The Standard Model

The Standard Model [31] is a gauge theory for electroweak and strong interactions. It is the simplest model capable of explaining all experimental data in elementary particle physics down to $10^{-19}m$.
The Standard Model Lagrangian is renormalizable, which forbids the coefficients of the interaction terms in a local Lagrangian density to have dimension of mass to a negative power. It is furthermore Poincaré invariant, and gauge invariant under the gauge group $SU(3)_c{\rm {x}} SU(2)_L{\rm {x}} U(1)_Y$, where $SU(3)$ is the strong colour group, $SU(2)_L$ corresponds to rotations in weak isospin space, and $U(1)$ to phase transformations. The gauge bosons are the gluon fields $G^a_{\mu}$, $a=1,..,8$ of the strong interaction and the electroweak bosons $A^i_{\mu}$, $i=1,2,3$ and $B_{\mu}$, with the corresponding coupling constants $g_s$, $g$ and $g'$, respectively. The Standard Model Lagrangian density is ${\cal{L}}_{SM}={\cal{L}}_{strong}+ {\cal{L}}_{electroweak}$, where

$${\cal{L}}_{strong}=\sum_j i\bar{q}_j\gamma^{\mu}D_{\mu}q_j- \frac{1}{2}trG_{\mu \nu}G^{\mu \nu}$$ (92)

where the strong parameters $G_{\mu \nu}$ and $G_{\mu}$ are given by (103). The electroweak piece of the Standard Model Lagrangian is given by

$${\cal{L}}_{SM}=\sum_k i\bar{\psi}_k\gamma_{\mu}D^{\mu}\psi_k... ...}F^{j \mu \nu}F^j_{\mu \nu}-\frac{1}{4}B^{\mu \nu}B_{\mu \nu},$$ (93)

where $D_{\mu}=\partial_{\mu}-ig{\vec{\sigma}}/2\vec{A}- ig'\frac{Y}{2}B_{\mu}$, $F^j_{\mu \nu}=\partial_{\mu}A^j_{\nu}-\partial_{\nu}A^j_{\mu}+ g\epsilon^{jkl}A^k_{\mu}A^l_{\nu}$, $j=1,2,3$, and $B_{\mu \nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$.

The matter fields of the Standard Model belong to irreducible representations of the gauge group. Under $SU(2)_L$ the fermions fields form multiplets

$$\psi_{L}= \left(\begin{array}{rcl} f\\ f'\nonumber\\ \e... ...L},{\hspace{5mm}}\psi_{R}=f_{R},{\hspace{4mm}}\psi'_{R}=f'_{R}$$

with the multiplet hypercharge given by $Y=2(Q-T_{3})$, where $T_{3}$ is the third isospin component.

On the basis of the observed mass spectrum of the fermion doublets $(u,d)_{L}, (\nu _{e},e)_{L}, (c,s)_{L}$,.. observed in nature, the fermions can be classified into three families, with one lepton flavoured and one quark flavoured doublet (in three colours) in each family. The basis for the present classification is that a fermion with a given charge in a given family is lighter than the corresponding fermion in a higher family. $m_{u} < m_{c} < m_{t}$ thus implies that we put u, c and t in the first, second and third family correspondingly.
For neutrinos, the one which couples most strongly to the electron is called $\nu_{e}$ and is placed in the first family, and so on. The fermion assignment of the Standard Model is free from anomalies, as each family is anomaly-free with respect to the gauge bosons.
The fermion ingredients of theory thus consist of three families of quarks and leptons, each family comprises left-handed doublets and right-handed singlets. Neglecting mixing (see below) one has

$$\left(\begin{array}{rcl} U\\ D\nonumber \end{array} \ri... ...ray}{rcl} N\\ E\nonumber \end{array} \right)_{L}=(1,2,-1),$$

and $U_{R}=(3,1,2/3)$, $D_{R}=(3,1,-1/3)$, $E_{R}=(1,1,-1)$, where $U = u, c, t$, $D = d, s, b$, $E = e, \mu, \tau$ and $N = \nu_{e}, \nu_{\mu}, \nu_{\tau}$. The two first entries in each parenthesis denote the dimensions of the SU(3)- and the SU(2)-representation, respectively, and the last entry denotes the U(1) hypercharge.

In the Standard Model one accomodates the observed families without explaining why nature squanders with these recurrent replicas of each particle. The families are treated on the same footing, i.e. the family scheme is regarded as consisting of a succession of families that are identical up to mass values and mixings (see below). Even in extensions of the Standard Model, like in Grand Unified theories, the fermion representation is simply repeated as many times as desired.

The constraints on the Standard Model Lagrangian are so powerful that the Lagrangian is forced to have a rather simple form. As a consequence, the Lagrangian exhibits several other symmetries which are not put into theory as a priori principles. The baryon number $B$ is conserved, just like the lepton number $L_a$ for each family. The baryon number however has a (suppressed) anomaly due to its interaction with the weak bosons, and a similar situation exists for the lepton number. Combinations of baryon and lepton numbers where these anomalies cancel out exist, like $L_{\tau}-L_{\mu}$, $L_{\mu}-L_e$,.. and $B$ - $L$ where $L=L_e+L_{\mu}+L_{\tau}$.

If there had been only one or two families, $CP$ and $T$ would had been symmetries of the total Standard Model Lagrangian. As it is, the Lagrangian for the whole Standard Model is not invariant under any of the discrete operations $C$, $P$ or $T$, neither is it flavour conserving.