Understanding Leptonic Lifetimes: Calculations and Corrections

[Orion1]

I attempted to calculate the leptonic lifetimes based upon Electroweak Theory, and I am inquiring as to what I have done wrong in my calculations?

Electron lifetime:
\tau_{e} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{e}^5}

\tau_{e} = 8.266 \cdot 10^5 \; \text{s}
\tau_{e} = \text{stable} - Wikipedia

Mass correction x-axis:
x = \frac{m_{e}^2}{m_{\mu}^2}

Mass correction function:
f(x) = 1 - 8x - 12x^2 \ln \left( \frac{1}{x} \right) + 8x^3 - x^4

Muon lifetime:
\tau_{\mu} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{\mu}^5 f\left( \frac{m_{e}^2}{m_{\mu}^2} \right)}

\tau_{\mu} = 2.18779092 \cdot 10^{-6} \; \text{s}
\tau_{\mu} = 2.19703421 \cdot 10^{-6} \; \text{s} - Wikipedia

Mass correction x-axis:
x = \frac{m_{e}^2}{m_{\tau}^2}

Tau lifetime:
\tau_{\tau} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{\tau}^5 f\left( \frac{m_{e}^2}{m_{\tau}^2} \right)}

\tau_{\tau} = 1.626 \cdot 10^{-12} \; \text{s}
\tau_{\tau} = 2.937 \cdot 10^{-13} \; \text{s} - Wikipedia
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Reference:
http://en.wikipedia.org/wiki/Electron"
http://en.wikipedia.org/wiki/Muon"
http://en.wikipedia.org/wiki/Tau_lepton"
http://physics.nist.gov/cgi-bin/cuu/Value?gf|search_for=fermi"
http://en.wikipedia.org/wiki/Fermi_coupling_constant"
http://en.wikipedia.org/wiki/Physical_constant"

 

[Hawkwind]

I attempted to calculate the leptonic lifetimes based upon Weak Theory, and I am inquiring as to what I have done wrong in my calculations?

Electron lifetime:
\tau_{e} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{e}^5}

In the electroweak theory, the electron is a stable particle. No decay exists. Thus, its lifetime is infinite and nothing else.

 

[Orion1]

In the electroweak theory, the electron is a stable particle. No decay exists. Thus, its lifetime is infinite and nothing else.

There is no doubt about that. What exactly is that electron equation measuring?

And what is wrong with the muon and tau lifetime equations?
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[Vanadium 50]

I attempted to calculate the leptonic lifetimes based upon Electroweak Theory

No you didn't. You took the result of a calculation (it looks like the muon lifetime) and started plugging other numbers in. What came out is, as you noticed, nonsense.

and I am inquiring as to what I have done wrong in my calculations?

See above.

 

[Parlyne]

It looks like you've calculated what the electron lifetime would be if there were a massless charged particle for it to decay into. But, since electric charge is conserved and the electron is the lightest charged particle, it must be stable.

For the muon, your number is off by less than 1%. So, offhand I'd guess that that's just the fact that the equation you're using is only the tree level result (that is, it ignores all higher order quantum corrections).

As for the tau, I think what you have is the lifetime corresponding to the leading-order partial decay width to an electron (and a neutrino and an antineutrino). The thing is, the tau can also decay to a muon (and a neutrino and an antineutrino) or to two quarks (and a neutrino). The tau's lifetime is the inverse of the sum of all partial widths; so, you need to include the partial widths for these other processes, as well.

 

[Orion1]

tau lifetime...


According to my understanding, there are actually two tau lifetimes depending on which decay channel the tau lepton follows, this is called the 'branching ratio':

B_{\tau} \left( \tau^{-} \rightarrow l^{-} \nu_{\tau} \overline{\nu}_{l} \right)

There are two tau lepton branching ratio decay channels:
B_{\tau} \left( \tau^{-} \rightarrow e^{-} \nu_{\tau} \overline{\nu}_{e} \right) = 0.1779 \pm 0.0012 - ref. 4 and 7
B_{\tau} \left( \tau^{-} \rightarrow \mu^{-} \nu_{\tau} \overline{\nu}_{\mu} \right) = 0.1731 \pm 0.0011 - ref. 4 and 7

The branching ratio is a percentage determined by experimental values only and mathematically defined as:
B_{\tau} = \frac{\tau_{\tau}}{\tau_{\mu}} \left( \frac{m_{\tau}^5}{m_{\mu}^5} \right) = 0.1797

