$2!=2\times 1$: Electroweak Interactions

In his paper ``A Theory of Leptons" published in Physical Review Letters in 1967, S. Weinberg says: ``Leptons interact only with photons and with the intermediate bosons which presumably mediate the weak interactions. What can be more natural than linking those one spin bosons in a multiplete of gauge fields?"

In this phrase we find the underlying idea for the model of Salam-Weinberg which we are going to develop now. We will certainly have to face several difficulties, already mentioned, as the mass difference between the photon and the intermediate bosons of the weak interactions. Besides, we must take into account that the electromagnetic and weak couplings are of very different intensity. The hope for success is based on the development of the theories with an spontaneously broken gauge symmetry which will allow to obtain a renormalizable model.

The first step in the construction of the model is the election of the symmetry group $G$.

The model must include the charged weak current $(J_{\mu}^{\pm})$, the electromagnetic interaction linked to the group $U(1)_{em}$ (that must be unbroken as $m_{\gamma}=0$) and should also include a neutral weak current $(J_{\mu}^{0})$ that emerges a posteriori. Summarizing, $G$ has to be a four parameter group. The proposal arising from the model that is called now Glashow-Salam-Weinberg electroweak model is

$$G\equiv SU(2)_{L} \times U(1)_{Y}$$ (289)

where $L$ stands for left and $Y$ for weak hypercharge Now we are going to introduce the model $2!$ in a constructive form. The fermion fields are described through their left-hand and right-hand components:

$$\psi_{L} = \frac{1}{2} (1 - \gamma_{5}) \psi$$ (290)

$$\psi_{R} = \frac{1}{2} (1 + \gamma_{5}) \psi$$ (291)

The $2!$ model is chiral in the sense that $\psi_{L}$ and $\psi_{R}$ behave differently under the gauge group $SU(2)_{L}\times U(1)_{Y}$. In fact the left components form doublets while the right components are singlets.

We begin analysing the charged weak current. As we saw when we dealt with the weak interaction, this kind of currents must have $V-A$ structure, that is to say, only the fermionic fields of left chirality will appear in its construction. In other words, the theory is a chiral theory. We will have as an example

$$J_{\mu}^{-} = \bar{\nu}_{e} \gamma_{\mu} (1 - \gamma_{5}) e^{-} = 2 \bar{\nu}_{e_{L}} \gamma_{\mu} e_{L}^{-}$$ (292)

It is clear that in each quark and lepton generation the same quantum numbers of $SU(2)_{L}\times U(1)_{Y}$ will be repeated. In particular, in the first generation or electron family $(e, \nu_{e}, u, d)$, the pairs $(e, \nu_{e})_{L}$ and $(u, d)_{L}$ will be doublets of $SU(2)_{L}$. That is to say, only the left parts carry contents of $SU(2)_{L}$ as a consequence of the $V-A$ structure.

When it is pretended that the symmetry $U(1)_{em}$ persists, it must be required that

$$SU(2)_{L} \times U(1)_{Y} \supset U(1)_{em}$$ (293)

and therefore the generator of $U(1)_{em}$ must be the sum of the generator of $U(1)_{Y}$ and of the diagonal generator of $SU(2)_{L}$. In other words

$$Q = T_{3} + Y$$ (294)

which is the well known relation of Gell-Mann-Nishijima, valid now also for leptons if they have the good assignment of $T_{3}$ and of $Y$. The quantum numbers of the first generation are summarized in the table

We can begin now to build a local (gauge) invariant Lagrangian under $SU(2)_{L}\times U(1)_{Y}$. In order to guarantee the local symmetry we must introduce vector gauge fields. Due to $SU(2)_{L}$, a triplet $W_{i}^{\mu} (i=1,2,3)$ will be needed and related to $U(1)_{Y}$ the field $Y^{\mu}$ will be necessary.

