Polarisation of the W-Pair System

W bosons can have helicity $\pm 1$ or 0. A W boson with helicity $\pm$1 is said to be transversely polarised and one with zero helicity is longitudinally polarised. This means that there are four possible final polarised states of the W boson pair; transverse-transverse (TT), longitudinal-longitudinal (LL), transverse-longitudinal (TL) and longitudinal-transverse (TL).

Of the final helicity combinations, all may be produced by both the s-channel and t-channel processes, except the final state where the W bosons have opposite helicity $\pm$1 and $\mp$1. These two final helicity states have angular momentum $J$$=$$2$ and are only accessible through the t-channel neutrino exchange process.

It is possible to write the helicity amplitude for ${\rm e}^{+}{\rm e}^{-}\rightarrow {\rm W}^{+}{\rm W}^{-}$ in terms of the t-channel neutrino exchange process and the s-channel TGC processes, including all 14 couplings parameters. For a final helicity state $\tau\tau^{\prime}$, where $\tau$ is the helicity of the ${\rm W}^{-}$ and $\tau^{\prime}$ that of the ${\rm W}^{+}$, and with initial helicity of the electron $\lambda = \pm \frac{1}{2}$, (In the limit of massless leptons, the helicity of the positron is $\lambda^{\prime} = -\lambda$), the helicity amplitude is given as:

$$F_{\lambda\tau\tau^{\prime}} = - \frac{e^{2}\lambda}{2}s [C^{(\nu)}(\lambda,t)M^{(\nu)}_{\lambda\tau\tau^{\prime}}(s,cos\theta_{\rm W})$$ (3.18)

$$+ \sum^{7}_{i=1}\left(C^{(\gamma)}_{i}(\lambda,s)+C^{(\rm Z)}_{i}(\lambda,s)\right) M_{i,\lambda\tau\tau^{\prime}}(s,\cos\theta_{\rm W})]$$

$s$ is the square of the centre-of-mass energy and $\theta_{W}$ is the angle between the electron direction and the ${\rm W}^{-}$ direction in the centre of mass frame, known as the W production angle. $t$ is the four-momentum transfer and is given by,

$$t = M^{2}_{W} - \frac{1}{2}s(1-\beta\cos\theta_{\rm W})$$ (3.19)

where,

$$\beta = (1-4M^{2}_{\rm W}/s)^{\frac{1}{2}}$$ (3.20)

Equation 3.18 for the helicity amplitude consists of three parts. The first, denoted by superscript $\nu$ is for W-pair production through neutrino exchange. The second and third are for W-pair production through photon and ${\rm Z}^{0}$ decays, and are denoted by superscript $\gamma$ and Z respectively. The TGC parts are each summed over the seven possible couplings given in the Lagrangian, equation 3.1. The $C$s are the terms which carry the dependence on the coupling value and hence there are seven each for the photon and ${\rm Z}^{0}$ TGCs. The $M$s give the helicity composition and W production angle for each of the different coupling terms, note that these are the same for the photon and ${\rm Z}^{0}$ TGC for each respective $i$.

The explicit expressions for each of the $C$s and $M$s can be calculated from the Feynman diagrams and written in terms of the couplings given in equation 3.1 [36,41,65]. They are shown in table 3.2, where in the ${\rm Z}^{0}$ propagator, $D_{Z}$ is approximated at $s > 4M^{2}_{\rm W}$ to be:

$$D_{Z} \simeq s-M^{2}_{Z}$$ (3.21)

and

$$a = \frac{-1+4\sin^{2}\theta_{w}}{4\sin\theta_{w}\cos\theta_{w}},~b = \frac{-1}{4\sin\theta_{w}\cos\theta_{w}}$$ (3.22)

The first column of table 3.2 gives the $C$s given in equation 3.18^3.3. The first row is for the neutrino exchange process. Due to the standard V$-$A constraint, the first term in this row will be zero when the electron spin $\lambda=+\frac{1}{2}$.

To calculate the total amplitude for a certain helicity combination you must multiply each term in the first column with the corresponding term in the column denoted with the required helicity. Each product must be summed together, then the final sum multiplied with the term at the top of the corresponding column. For helicity combinations with $\tau^{\prime} = 0$ and $\tau = \pm 1$, the last column can be used with $\tau\rightarrow\tau^{\prime}$, $\tau^{\prime}\rightarrow\tau$ and $\epsilon \rightarrow -\epsilon$.