Mass correction x-axis:
x = \frac{m_{e}^2}{m_{\tau}^2}

Mass correction:
\Delta_{m} = f(x)^{-1} = (1 - 8x - 12x^2 \ln \left( \frac{1}{x} \right) + 8x^3 - x^4)^{-1} = 1.00000066165937

W boson propagator correction:
\Delta_{W} = 0.9997

QED radiative correction:
\Delta_{\gamma} = 1.0001

Tau lifetime:
\boxed{\tau_{\tau} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{\tau}^5} B_{\tau} \Delta_{m} \Delta_{W} \Delta_{\gamma}}

Tau lifetime:
\boxed{\tau_{\tau} = 2.923 \cdot 10^{-13} \; \text{s}}

Interesting to note that reference 3 cites:
\tau = 1.632 \pm 0.0012 \cdot 10^{-12} \; \text{s} \cdot B_{\tau} \Delta_{m} \Delta_{W} \Delta_{\gamma} - ref. 3
\tau = 1.626 \cdot 10^{-12} \; \text{s} - Orion1 - post #1

Reference data:
\tau_{\tau} = 2.9 \cdot 10^{-13} \; \text{s} - Wikipedia
\tau_{\tau} = 2.901 \pm 0.04 \cdot 10^{-13} \; \text{s} - ref. 2
\tau_{\tau} = 2.930 \pm 0.053 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.875 \pm 0.038 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.901 \pm 0.04 \cdot 10^{-13} \; \text{s}

\tau_{\tau} = 2.890 \pm 0.028 \cdot 10^{-13} \; \text{s} - ref. 3
\tau_{\tau} = 2.901 \pm 0.015 \cdot 10^{-13} \; \text{s} - ref. 4
\tau_{\tau} = 2.890 \pm 0.018 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.904 \pm 0.032 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.879 \pm 0.031 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.890 \pm 0.019 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.901 \pm 0.015 \cdot 10^{-13} \; \text{s}

\tau_{\tau} = 2.7 \cdot 10^{-13} \; \text{s} - ref. 6
\tau_{\tau} = 2.935 \pm 0.031 \cdot 10^{-13} \; \text{s} - ref. 7
\tau_{\tau} = 2.937 \pm 0.027 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.978 \pm 0.011 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.969 \pm 0.036 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.974 \pm 0.038 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.884 \pm 0.056 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.919 \pm 0.058 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.935 \pm 0.031 \cdot 10^{-13} \; \text{s}
\tau_{\tau} = 2.937 \pm 0.027 \cdot 10^{-13} \; \text{s}
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Reference:
http://en.wikipedia.org/wiki/Tau_lepton"
http://hal.in2p3.fr/docs/00/00/47/75/PDF/democrite-00001181.pdf"
http://www.slac.stanford.edu/pubs/slacpubs/9750/slac-pub-9918.pdf"
http://arxiv.org/PS_cache/hep-ex/pdf/9710/9710026v1.pdf"
http://courses.washington.edu/phys55x/Physics%20557_lec9.htm"
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-2446.pdf"
http://hal.archives-ouvertes.fr/docs/00/00/50/28/PDF/democrite-00001434.pdf"

 

[Hawkwind]

According to my understanding, there are actually two tau lifetimes depending on which decay channel the tau lepton follows, this is called the 'branching ratio':

Usually, "lifetime" uniquely denotes a measurable quantity: you have a large number of samples and measure the time until a certain fraction of the original size remains.
Theoretically, you calculate this quantity by taking into account all possible decay channels because each of them contributes to the lifetime.
The decay probabilities for the various channels are rather expressed via "branching ratios".

 

[Orion1]

How do I modify the equation to account for all possible decay channels?
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[Hawkwind]

How do I modify the equation to account for all possible decay channels?
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That's a nontrivial task. In principle, you have to calculate the probability amplitude for each channel separately up to the desired order of accuracy. You have to add them up, square the sum and make the usual integration over the phase space of the various final states. You will get something like the total decay rate; the lifetime being invers proportional to the decay rate.

In principle, you would proceed like for instance indicated here
http://rjs.phys.uvic.ca/sites/rjs.phys.uvic.ca/files/lec6.pdf

Do you know the Feynman rules of electroweak theory ?
Otherwise, you are hopelessly lost (imho).