In terms of these gauge fields it is immediate to write the covariant derivatives of the fermionic fields. For the first family it results

$$D_{\mu} \left( \begin{array}{c} u d \end{array} \right)_{L} = [\partial_{\mu} - i g_{1} \frac{1}{6} Y_{\mu} - i g_{2} \fra... ...{2}\cdot\vec{W}_{\mu}] \left( \begin{array}{c} u d \end{array} \right)_{L}$$ (295)

$$D_{\mu} u_{R} = [\partial_{\mu} - i g_{1} \frac{2}{3} Y_{\mu}] u_{R}$$ (296)

$$D_{\mu} d_{R} = [\partial_{\mu} - i g_{1} \frac{1}{2} Y_{\mu}] d_{R}$$ (297)

$$D_{\mu} \left( \begin{array}{c} \nu_{e} e \end{array} \right)_{L} = [\partial_{\mu} - i g_{1} \frac{1}{2} Y_{\mu} - i g_{2} \fra... ...ot\vec{W}_{\mu}] \left( \begin{array}{c} \nu_{e} e \end{array} \right)_{L}$$ (298)

$$D_{\mu} e_{R} = [\partial_{\mu} - i g_{1} Y_{\mu}] e_{R}$$ (299)

where

$$g_{1} \Leftarrow U(1)_{Y} \;;\; g_{2} \Leftarrow SU(2)_{L}$$ (300)

We then have that the Lagrangian $2!$ for the first generation of quarks and leptons is

$${\cal L} = -\frac{1}{2} (\bar{u}\;\bar{d})_{L} \gamma^{\mu} D_{\mu} \left... ...mma^{\mu} D_{\mu}u_{R} - \frac{1}{2} \bar{d}_{R} \gamma^{\mu} D_{\mu} d_{R}$$

$$-\frac{1}{2} (\bar{\nu}_{e}\;\bar{e})_{L} \gamma^{\mu} D_{\mu... ... e \end{array} \right)_{L} - \frac{1}{2} \bar{e}_{R}\gamma^{\mu} D_{\mu}e_{R}$$

$$- \frac{1}{4} \vec{W}^{\mu\nu}\cdot\vec{W}_{\mu\nu} - \frac{1}{4} Y^{\mu\nu} Y_{\mu\nu}$$ (301)

where the field tensors are

$$W_{i}^{\mu\nu} = \partial^{\mu}W_{i}^{\nu} - \partial^{\nu}W_{i}^{\mu} + g_{2} \epsilon_{ijk} W_{j}^{\mu} W_{k}^{\nu}$$ (302)

and

$$Y^{\mu\nu} = \partial^{\mu}Y^{\nu} - \partial^{\nu}Y^{\mu}$$ (303)

respectively.

Some of the interactions present in (303) can be summarized in the expression

$${\cal L}_{INT} = g_{1} J_{Y}^{\mu} Y_{\mu} + g_{2} \vec{J}^{\mu}\cdot\vec{W}_{\mu}$$ (304)

where the currents (of the first family) are

$$J_{Y}^{\mu} = \frac{1}{6} (\bar{u}\;\bar{d})_{L} \gamma^{\mu} \left( \begin{... ...bar{u}_{R} \gamma^{\mu} u_{R} - \frac{1}{3} \bar{d}_{R} \gamma^{\mu} d_{R}$$

$$-\frac{1}{2} (\bar{\nu}_{e}\;\bar{e})_{L} \gamma^{\mu} \left(... ...ay}{c} \nu_{e} e \end{array} \right)_{L} - \bar{e}_{R} \gamma^{\mu} e_{R}$$ (305)

and

$$J_{i}^{\mu} = (\bar{u}\;\bar{d})_{L} \gamma^{\mu} \frac{\t... ..., \left( \begin{array}{c} \nu_{e} e \end{array} \right)_{L}$$ (306)

The next question refers to the mass of the massive fermions. We could propose first a typical Dirac mass Lagrangian as

$${\cal L} = -m \bar{\psi} \psi = - m (\bar{\psi}_{L} \psi_{R} + \bar{\psi}_{R} \psi_{L})$$ (307)

but it is unacceptable as the left part of the fermions $\psi_{L}$ is a doublet of $SU(2)$ and the right part $\psi_{R}$ is a singlet of $SU(2)$. Therefore, (309) does not respect the symmetry that we are requiring to the model. As we will now see, the masses of the fermions arise through the latent realization of the symmetry.

