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Multichannel particle decay survival probability 
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#1
Dec1212, 07:23 AM

P: 16

Particle can decay through many channels with probabilities [itex]p_i[/itex], where in each channel its decay time is different [itex]\tau_i[/itex]. It always decays through one of the channels.
Particle decays according to exponential law where probability to decay in time [itex]t[/itex] is [tex] P^{(i)}_d(t)={1\over\gamma\tau_i}\exp\left({{t\over\gamma\tau_i}}\right)\;. [/tex] What is the total probability for a particle to survive a given time [itex]t[/itex] (so it does not decay in any channel)? 


#2
Dec1212, 08:24 AM

P: 34

Why not integrate all the decay probabilities from 0 to t, add them up and then subtract the resulting probability from 1 to find the probability of the particle surviving?



#3
Dec1212, 09:08 AM

P: 4

Please suggest me a good book of nuclear and particle physics. My syllabus is nuclear reaction, nuclear fission and fusion, elementary particles, particle accelerator and detector, nuclear astrophysics.



#4
Dec1212, 09:11 AM

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P: 11,576

Multichannel particle decay survival probability
How can you have different lifetimes? Mixing of neutral mesons can give this on the level of ~1% in an effective description, but you have to consider both states there. 


#5
Dec1212, 12:26 PM

P: 16

@MarekS
[tex] P_s(t)=1\sum_i p_i \int\limits_0^t{1\over\gamma\tau_i}\exp\left({t'\over\gamma\tau_i}dt'\right) [/tex] Is this what you meant? @Myphyclassnot: for nuclear and particle physics see e.g. Griffiths, Povh, Perkins, Halzen & Martin, Peskin & Schroeder, Bjorken & Drell ... @mfb: yes, i meant proper lifetime, e.g. [itex]K^+[/itex] can decay via multiple decay channels, see this for listing of all decay modes. 


#6
Dec1212, 02:12 PM

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P: 11,576




#7
Dec1312, 04:44 AM

P: 34




#8
Dec1512, 08:53 AM

P: 16

[tex] {1\over\tau_{\rm tot}}=\sum\limits_i{1\over\tau_i}\;. [/tex] And the probability for such a particle to live for a time [itex]t[/itex] is [tex] P_s(t)=\exp\left({t\over\gamma\tau_{\rm tot}}\right)= \prod_i \exp\left({t\over\gamma\tau_i}\right)\;. [/tex] This is the probability for a particle to survive all channels at once: the first AND the second AND the third etc. And the probabilities for particular channels are not taken into account. Is this really the case? The resulting probability is also very low. This is the way I simulate the process:



#9
Dec1512, 10:19 AM

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P: 11,576

I do not think ##\tau_i## is used somewhere, as it is misleading  this time has no physical meaning. ##\Gamma_i## is the common way to give decay rates for individual channels, if that is done at all (lifetime of the particle and branching fraction is more common).



#10
Dec1912, 06:22 PM

P: 16

@mfb: Thank you for your answer, this really seems to be the case. Could you suggest some literature, a standard book or an article on this issue. I just can't seem to find this anywhere. Thanks again!



#11
Dec2012, 07:10 AM

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P: 11,576

I think there should be lecture notes and books for particle physics and monte carlo generators, but I don't know any specific one.



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