So, for example, the Standard Model amplitude, $F^{\lambda}_{\tau\tau^{\prime}}$, for a pair of W bosons with spin $\tau = \tau^{\prime} = +1$, with the initial electron spin $\lambda = -\frac{1}{2}$, would be:

$$F^{\frac{1}{2}}_{+1,+1} = \frac{e^{2}s}{2}\sin\theta_{\rm W}\left[(\frac{-2}{4t\sin^{2}\theta_{w}})(\cos\theta_{\rm W}-\beta)\right.$$ (3.23)

$$+ \left.(\frac{-2}{s}+\frac{2\cot\theta_{w}}{D_{Z}}(a+b))(-\beta)\right]$$

The equivalent term for an initial electron spin of $\lambda=+\frac{1}{2}$ is as in equation 3.24. Notice how the terms due to the neutrino exchange are now absent due to the fact that right handed neutrinos cannot be produced.

$$F^{-\frac{1}{2}}_{+1,+1} = \frac{-e^{2}s}{2}\sin\theta_{\rm W}\left[(\frac{-2}{s}+\frac{2\cot\theta_{w}}{D_{Z}}(a-b))(-\beta)\right]$$ (3.24)

Another important thing to note from table 3.2 is the column for W boson helicities $\tau = -\tau^{\prime} = \pm1$, this shows explicitly that these combinations can only be produced via the neutrino exchange process.

Table 3.2: The elements needed to construct the helicity amplitudes for the W-pair production process.

The W-pair production differential cross-section due to both the neutrino exchange and the TGC channels can be written in terms of the helicity amplitudes:

$$\frac{d\sigma( {\rm e}^{+}{\rm e}^{-} \rightarrow {\rm W}^{+}{\rm... ...da\tau\tau^{\prime}}\left\vert F^{\lambda}_{\tau\tau^{\prime}}\right\vert^{2}~,$$ (3.25)

where the ${\rm W}^{\pm}$ centre-of-mass momentum $\vert\vec{P}\vert = \sqrt{s/4-M^{2}_{\rm W}}$.

As well as the total cross-section, predictions about the polarised cross-sections can also be made, for example, the production of pairs of transverse W bosons $\sigma_{\rm TT}$. Figure 3.5 shows how the total cross-section and the total polarised cross-sections behave as a function of centre-of-mass energy, $\sqrt{s}$. The range is from the threshold of W-pair production, through the energy range of LEP-2 (162-202 GeV), and beyond.

Figure 3.5: The Standard Model total production cross-section for W pairs as a function of centre-of-mass energy. Also shown is the total cross-section for the production of W-pairs with different polarisation states. Transverse-transverse (TT), longitudinal-longitudinal (LL) and transverse-longitudinal (TL+LT).

It can be seen that the total cross-section and each for the different polarisation states rise rapidly from the threshold value, however, they all peak at different values of $\sqrt{s}$, with the total cross-section for W-pair production peaking at about 200 GeV. From table 3.2, it can be shown that the cross-section for transverse-longitudinal (TL) W-pairs is always equal to that of longitudinal-transverse (LT) W-pairs, even in the presence of anomalous couplings.

Figure 3.6: The total production cross-section for W pair production as a function of centre-of-mass energy in the presence of the anomalous TGC $\Delta\kappa_{\gamma}$=+1. Also shown is the total cross-section for the production of W-pairs with different polarisation states.

Figure 3.6 shows the total cross-sections as a function of centre-of-mass energy in the presence of an anomalous coupling of $\Delta\kappa_{\gamma}$=+1. The cross-section blows-up with energy, which would violate unitarity unless some non-Standard Model process occurs at some higher energy value, $\Lambda_{NP}$. With a non-zero value of $\Delta\kappa_{\gamma}$, as the centre-of-mass energy increases the W-pairs produced become almost entirely longitudinally polarised, so the dominant polarisation state becomes LL. With the presence of any of the anomalous couplings, as $\sqrt{s}$ is increased, the polarisation of the W-pairs becomes dominated by just one of the polarisation states. Some of the polarisation states are completely insensitive to certain anomalous couplings. Table 3.3 shows which polarisation states are sensitive to which anomalous couplings and which state dominates when $\sqrt{s}$ becomes large.