The symmetry $SU(2)_{L}\times U(1)_{Y}$ must be realized latent in a way to give place to massive gauge fields. Nevertheless, this process must be carefully done in order to guarantee that one of the final gauge fields be correspondent with the massless photon. In other words, the symmetry $SU(2)_{L}\times U(1)_{Y}$ cannot be fully broken. We must act in such a way that

$$SU(2)_{L} \times U(1)_{Y} \rightarrow U(1)_{em}$$ (308)

As a consequence of the spontaneous symmetry breaking according to the scheme (310), from the four initial gauge fields $W_{1}^{\mu}, W_{2}^{\mu}, W_{3}^{\mu}, Y^{\mu}$ one ends with three massive fields and one massless identifiable with the photon.

The photon, $A^{\mu}$, is a combination of the neutral fields $W_{3}^{\mu}$ and $Y^{\mu}$ meanwhile the neutral combination orthogonal to the photon corresponds to the intermediate boson of the neutral weak interactions, $Z^{\mu}$ , which we mentioned above and should be massive. Through the introduction of a mixing angle: the Weinberg angle, we then write

$$A^{\mu} = \;\sin \theta_{W} W_{3}^{\mu} + \cos \theta_{W} Y^{\mu}$$ (309)

$$Z^{\mu} = \;\cos \theta_{W} W_{3}^{\mu} - \sin \theta_{W} Y^{\mu}$$ (310)

It is useful to introduce the combinations of charged fields

$$W_{\pm}^{\mu} = \frac{1}{\sqrt{2}} (W_{1} \mp i W_{2}^{\mu})$$ (311)

which are the bearers of the charged weak interactions. In summary, the physical gauge fields are $W_{+}^{\mu}, W_{-}^{\mu}, Z^{\mu}$ and $A^{\mu}$.

In terms of these mediating fields of the electroweak forces, the interactions are written

$${\cal L}_{INT} = \frac{1}{2 \sqrt{2}} [J_{-}^{\mu} W_{\mu+} + J_{+}^{\mu} W_{\mu-}]$$

$$+ [(g_{2} \cos \theta_{W} + g_{1} \sin \theta_{W}) J_{3}^{\mu} - g_{1} \sin \theta_{W} J_{em}^{\mu}] Z_{\mu}$$

$$+ [g_{1} \cos \theta_{W} J_{em}^{\mu} + (g_{1} \cos \theta_{W} - g_{2} \sin \theta_{W}) J_{3}^{\mu}] A_{\mu}$$ (312)

where we have introduced the notation

$$J_{\pm}^{\mu} = 2 (J_{1}^{\mu} \mp i J_{2}^{\mu})$$ (313)

$$J_{em}^{\mu} = J_{3}^{\mu} + J_{Y}^{\mu}$$ (314)

so that in order to identify the last term of (314) with the usual electromagnetic interaction

$${\cal L}_{INT}^{em} = e J_{em}^{\mu} A_{\mu}$$ (315)

we must impose the relation

$$e = g_{1} \cos \theta_{W} = g_{2} \sin \theta_{W}$$ (316)

which we identify with the electric charge of the electron. This relation allows to write again (314) as

$${\cal L}_{INT} = \frac{e}{2 \sqrt{2} \sin \theta_{W}} (W_{+}^{\mu} J_{\mu-} + W_{-}^{\mu} J_{\mu+})$$

$$+ \frac{e}{2 \cos \theta_{W} \sin \theta_{W}} Z^{\mu} J_{\mu NC} + e A^{\mu} J_{\mu em}$$ (317)

where there appears the notation

$$J_{NC}^{\mu} = 2 (J_{3}^{\mu} - sin^{2} \theta_{W} J_{em}^{\mu})$$ (318)

Let us return now to the explicit analysis of the spontaneous symmetry breaking (310). We must trigger the Higgs mechanism. In order to do it we introduce scalar fields, logically non trivial under the group $SU(2)_{L}\times U(1)_{Y}$, so that its selfinteraction leads to a realization of the symmetry $\grave{a}\; la$ Nambu-Goldstone. The simplest election is a complex scalar field doublet of $SU(2)$:

$$\Phi = \left( \begin{array}{c} \phi^{0} \phi^{-} \end{array} \right)$$ (319)

with $\phi^{0}$ and $\phi^{-}$ complex. The hypercharge (charge $Y$) of $\Phi$ must be $-1/2$. The interaction necessary to make the symmetry latent is written

$$V(\Phi^{\dagger} \Phi) = \lambda (\Phi^{\dagger} \Phi - v)^{2}$$ (320)

that is to say, we will have

$$<0\mid \Phi \mid 0> = \left( \begin{array}{c} \sqrt{v} 0 \end{array} \right)$$ (321)

election that as we will later see guarantees the symmetry $U(1)_{em}$.

