traditional ‘wine bottle’ potential of the Higgs field. There are infinite number of degenerate minima states after spontaneous symmetry breaking. Shifting one of these states to the zero point of the vacuumhidesthe broken symmetry and the previously massless Goldstone Bosons acquire mass.

and a vacuum potential term:

V(Φ) =µΦ^†Φ +λ(Φ^†Φ)² (6.7) where µand λ are constants with λ >0 so the vacuum potential is bounded from below [78]. When µ <0, the vacuum potential has a non-trivial zero value, as seen in figure 6.4. The degenerate set of zero values are found at a magnitude v₀ =

q−µ

2λ. Choosing one of these possible minima - traditionally the one laying on the Re(Φ) axis - breaks the symmetry.

Performing small perturbations around the newly chosen vacuum state^∗ results in mass terms for all the gauge vector bosons as well as a massless scalar boson for each, known as Goldstone Bosons. Additionally, one also obtains a massive scalar field, theHiggs Field. By applying a carefully chosen gauge transformation,

∗ Small perturbations around the vacuum are excited states which, in Quantum Mechanics are interpreted as particles.

known as the unitarity gauge, these non-physical massless Goldstone bosons can be ‘rotated’ away and in their place we gain the longitudinal polarisation of the massive SU(2) vector bosons.

However, the vacuum is invariant under simultaneous SU(2)_z, U(1) rotations ofφ and ¹₂φ respectively (see equation 6.4). It is this part of the gauge group which remains massless and provides the required massless photon.