The TGC Lagrangian

Any system is described fully by its Lagrangian. The general couplings of two charged vector bosons with a neutral vector boson can be described by the effective Lagrangian given in equation 3.1 [35,36,41,42]. Where $e$ is the positron charge and $\theta_{w}$ is the weak mixing angle of the Standard Model.

The Lagrangian contains 14 separate terms and each term has a coupling parameter, indicated in red. The coupling parameters are known as the Trilinear Gauge Couplings, or TGCs. Many of the terms in the Lagrangian (3.1) would give cross sections which diverge with the energy scale, $\sqrt{s}$. This would lead to unitarity violation. As this is not possible, there then would have to be new physics interactions occurring to counter the effect. Thus, within the Standard Model, the values of the coupling parameters which violate unitarity are zero.

$$\cal{L}_{\rm TGC} = ie\textcolor{red}{g^{\gamma}_{1}}(A_{\mu}(\partial_{\mu}W_{-\nu}-... ...^{+}_{\nu} - A_{\mu}(\partial^{\mu}W^{+\nu}-\partial^{\nu}W^{+\mu})W^{-}_{\nu})$$

$$+ ie\textcolor{red}{\kappa_{\gamma}}(\partial_{\mu} A_{\nu}-\partial_{\nu}A_{\mu})W^{+\mu}W^{-\nu}$$

$$+ ie\cot\theta_{w}\textcolor{red}{g^{Z}_{1}}(Z_{\mu}(\partial_{\mu}... ...^{+}_{\nu} - Z_{\mu}(\partial^{\mu}W^{+\nu}-\partial^{\nu}W^{+\mu})W^{-}_{\nu})$$

$$+ ie\cot\theta_{w}\textcolor{red}{\kappa_{Z}}(\partial_{\mu} Z_{\nu}-\partial_{\nu}Z_{\mu})W^{+\mu}W^{-\nu}$$

$$+ ie\frac{\textcolor{red}{\lambda_{\gamma}}}{M^{2}_{W}} ((\partial_... ...{+\nu}-\partial_{\nu}W^{+\rho})(\partial_{\nu}W^{-\mu}-\partial_{\mu}W^{-\nu}))$$

$$+ ie\cot\theta_{w}\frac{\textcolor{red}{\lambda_{Z}}}{M^{2}_{W}}((\... ...{+\nu}-\partial_{\nu}W^{+\rho})(\partial_{\nu}W^{-\mu}-\partial_{\mu}W^{-\nu}))$$

$$- e\textcolor{red}{g^{\gamma}_{4}}W^{+}_{\nu}W^{-}_{\mu}(\partial^{\mu}A^{\nu} + \partial^{\nu}A^{\mu})$$

$$- e\cot\theta_{w}\textcolor{red}{g^{Z}_{4}}W^{+}_{\nu}W^{-}_{\mu}(\partial^{\mu}Z^{\nu} + \partial^{\nu}Z^{\mu})$$ (3.1)

$$+ e\textcolor{red}{g^{\gamma}_{5}}\epsilon_{\mu\nu\rho\sigma}((\partial^{\rho}W^{-\mu})W^{+\nu} - (\partial^{\rho}W^{+\nu})W^{-\mu})A^{\sigma}$$

$$+ e\cot\theta_{w}\textcolor{red}{g^{Z}_{5}}\epsilon_{\mu\nu\rho\sig... ...\partial^{\rho}W^{-\mu})W^{+\nu} - (\partial^{\rho}W^{+\nu})W^{-\mu})Z^{\sigma}$$

$$+ ie\textcolor{red}{\tilde{\kappa}_{\gamma}}W^{+}_{\nu}W^{-}_{\mu}\... ...epsilon^{\mu\nu\rho\sigma}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})$$

$$+ ie\frac{\textcolor{red}{\tilde{\lambda}_{\gamma}}}{M^{2}_{W}}((\p... ...psilon^{\nu\rho\sigma\ell}(\partial_{\sigma}A_{\ell}-\partial_{\ell}A_{\sigma})$$

$$+ ie\cot\theta_{w}\textcolor{red}{\tilde{\kappa}_{Z}}W^{+}_{\nu}W^{... ...epsilon^{\mu\nu\rho\sigma}(\partial_{\rho}Z_{\sigma}-\partial_{\sigma}Z_{\rho})$$

$$+ ie\cot\theta_{w}\frac{\textcolor{red}{\tilde{\lambda}_{Z}}}{M^{2}... ...psilon^{\nu\rho\sigma\ell}(\partial_{\sigma}Z_{\ell}-\partial_{\ell}Z_{\sigma})$$

The Standard Model values of the TGC parameters $\kappa_{\gamma}$, $\kappa_{z}$, $g^{\gamma}_{1}$ and $g^{z}_{1}$ are one, all other parameters are set to zero. This leaves the Lagrangian shown in equation 3.2, which describes the trilinear gauge boson interaction within the Standard Model. Table 3.1 shows the properties of all the 14 TGC parameters.