Due to the properties under $SU(2)_{L}\times U(1)_{Y}$ of the field $\Phi$, its presence in the Lagrangian of the theory will be of the form

$${\cal L}_{H} = - (D_{\mu}\Phi)^{\dagger} D^{\mu}\Phi - V(\Phi^{\dagger} \Phi)$$ (322)

with the covariant derivative given by

$$D_{\mu} \Phi = (\partial_{\mu} + i \frac{g_{1}}{2} Y_{\mu} - i g_{2} \frac{\vec{\tau}}{2}\cdot \vec{W}_{\mu}) \Phi$$ (323)

At this point it is easy to understand the appearance of the masses of $W^{\pm}$ and of $Z$. After the spontaneous symmetry breaking, shifting from the field $\Phi$ to the field $\chi$ satisfying

$$< 0 \mid \chi \mid 0 > = 0$$ (324)

the mass Lagrangian of the gauge bosons will results from the covariant derivatives present in (324) that provide

$${\cal L}_{G-H} = [(g_{2} \frac{\vec{\tau}}{2}\cdot \vec{W}^{\mu} - g_{1} \frac{1}{2} Y^{\mu}) \Phi]^{\dagger}\cdot$$

$$(g_{2} \frac{\vec{\tau}}{2}\cdot \vec{W}_{\mu} - g_{1} \frac{1}{2} Y_{\mu}) \Phi$$ (325)

We remember then, that

$$g_{2} \frac{\vec{\tau}}{2}\cdot \vec{W}^{\mu} - g_{1} \fra... ...2} W_{3}^{\mu} - \frac{g_{1}}{2} Y^{\mu} \end{array} \right)$$ (326)

that written in terms of $W^{\pm}$ and $Z$ it is written

$$g_{2} \frac{\vec{\tau}}{2}\cdot \vec{W}^{\mu} - g_{1} \fra... ...W} - \cos^{2}\theta_{W} Z^{\mu} - A^{\mu} \end{array} \right)$$ (327)

allows one to conclude, using (327) that

$${\cal L}_{GM} = -\frac{1}{2} (g_{2} \sqrt{v})^{2} W_{+}^{... ...\frac{g_{2} \sqrt{v}}{\cos \theta_{W}})^{2} Z^{\mu} Z_{\mu}$$ (328)

where it is immediate to read

$$M_{W}^{2} = g_{2}^{2} v$$ (329)

$$M_{Z}^{2} = \frac{g_{2}^{2} v}{\cos^{2}\theta_{W}} = \frac{M_{W}^{2}}{\cos^{2}\theta_{W}}$$ (330)

In order to make things clearer, it is convenient to return again to the exponential parametrization of the field $\Phi$ , that is

$$\Phi(x) = e^{i \frac{\vec{\tau}\cdot\vec{\sigma}(x)}{\sqrt{... ...t( \begin{array}{c} \sqrt{v} + H(x) 0 \end{array} \right)$$ (331)

where we remember that the fields $\vec{\sigma}(x)$ are irrelevant as they are ``eaten" by the bosons $W^{\pm}$ and $Z$ to acquire the longitudinal polarization when they turn massive. In terms of the parametrization (333), the Lagrangian of the Higgs field results

$${\cal L}_{G-H} = - \frac{1}{2} \partial^{\mu}H \partial_{\mu}H - \lambda (\sqrt{v} H + \frac{1}{2} H^{2})^{2}$$

$$- \frac{1}{2} g_{2}^{2} (\sqrt{v} + H)^{2} W_{+}^{\mu} W_{\mu-} - \frac{1}{2} (g_{1}^{2} + g_{2}^{2}) (\sqrt{v} + H)^{2} Z^{\mu} Z_{\mu}$$ (332)

where it is evident that

$$m_{H}^{2} = \lambda v$$ (333)

