The amplitude of B decay into tau neutrino

[Safinaz]

Hi there,

In Reference as hep-ph/0306037v2, we see the effective Hamiltonian of $B \to l \nu$ equ. 1, which has the SM and the NP parts. In equ. 4, it seems that $m_l$ comes from $P^\mu_B$ equ. 3.

The question that how $P^\mu_B$ yields $m_l$ ? Where $P^\mu_B = ( E_B, 0, 0, \bf{p} )$ , or $P^\mu_B = ( m_B, 0, 0, 0)$ in B rest frame ..

Bests.

 

[Hepth]

$P^{\mu} \bar{\ell} \gamma_{\mu} (1-\gamma_5) \nu = (p^{\mu}_{\ell} + p^{\mu}_{\bar{\nu}} ) \bar{\ell} \gamma_{\mu} (1-\gamma_5) \nu$
$= \bar{\ell} (\not p_{\ell} + \not p_{\bar{\nu}} )(1-\gamma_5) \nu$

and
$\bar{\ell} \not p_{\ell} = -m_{\ell} \bar{\ell}$

 

[Safinaz]

Many thanks .. May I ask another question about " the vacuum saturation approximation ", that why

$< 0 | \bar{u} \gamma_\mu b | B > ~ and ~ < 0 | \bar{u} b | B > = 0$ ? , and for $[ \bar{l} \gamma^\mu (1-\gamma_5) \nu ]$, why didn't say:
$< l^- \bar{ \nu} | \bar{l} \gamma^\mu (1-\gamma_5) \nu | 0 >$ ?