$$\cal{L}_{\rm TGC} = ie(A_{\mu}(\partial_{\mu}W_{-\nu}-\partial_{\nu}W_{-\mu})W^{+}_{\nu} - A_{\mu}(\partial^{\mu}W^{+\nu}-\partial^{\nu}W^{+\mu})W^{-}_{\nu})$$

$$+ ie(\partial_{\mu} A_{\nu}-\partial_{\nu}A_{\mu})W^{+\mu}W^{-\nu}$$

$$+ ie\cot\theta_{w}(Z_{\mu}(\partial_{\mu}W_{-\nu}-\partial_{\nu}W_{... ...^{+}_{\nu} - Z_{\mu}(\partial^{\mu}W^{+\nu}-\partial^{\nu}W^{+\mu})W^{-}_{\nu})$$

$$+ ie\cot\theta_{w}(\partial_{\mu} Z_{\nu}-\partial_{\nu}Z_{\mu})W^{+\mu}W^{-\nu}$$ (3.2)

The first six couplings of the Lagrangian, equation 3.1, respect the discrete parity (P) and charge (C) symmetries. The first term of the Lagrangian is for a photon coupling to two W bosons. It is called the minimal coupling term. The value of the $g^{\gamma}_{1}$ determines the charge of the positive W boson, ${\cal{C}}_{W}$, in units of the positron charge, $e$, and therefore has a value of one, equation 3.3.

$${\cal{C}}_{W} = e \mbox{$g^{\gamma}_{1}$}$$ (3.3)

The second photon TGC, $\kappa_{\gamma}$ is called the anomalous magnetic moment of the W [43,44]. $\kappa_{\gamma}$ and $\lambda_{\gamma}$ are related to the magnetic dipole moment of the ${\rm W}^{+}$, $\mu_{W}$ [45,46], as in equation 3.4.

$$\mu_{W} = \frac{e}{2M_{W}}(1 + \kappa_{\gamma} + \lambda_{\gamma})$$ (3.4)

Both these two photon TGCs, $\kappa_{\gamma}$ and $\lambda_{\gamma}$, are also related to the electric quadrupole moment of the ${\rm W}^{+}$, $Q_{W}$, as in equation 3.5.

$$Q_{W} = -\frac{e}{M_{W}}(\kappa_{\gamma} - \lambda_{\gamma})$$ (3.5)

Of the remaining eight couplings, $g^{\gamma}_{5}$ and $g^{z}_{5}$, violate both C and P symmetry, but respect combined CP-invariance. The other six couplings all violate CP. $g^{\gamma}_{4}$ and $g^{z}_{4}$^3.1 violate charge conjugation symmetry. However, if $g^{\gamma}_{4}$ or $g^{\gamma}_{5}$ are non-vanishing at $q^{2}=0$, the photon part of the Lagrangian, equation 3.1, will not be electromagnetically gauge invariant [41].

The remaining four couplings $\tilde{\kappa}_{\gamma}$, $\tilde{\kappa}_{z}$, $\tilde{\lambda}_{\gamma}$ and $\tilde{\lambda}_{z}$ all violate parity. The photon P and CP-violating couplings, $\tilde{\kappa}_{\gamma}$ and $\tilde{\lambda}_{\gamma}$, are related to the electric dipole moment of the ${\rm W}^{+}$, $d_{W}$, as in equation 3.6.

$$d_{W} = \frac{e}{2M_{W}}(\tilde{\kappa}_{\gamma} + \tilde{\lambda}_{\gamma})$$ (3.6)

$\tilde{\kappa}_{\gamma}$ and $\tilde{\lambda}_{\gamma}$ are also related to the magnetic quadrupole moment, $\tilde{Q}_{W}$, of the ${\rm W}^{+}$as follows:

$$\tilde{Q}_{W} = -\frac{e}{M_{W}}(\tilde{\kappa}_{\gamma} - \tilde{\lambda}_{\gamma})$$ (3.7)

Table 3.1: Properties of the 14 TGC parameters. Dim. is the dimension of the operator needed to induce each coupling.

Coupling	Dim.	SM Value	C-Conserving	P-Conserving	CP-Conserving
$g^{\gamma}_{1}$, $g^{z}_{1}$	4	1	$\surd$	$\surd$	$\surd$
$\kappa_{\gamma}$,  $\kappa_{z}$	4	1	$\surd$	$\surd$	$\surd$
$\lambda_{\gamma}$,  $\lambda_{z}$	6	0	$\surd$	$\surd$	$\surd$
$g^{\gamma}_{4}$, $g^{z}_{4}$	6	0	$\times$	$\surd$	$\times$
$g^{\gamma}_{5}$, $g^{z}_{5}$	6	0	$\times$	$\times$	$\surd$
$\tilde{\kappa}_{\gamma}$,  $\tilde{\kappa}_{z}$	4	0	$\surd$	$\times$	$\times$
$\tilde{\lambda}_{\gamma}$,  $\tilde{\lambda}_{z}$	4	0	$\surd$	$\times$	$\times$

The Lagrangian given in equation 3.1 only contains the lowest dimension operators, up to dimension six. As the strength of the coupling is generally suppressed by factors like $(\sqrt{s}/\Lambda_{NP})^{d-4}$ [12], where $\Lambda_{NP}$ is the scale of new physics and $d$ is the dimension of the operator, neglecting operators of dimension higher than six is a valid assumption at LEP energies^3.2.

A further consequence of higher dimensional operators would be to render the photon part of the effective Lagrangian gauge invariant, even in the presence of non-vanishing, C-violating photon couplings, $g^{\gamma}_{4}$ and $g^{\gamma}_{5}$ [35].