Electroweak interaction

approach.

In the SM, the gauge invariance principle is also applied using more complicated symmetries than U(1). In principle it is possible to create interactions with different representations of different Lie groups. In the context of SM, the representations of the non-AbelianSU(2)andSU(3) groups⁽⁸⁾are used to describe the weak and strong interactions, respectively. A recipe how to make the Lagrangian invariant under transformations of non-Abelian groups was determined by Yang and Mills [4]. The creation of a Lagrangian invariant under theSU(N)local gauge transformations is achieved as follows.

Firstly, the covariant derivative that ensures the local gauge invariance ofL gains the form of:

∂_µ 7→D_µ= ∂_µ+igA^a_µT^a, (1.6)

whereA^a_µare the interacting fields andT^athe generators of the representation of theSU(N)group, wherea=1. . .(N²−1). Each interacting field corresponds to a new gauge boson. The field-strength tensor for non-Abelian gauge theories is also more complex:

F_µν^a =∂_µA_ν^a−∂_νA^a_µ−gf^abcA^b_µA^c_ν, (1.7) where thef^abcare the structure constants of the representation of the group, defined by the commutation relations of the generators: [T^a,T^b]= f^abcT^c(9). The defining characteristic of non-Abelian groups is that f^abc , 0. Consequently, the free-field termF_µν^aF_a^µν now contains third- and fourth-power gauge field terms. These terms correspond to self-interactions of the gauge bosons, a phenomenon not present in electrodynamics.

The full SM Lagrangian is invariant under a combinedSU(3)_C⊗SU(2)_L ⊗U(1)_Ylocal gauge symmetry. TheSU(3)_Cstands for the colour symmetry of the Quantum Chromodynamics (QCD), the theory of strong interactions. TheSU(2)_L⊗U(1)_Yis the symmetry of the unified EW theory. The requirement of the local gauge invariance gives rise to the corresponding interactions terms in the Lagrangian. We will further discuss these symmetries and the interactions in the following sections.

1.3 Electroweak interaction

The electromagnetic and weak interactions are unified in the SM into a single theory, foundations of which were formulated by Glashow [5], Weinberg [6] and Salam [7]. The idea of theGWStheory is that both weak and electromagnetic interactions are manifestations of a single underlying interaction. Let us first begin with how the weak interactions are embedded in the SM. Experimental evidence, such as the experiment by Wu, studying βdecays of radioactive⁶⁰Co isotope [8], showed that parity was violated in weak interactions. In this experiment, the electrons in the βdecays were preferentially emitted in a direction opposite to the spin projection of the nuclei. The emitted electron spin projection was preferentially opposite to its momentum direction⁽¹⁰⁾. A hypothesis to explain the parity violation was formulated which suggested that the electron and neutrino in the decay were coupled via avector-axial

(8)TheSU(n)is the group of unitary matrices of dimensionn×nwith determinant equal to one. The generators of representations of this group aren×nmatrices that are anti-hermitian and with a trace equal to zero. The generators are the crucial piece to the formulation of a corresponding gauge-invariant Lagrangian and the induced interaction.

(9)[T^a,T^b]=T^aT^b−T^bT^ais the commutator of two matrices.

(10)The projection of spin into momentum is commonly referred to ashelicity.

1. The Standard Model of elementary particles

(V-A) coupling [9]:

¯

eγ^µ(1−γ⁵)ν_e, (1.8)

witheandν_ebeing the Dirac bi-spinors for electron and neutrino. This coupling can be rewritten into

¯

e_Lγ^µν_e,L, wheree_L andν_e,Lare the left-component chirality projections of the electron and neutrino bi-spinor. The chirality states are defined by the projection operator(1−γ⁵)⁽¹¹⁾, i.e. the chirality projections⁽¹²⁾of a bi-spinor are defined as:

Ψ_L ≡(1−γ⁵)Ψ, Ψ_R ≡ (1+γ⁵)Ψ. (1.9) From this point, chirality projections will be referred to as left-handed and right-handed components.

The V-A theory did explain the parity violation, however it suffered from the fact it was not a renormalisable theory. In the SM, the weak interaction is described by a renormalisable gauge theory based on SU(2) group, that also includes the vector-axial coupling of the fermions, however the left-handed fermion fields couple to gauge boson fields. In a low-momentum-transfer limit, such as in the βdecays, this gauge theory yields the original V-A theory.

The coupling of left-handed weakly-interacting particles is reflected in the SM by the fact that the left-handed components of fermion fields are grouped in SU(2) doublets, while right-handed components are SU(2) singlets. For the leptons, the doublets and singlets are as follows:

*

where`is one of thee,µ,τbi-spinors andν<sub>`</sub>is the neutrino bi-spinor from the corresponding lepton generation.

For quarks, the situation is more complicated due to the fact that eigenstates of the weak interaction are not mass eigenstates, leading to flavour violation in charged weak interactions. The relation between the weak eigenstates and the mass eigenstates is given by theCabibbo-Kobayashi-Maskawa(CKM) matrix, a complex unitary 3×3 matrix:

*

where thed⁰,s⁰andb⁰are the weak eigenstates and thed,s,bare the mass eigenstates. The probability of a quarkitransitioning to a quarkjis proportional to|Vi j|². The matrix has four physical parameters⁽¹³⁾, a common choice of parametrisation is via three mixing angles and one CP-violating complex phase.

