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BR of semileptonic B meson decay

  1. Mar 16, 2015 #1
    Hi there,

    In a reference as 1303.5877v1 [hep-ph ] the SM branching ratio of ## B \to \tau \nu ## is given by:

    ## \frac{m_B G_F^2 m_\tau^2 \tau_B f^2_B } { 8 \pi } V_{ub}^2 ( 1 - \frac{m_\tau^2}{m_B^2} )^2 ## . In the SM model the value of this BR ## \sim 0.7 \times 10^{-4} ## .

    But I don't understand how the BR is of order 10^-4, while the mean life time ## \tau_B \sim 10^{-12} s ## and the Fermi constant ## \sim 10^{-5} GeV ^{ -2} ## ?
     
    Last edited: Mar 16, 2015
  2. jcsd
  3. Mar 16, 2015 #2

    ChrisVer

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    [itex]10^{-12} ~s \sim \mathcal{O}(10^{11} ~GeV^{-1}) [/itex]

    *edit*

    Well I got this from wolframalpha, but I think it's wrong....it's not always correct with its conversions...

    eg. here:
    http://www.saha.ac.in/theory/palashbaran.pal/conv.html
    I find [itex] 10^{-12} ~s \approx 1.52 \times 10^{12} ~GeV[/itex]

    I guess the right way is to set the natural units [itex]\hbar= 6.58 \times 10^{-25} ~GeV~s=1[/itex] so from that you can calculate by yourself the relation between energy and seconds.
    You get:
    [itex] 10^{-12} s = \frac{1}{6.58} 10^{13} ~GeV^{-1}= 1.52 \times 10^{12}~GeV^{-1}[/itex]
     
    Last edited: Mar 16, 2015
  4. Mar 16, 2015 #3

    ChrisVer

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    The units then work fine, since the Branching ratio is immediately dimless...
    [itex] GeV^{-4}_{G_F} \times GeV_{m_E} \times GeV^{-1}_{\tau_B} \times GeV^2_{m_\tau} \times GeV^2_{f_B} =1[/itex]

    and then you have [itex]10^{-10}[/itex] from [itex]G_F^2[/itex],
    [itex]10^{12}[/itex] from [itex]\tau_B[/itex],
    [itex]10^{-5} \text{-} 10^{-6}[/itex] from [itex]|V_{ub}|^2[/itex]
    so the orders of magnitude can work out. I don't know what values they used for the masses and the coupling constant f
     
    Last edited: Mar 16, 2015
  5. Mar 16, 2015 #4
    Thanx ..
     
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