Let us go now to the masses of quarks and leptons. It is clear from above that here those masses also arise from the coupling of the fermionic fields with the Higgs field $\Phi$, when we pass to the field $\chi\equiv H$ which verifies (326). The necessary couplings, of the Yukawa type, are written, for example for the quarks of the first generation

$${\cal L}_{YKW} = - h [(\bar{u} \bar{d})_{L} \left( \beg... ... \left( \begin{array}{c} u d \end{array} \right)_{L} ]$$ (334)

which evidently are invariant $SU(2)_{L}$ and $U(1)_{Y}$ remembering that $Y_{u_{R}} = 2/3$, $Y_{\Phi} = -1/2$ and $Y_{u_{L}} = Y_{d_{L}} = 1/2$.

The Yukawa Lagrangian (336) can be written

$${\cal L}_{YKW} = - h [\bar{u}_{L} u_{R} + \bar{u}_{R} u_{L}] (\sqrt{v} + H)$$

$$= - m_{u} \bar{u} u - \frac{m_{u}}{\sqrt{v}} \bar{u} u H$$ (335)

which allows to conclude that $h$ could be considered as defined by

$$m_{u} = h \sqrt{v}$$ (336)

and that the Higgs couples to quark $u$ (and to fermions in general) in a way proportional to the fermion mass.

At this point it is necessary to generalize the technique to give mass to all the quarks and all the massive leptons. To do that, another Higgs doublet is introduced, this time with $Y=+1/2$, which works for the low members of the doublets, the fermions of type $\lq\lq d''$. For this reason, the currents $J_{\mu}^{\pm}$ which couple with $W_{\mu}^{\pm}$ are not diagonal in the generations, although neutral currents are.

We then introduce the usual notation

$$Q_{i L} = \left\{ \left( \begin{array}{c} u d \end{array} \right)_{L}... ...ght)_{L}\;;\; \left( \begin{array}{c} t b \end{array} \right)_{L} \right\}$$ (337)

$$L_{i L} = \left\{ \left( \begin{array}{c} \nu_{e} e \end{array} \righ... ...; \left( \begin{array}{c} \nu_{\tau} \tau \end{array} \right)_{L} \right\}$$ (338)

$$U_{i R} = \{ u_{R}\;;\;c_{R}\;;\;t_{R} \}$$ (339)

$$D_{i R} = \{ d_{R}\;;\;s_{R}\;;\;b_{R} \}$$ (340)

$$\ell_{i R} = \{ e_{R}\;;\;\mu_{R}\;;\;\tau_{R} \}$$ (341)

so that the Yukawa Lagrangian is written

$${\cal L}_{YKW} = -h_{ij}^{u} \bar{Q}_{i L} \Phi U_{j R}... ...j}^{\ell} \bar{L}_{ij} \tilde{\Phi} \ell_{j R} + h.c. ]$$ (342)

with

$$\tilde{\Phi} = i \tau_{3} \Phi^{\ast} = \left( \begin{array}{c} \phi^{+} -\phi^{0\ast} \end{array} \right)$$ (343)

It is clear from (344) that the coefficients $h_{ij}^{f}$ are not diagonal in generation and besides, via the latent symmetry a non diagonal mass matrix will arise

$$M_{ij}^{f} = h_{ij}^{f} \sqrt{v}$$ (344)

The necessary diagonalization will provoke the mixing between the quark generations.

We introduce the notation

$$\Upsilon_{L}^{u} = \left( \begin{array}{c} u_{L} c_{L} \... ...u \tau \end{array} \right) \;;\;\Upsilon_{R}^{f}\;;\;\cdots$$ (345)

so that the diagonalization of $M^{f}$

$$({\cal U}_{L}^{f})^{\dagger} M^{f} {\cal U}_{R}^{f} = M_{diag}^{f}$$ (346)

which implies ${\cal U}_{L}^{f} = {\cal U}_{R}^{f}$ because $M^{f}$ is hermitic and linked to the basis change

$$\Upsilon_{L}^{f} \rightarrow {\cal U}_{L}^{f} \Upsilon_{L}^... ...Upsilon_{R}^{f} \rightarrow {\cal U}_{R}^{f} \Upsilon_{R}^{f}$$ (347)

Here we understand that the neutral currents are not affected because they are charge diagonal.

