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I'm trying to understand how the sum over spin polarizations is treated when using the Narrow Width Approximation(NWA) with a spin 1 resonance. For a spin zero resonance there is no such problem and the factorization is quite straightforward. I'll go through some details to explain where my problem is.

It'll be easier to talk about a specific example so I'm going to look at the decay of a top quark via a W boson, which is a spin 1 resonance. The decay follows like this ##t(k)\to b(p_1) W^*(q) \to b(p_1) f(p_2) \overline{f}'(p_3)## where ##f## and ##f'## are any fermion. So we'd expect that close to the resonance peak we can factorize the width as

$$\Gamma(t\to b f \overline{f}')=\Gamma(t\to b W) BR(W\to f\overline{f}')$$

that is the production and decay of a real W. My aim is to understand how this factorization occurs here for a spin 1 resonance.

Starting from the decay width formula we have,

$$\Gamma_{t\to b f\overline{f}'}=

\frac{1}{2 m_{t}}\frac{1}{2}\sum_{spin}

|{\mathcal M}(t\to b f \overline{f}')|^2

dLIPS$$

where

$$dLIPS

=

\frac{d^3 \vec{p}_{1}}{2 E_{p_1}(2\pi)^3}

\frac{d^3 \vec{p}_{2}}{2 E_{p_2}(2\pi)^3}

\frac{d^3 \vec{p}_{3}}{2 E_{p_3}(2\pi)^3}

(2\pi)^4 \delta^4(p_1+p_2+p_3-k)$$

is the lorentz invariant phase space.

So my first job is to factorize the amplitude into ##t\to b W## and ##W\to f\overline{f}'## parts. We can write the amplitude as,

$${\mathcal M}(t\to b f \overline{f}')=

{\mathcal M}_1^\mu

\,\frac{(-g_{\mu\nu}+q_\mu q_\nu / M_W^2)}{q^2-M_W^2+iM_W\Gamma_W}

\,{\mathcal M}_2^\nu

=

{\mathcal M}_1^\mu

\,\sum_\lambda \frac{\epsilon^*_\mu(q,\lambda)

\epsilon_\mu(q,\lambda)}

{q^2-M_W^2+iM_W\Gamma_W}

\,{\mathcal M}_2^\nu$$

where ##{\mathcal M}_{1,2}## come from the ##t-b-W## vertex and ##W-f-f'## vertex respectively and I've used the polarization sum over the W polarization vector identity, ##\sum_\lambda \epsilon^*(q,\lambda)\epsilon(q,\lambda)=(-g_{\mu\nu}+q_\mu q_\nu/M_W^2)##, in the second equality where ##\lambda## is the W polarization. To simplify the notation I'm going to define as follows,

$${\mathcal M}^\lambda_{t\to b W}=

{\mathcal M}_1^\mu \epsilon^*_\mu(q,\lambda)$$

and

$${\mathcal M}^\lambda_{W\to f\overline{f}'}=

{\mathcal M}_2^\mu \epsilon_\mu(q,\lambda)

$$

So the amplitude becomes,

$$

{\mathcal M}(t\to b f \overline{f}')

=

\sum_\lambda

\frac{{\mathcal M}^\lambda_{t\to b W}{\mathcal M}^\lambda_{W\to f\overline{f}'}}

{q^2-M_W^2+iM_W\Gamma_W}

$$

Now squaring the amplitude we have,

$$|{\mathcal M}(t\to b f \overline{f}')|^2

=

\left|\sum_\lambda

{\mathcal M}^\lambda_{t\to b W}{\mathcal M}^\lambda_{W\to f\overline{f}'}\right|^2

\frac{1}{{(q^2-M_W^2)^2+M_W^2\Gamma_W^2}}

$$

And this is where I get stuck. The propagator part is easy to deal with in the NWA it will change to a delta function like this,

$$

\frac{1}{{(q^2-M_W^2)^2+M_W^2\Gamma_W^2}}\to

\frac{2\pi\,\delta(q^2-M_W^2)}{2M_W^2\Gamma_W^2}

$$

but I'm not sure how to deal with the square of the polarization summed amplitude,

$$\left|\sum_\lambda

{\mathcal M}^\lambda_{t\to b W}

{\mathcal M}^\lambda_{W\to f\overline{f}'}\right|^2

$$

Ideally I want to show that this can be written as something like,

$$

\left(\sum_\gamma\left|{\mathcal M}^\gamma_{t\to b W}\right|^2\right)

\left(

\frac{1}{3}\sum_\lambda \left|{\mathcal M}^\lambda_{W\to f\overline{f}'}\right|^2

\right)

$$

so that the decay width will factorize into, ##\Gamma(t\to b f\overline{f}')=\Gamma(t\to b W)BR(W\to f\overline{f}')##.

So my question is how do I show that, ##\left|\sum_\lambda

{\mathcal M}^\lambda_{t\to b W}

{\mathcal M}^\lambda_{W\to f\overline{f}'}\right|^2

=

\left(\sum_\gamma\left|{\mathcal M}^\gamma_{t\to b W}\right|^2\right)

\left(

\frac{1}{3}\sum_\lambda \left|{\mathcal M}^\lambda_{W\to f\overline{f}'}\right|^2

\right)

## ?

I'm sorry for all the details but I wanted to give the full context of my question.

Thanks

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# Narrow Width Approximation for a spin 1 resonance

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