Table 3.3: Table showing which final helicity states are sensitive to each anomalous coupling. A tick indicates that the final state is sensitive to the corresponding coupling. The subscript D indicates which helicity state becomes dominant at very high energy in the presence of the corresponding anomalous coupling.

Coupling	$\Delta g^{z}_{1}$	$\Delta\kappa_{\gamma}$	$\lambda$	$g^{z}_{5}$	$g^{z}_{4}$	$\tilde{\kappa}_{z}$	$\tilde{\lambda}_{z}$
TT	$\surd$	$\times$	$\surd_{D}$	$\times$	$\times$	$\surd$	$\surd_{D}$
LL	$\surd_{D}$	$\surd_{D}$	$\times$	$\times$	$\times$	$\times$	$\times$
TL	$\surd$	$\surd$	$\surd$	$\surd_{D}$	$\surd_{D}$	$\surd_{D}$	$\times$

Plots of the differential cross-section of W-pair production as a function of the W production angle have been made at centre-of-mass energy 189 GeV, as this corresponds to the data sample considered in this thesis. Figure 3.7 shows the total differential cross-sections and polarised differential cross-sections for the Standard Model and anomalous C and P-conserving couplings. The anomalous couplings have been set at values $\pm$1.

Figure 3.7: The differential cross-section of W-pair production as a function of the ${\rm W}^{-}$ production angle, $\cos\theta_{\rm W}$, at 189 GeV. Shown is the case for the Standard Model and also with various CP-conserving anomalous couplings implemented.

Figure 3.8 shows the total differential cross-sections and polarised differential cross-sections for the Standard Model and anomalous CP-violating couplings. The anomalous couplings have been set at values $\pm$1.

Figures 3.7 and 3.8 show explicitly the observations made in table 3.3, which were derived by looking at table 3.2. It immediately can be seen that the LL differential cross-section is insensitive to $\lambda$ and all the CP-violating couplings. TT is insensitive to $\Delta\kappa_{\gamma}$ and $g^{z}_{4}$. For the CP-violating couplings the negative coupling has an identical effect as the positive coupling.

Figure 3.8: The differential cross-section of W-pair production as a function of the ${\rm W}^{-}$ production angle, $\cos\theta_{\rm W}$, at 189 GeV. Shown is the case for the Standard Model and also with various CP-violating anomalous couplings implemented.

By integrating over $\cos\theta_{\rm W}$, the fraction of TT, LL and TL+LT W-pairs can be calculated. This has been done for figures 3.7 and 3.8 and the results are displayed in table 3.4. Also shown in this table is the proportion of each helicity state at a much higher energy than LEP-2 has run at. This is to indicate once again which helicity state dominates the total cross-section.

Table 3.4: The fraction of W-pairs with each polarisation state for the Standard Model, and with various anomalous couplings implemented. The first column is at $\sqrt{s} = 189$ GeV and the second is at $\sqrt{s} = 1$ TeV.

Coupling	189 GeV			1 TeV
	TT	LL	LT+TL	TT	LL	LT+TL
SM	0.593	0.094	0.313	0.975	0.007	0.019
$\Delta g^{z}_{1}$= $+$1	0.448	0.165	0.385	0.008	0.901	0.091
$\Delta g^{z}_{1}$= $-$1	0.426	0.181	0.393	0.008	0.901	0.091
$\Delta\kappa_{\gamma}$= $+$1	0.589	0.103	0.308	0.010	0.966	0.024
$\Delta\kappa_{\gamma}$= $-$1	0.516	0.152	0.332	0.010	0.965	0.026
$\lambda$ = $+$1	0.653	0.071	0.276	0.988	0.000	0.012
$\lambda$ = $-$1	0.609	0.067	0.334	0.987	0.000	0.013
$g^{z}_{4}$= $\pm$1	0.542	0.085	0.372	0.149	0.003	0.848
$\tilde{\kappa}_{z}$= $\pm$1	0.383	0.033	0.584	0.045	0.001	0.954
$\tilde{\lambda}_{z}$= $\pm$1	0.567	0.054	0.379	0.987	0.000	0.013