On the contrary, for the charged currents, for example for

$$J_{-}^{\mu} = [\bar{\Upsilon}_{L}^{u} \gamma^{\mu} \Upsilo... ... \bar{\Upsilon}_{L}^{\nu_{e}} \gamma^{\mu} \Upsilon_{L}^{e}]$$ (348)

the change of basis (349) leads to

$$J_{-}^{\mu} = 2 [\bar{\Upsilon}_{L}^{u} \gamma^{\mu} ({\c... ...^{\nu_{e}} \gamma^{\mu} {\cal U}_{L}^{e} \Upsilon_{L}^{e}]$$ (349)

We must emphasize that due to the supposed zero mass for the neutrinos, the matrix ${\cal U}_{L}^{e}$ in the second term can be eliminated redefining $\Upsilon_{L}^{\nu_{e}}$. In the first term the mixing matrix of Cabibbo-Kobayashi-Maskawa is clearly defined

$$V_{CKM} = ({\cal U}_{L}^{u})^{\dagger} {\cal U}_{L}^{d}$$ (350)

which in the case of three families depends on three angles $\theta_{1}, \theta_{2},\theta_{3}$ and a phase $\delta$ which can be, in principle, experimentally determined.

It is interesting at this point to present a test of coherence of the spontaneous symmetry breaking mechanism in the sense of verifying that the condition (310) has been respected. When the symmetry is latent, the Higgs field verifies (323). Therefore, the application of an operation of $SU(2)_{L}$ implies

$$G_{1} <0\mid \Phi \mid 0> = \frac{\tau_{1}}{2} <0\mid \Phi \mid 0> =\frac{1}{2} \left( \begin{array}{c} 0 \sqrt{v} \end{array} \right) \neq 0$$ (351)

$$G_{2} <0\mid \Phi \mid 0> = \frac{\tau_{2}}{2} <0\mid \Phi \mid 0> =\frac{1}{2} \left( \begin{array}{c} 0 i \sqrt{v} \end{array} \right) \neq 0$$ (352)

$$G_{3} <0\mid \Phi \mid 0> = \frac{\tau_{3}}{2} <0\mid \Phi \mid 0> =\frac{1}{2} \left( \begin{array}{c} \sqrt{v} 0 \end{array} \right) \neq 0$$ (353)

meanwhile the transformation $U(1)_{Y}$ gives place to

$$G_{Y} <0\mid \Phi \mid 0> = -\frac{1}{2} \left( \begin{array}{c} \sqrt{v} 0 \end{array} \right) \neq 0$$ (354)

That is to say none of the generators of $SU(2)_{L}\times U(1)_{Y}$ annihilates the vacuum. Nevertheless, the generator of $U(1)_{em}$ verifies

$$Q <0\mid \Phi \mid 0> = (I_{3} + Y) <0\mid \Phi \mid 0> = 0$$ (355)

that is to say $Q\equiv Q_{em}$ leaves the vacuum invariant and $U(1)_{em}$ is preserved as we wanted. In other words, it is guaranteed that there exists in the theory the massless photon.

Let us try now a summary of $2!$ in terms of the characteristic parameters. In order to do that we write the limit of the model for values of the momentum much lower than the mass of the gauge bosons. In this case we will have to be able to contrast the resulting expression with the Fermi effective Lagrangian at low energies. That is to say

$${\cal L}_{eff} = \frac{G_{F}}{\sqrt{2}} [J_{\lambda}^{CC} ... ...\frac{\rho}{2} J_{\lambda}^{NC} (J^{\lambda NC})^{\dagger}]$$ (356)

where we include the parameter $\rho$ which measures the relative weight between the charged and neutral currents.