In the SM, the elements of the CKM matrix are free parameters and are measured experimentally. An interesting fact about the CKM matrix is that it was proposed before either of the third-generation quarks were discovered. This is because already in 1964, Cronin and Fitch observed violation of

(11)γ⁵=iγ¹γ²γ³γ⁴, whereγⁱare the Dirac gamma matrices. The1operator denotes the 4×4 unit matrix.

(12)A left-handed helicity projection of state in general contains non-zero contributions of both left-handed and right-handed chirality states. For massless particles and in ultra-relativistic limit, the helicity and chirality states coincide.

(13)AN×Nunitary matrix does haveN²real degrees of freedom. However a complex phase can be absorbed by each of the six quark fields, except for one overall phase. This yieldsN²−(2N−1)physical parameters.

1.3. Electroweak interaction

combined charge-conjugation and parity (CP) violation in Kaon decays [10]. The quark mixing matrix is a natural construct to include such CP violation, however a 2-generation mixing 2×2 unitary matrix leads to only one real-value mixing angle, which cannot induce CP violation, hence why CKM matrix assumes at least three quark generations.

Going back to the coupling of the fermions in the EW sector, the up- and down-type quark bi-spinors are organised in left-handed doublets and right-handed singlets as follows:

* , u d⁰+

-^L

, (u)_R, (d)_R, (1.12)

where the left-handed doublet mixes the quark fields according to the CKM matrix. No weak mixing is introduced in the right-handed singlets since no weak interaction is involved. Finally, the kinetic term of the Lagrangian is:

L =Ψ¯_LiD/_LΨ¯_L +Ψ¯_RiD/_RΨ¯_R, (1.13) whereΨ_L (Ψ_R) are the left-handed doublets (right-handed singlets) from Eq.1.10and1.12, for each of the three generations of quarks and leptons. The gauge-covariant derivativeD_µmust be different for left-handed and for right-handed components, since the right-handed components do not interact via charged weak currents. At the same time, both left-handed and right-handed components of charged leptons and quarks must transform according to a non-trivialU(1)representation since these particles do interact electromagnetically. The SU(2) andU(1) gauge symmetries give rise to the following covariant derivative for left-handed states:

D_µ= ∂_µ+ig

2σ_iW_µⁱ +ig⁰

2Y_WB_µ, (1.14)

whereσ_iare Pauli’s matrices, the representation ofSU(2)group, andW_µⁱ are the corresponding gauge bosons. TheU(1)interaction is inserted via the last term, whereB_µ is the gauge boson of this group, and an implicit 2x2 identity matrix is present. TheY_W =2(Q−I₃)is the weak hyper-charge defined via the charge of the particleQand third component of the iso-spin I₃, which is±¹₂ for left-handed doublets and zero for right-handed singlets. For right-handed components states, we expect trivial representation⁽¹⁴⁾forSU(2), in other words, theWµⁱ fields should not enter the covariant derivative, and only a term withB_µshould be present. The immediate problem faced is that neither left-handed nor right-handed neutrino components should interact electromagnetically. For right-handed neutrinos, this is achieved trivially thanks toY_W =0. However, for left-handed neutrinos, this is not true and a coupling toB_µis present. The solution in the GWS theory is that the fields describing real bosons of the weak and electromagnetic interactions are linear combinations of theW_µⁱ andB_µfields:

W_µ^±= 1

√2(W_µ¹∓W_µ²), (1.15)

A_µ =W_µ³sinθ_W +B_µcosθ_W, (1.16) Z_µ⁰ =W_µ³cosθ_W −B_µsinθ_W. (1.17)

(14)A trivial representation of a group maps all group elements to unity. The generators of a trivial representation are zero.

1. The Standard Model of elementary particles

The charged flavour-changing weak currents are mediated by theW^±bosons and the neutral weak current and the electromagnetic current is mediated by theZ_µand A_µbosons, respectively. Finally, the coupling constantsgandg⁰are not independent in the GWS theory, but are bound together via the weak mixing angleθ_W that also enters the mixing of fields ofW³and B– expressing that the electromagnetic and weak interactions are manifestations of a single interaction with a single coupling constant:

gsinθ_W =g⁰cosθ_W =e (1.18)

The angleθ_W is a free parameter of the SM and is determined experimentally. In addition, because the coupling constants in QFT are not constants but functions of an energy scale at which they are probed, the same applies toθ_W. The running of coupling constants is discussed in more detail in Sec.1.5.

At this point, it is worth noting a very important limitation of the EW theory as presented up until this point. Experiments sensitive to the mass of theW^±andZ⁰boson give evidence that these are massive particles. Adding mass terms for gauge fields to the Lagrangian will break theSU(2)_L⊗U(1)_Y symmetry. The problem of these contradictory requirements in the SM is resolved by mechanism of spontaneous symmetry breaking(SSB), described in the following section.