Let us consider the propagator of $W$ or $Z$ in the mentioned limit

$$-i \frac{g_{\lambda\sigma} + \frac{q_{\lambda} q_{\sigma}}... ...^{2}}{\rightarrow} -i \frac{g_{\lambda\sigma}}{M_{W/Z}^{2}}$$ (357)

which shows that in that limit

$${\cal L}^{CC} \rightarrow (\frac{g_{2}}{2 \sqrt{2}})^{2} \frac{1}{M_{W}^{2}} J_{\lambda}^{CC} (J^{\lambda CC})^{\dagger}$$ (358)

$${\cal L}^{NC} \rightarrow (\frac{g_{2}}{4 \cos \theta_{W}})^{2} \frac{1}{M_{Z}^{2}} J_{\lambda}^{NC} (J^{\lambda NC})^{\dagger}$$ (359)

These last two expressions, when compared to the effective Lagrangian (358) allow to make the identifications

$$\frac{G_{F}}{\sqrt{2}} = \frac{g_{2}^{2}}{8 M_{W}^{2}}$$ (360)

and

$$\frac{\rho}{2} \frac{G_{F}}{\sqrt{2}} = \frac{g_{2}^{2}}{16 \cos^{2} \theta_{W} M_{Z}^{2}}$$ (361)

identities that make contact with the Fermi phenomenological constant of the weak interactions. From (363) we have

$$\rho = \frac{M_{W}^{2}}{M_{Z}^{2} \cos^{2} \theta_{W}}$$ (362)

which remembering (332) allows to conclude that in the presence of an $SU(2)_{L}$ Higgs doublet to give mass to the gauge bosons, the model $2!$ predicts

$$\rho = 1$$ (363)

We bring here the experimental values of some of the parameters as a reference

$$\sin^{2} \theta_{W} \simeq 0.2325\;;\;M_{W} \simeq 80.22 GeV\;;\; M_{Z} \simeq 91.173 GeV\;;\; \Rightarrow \rho \simeq 1$$

We also make a balance of the free parameters, to be fixed as the above mentioned from the experimental data:

• couplings:
  
  $$g_{2}\;,\;v\;,\;\sin^{2} \theta_{W}$$
  
  
  or
  
  $$\alpha\;,\;G_{F}\;,\;\sin^{2} \theta_{W}$$
  
  

• mixing of generations of quarks $\equiv$ Cabibbo-Kobayashi-Maskawa matrix:
  
  $$\theta_{1}\;,\;\theta_{2}\;,\;\theta_{3}\;,\;\delta$$
  
  

• masses:
  
  $$m_{H}\;,\;m_{f}[or h_{f}] (f=e,\mu,\tau,u,d,c,s,t,b,[\nu's?])$$
  
  

It can be thought that these parameters should be predicted from some super (or sub) structure underlying the $2!$ model. In any case it is a too large number of free parameters. Among them we can call basic constants to the three couplings above (sometimes $M_{Z}$ is choosen instead of $\sin^{2} \theta_{W}$).

The electromagnetic constant $\alpha$ is defined without ambiguity and very accurately at $Q^{2} = 0$ from Thomson scattering. This constant, as all gauge coupling constants, is running with $Q^{2}$. The Fermi constant $G_{F}$ is also a very precise quantity measured in the $\mu$ decay. Finally $M_{Z}$ has recently been determined very accurately in LEP with an error less than $0.2 \%$.

It should be noticed that $\alpha_{s}$, the strong coupling constant (the QCD coupling) also intervenes in the prediction of several observables.

Nowadays, the most important unknown is the masses of the Higgs boson. Its virtual contribution (through quantum loop corrections) modify the predicted values for almost any observable.

To try a summary of the success of the standard electroweak model $SU(2)_{L}\times U(1)_{Y}$ would imply to multiply several times the content of this course. Let us only say that the theory works dramatically well for our reality, in spite of the existence of valid theoretical and structural reasons as the presence of all the free parameters cited above to transcend the model in some direction. Nevertheless, we have chosen to present a verification, one of the first, which has a relaxing flavor at the time of the lecture:

Figure 19: A,B,...,Z Collaboration

Part of the first page of the Review of Particle Properties where the gauge bosons $W$ and $Z$, mediators of the weak interactions, is also presented as Table 20.

Figure 20: $W$ and $Z$ parameters

Finally we include in Table 1 the usual set of physical observables together with their Standard Model predictions and their present and future experimental uncertainties.

Table 1: Standard Model predictions