 

[Parlyne]

That's a nontrivial task. In principle, you have to calculate the probability amplitude for each channel separately up to the desired order of accuracy. You have to add them up, square the sum and make the usual integration over the phase space of the various final states. You will get something like the total decay rate; the lifetime being invers proportional to the decay rate.

In principle, you would proceed like for instance indicated here
http://rjs.phys.uvic.ca/sites/rjs.phys.uvic.ca/files/lec6.pdf

Do you know the Feynman rules of electroweak theory ?
Otherwise, you are hopelessly lost (imho).

A small correction - you only need to sum amplitudes for processes with the same final state particles. If the final states contain different particles, you only need to sum decay widths after the squaring and integration, as there can be no interference.

 

[Hawkwind]

A small correction - you only need to sum amplitudes for processes with the same final state particles. If the final states contain different particles, you only need to sum decay widths after the squaring and integration, as there can be no interference.

You are completely right - thanks for your remark.

 

[lpetrich]

That discrepancy is, in part, because a tau lepton is massive enough to decay into hadrons as well as into electrons or muons. I went to the Particle Data Group to find the decay rates of tau leptons into electrons, muons, and hadrons.

The tau's mass, 1777 MeV, is barely below the lightest charmed particle's mass, 1870 MeV (D+), so a tau decaying into quarks would make only up-down or up-strange.

I'll use Orion1's electron->massless decay rate and particle-mass correction as a base for my numbers.

Electron-only mean life (decay into e, nu-e, nu-tau, possibly also photon):
Calculated: 1.63*10^-12 s
Observed: 1.61*10^-12 s

Muon-only mean life (decay into mu, nu-mu, nu-tau, possibly also photon):
Calculated: 1.67*10^-12 s
Observed: 1.64*10^-12 s

Hadron-only mean life (decay into quarks and nu-tau):
Calculated: 5.43*10^-13 s
Observed: 4.51*10^-13 s

The decay rate into quarks is much quicker than that into each flavor of lepton because quarks come in three colors.

I didn't bother to try to untangle the hadron modes to add up modes with different final strangeness. But if anyone wants to try, here is my prediction from a simple quark model:
up+down (strangeness 0): 95%
up+strange (strangeness 1): 5%

 

[Vanadium 50]

The problem here is that Orion didn't do any calculation - he Wikiscraped a formula, and then started blindly plugging numbers in. We shouldn't be surprised that the results are nonsensical. The procedure itself was nonsensical.

The calculation in question is for \frac{1}{\Gamma(l \rightarrow l_0 + \nu + \overline{\nu})} where l indicates a lepton and l_0 an imaginary massless lepton. For a muon, this partial width is close to the total width, because the dominant decay is to the almost massless electron.

For the other two leptons, this is not true: there is no lighter lepton than the electron, and there is no single dominant decay of the tau. That's how I know Orion1 didn't do any calculation - he didn't even set up the problem, much less calculate anything.

 

[Orion1]

Is it possible to derive the branching ratios based on theory rather than experiment, an 'a priori' approach to Electroweak Theory, without using experimentally determined lifetime values?

Experimentally determined branching ratio:
B_{\tau} = \frac{\tau_{\tau}}{\tau_{\mu}} \left( \frac{m_{\tau}^5}{m_{\mu}^5} \right) = 0.1797

Using experimentally determined lifetime values to predict a theoretical lifetime value seems to be circular logic 'petitio principii' to me, and still does not seem to prove 'ab initio' or first theoretical principle, and does not seem to rely on basic and established laws of nature without additional assumptions or special models.
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[Parlyne]

Is it possible to derive the branching ratios based on theory rather than experiment, an 'a priori' approach to Electroweak Theory, without using experimentally determined lifetime values?

Experimentally determined branching ratio:
B_{\tau} = \frac{\tau_{\tau}}{\tau_{\mu}} \left( \frac{m_{\tau}^5}{m_{\mu}^5} \right) = 0.1797

Using experimentally determined lifetime values to predict a theoretical lifetime value seems to be circular logic 'petitio principii' to me, and still does not seem to prove 'ab initio' or first theoretical principle, and does not seem to rely on basic and established laws of nature without additional assumptions or special models.
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At some level, you will always need to include the results of some sort of experimental observation. The standard model of particle physics has around 20 parameters (mostly particle masses) that are not (at least, as yet) derivable from any underlying physics. That said, taking these as given, it is possible to predict the branchings of a particular particle's decay from theory without needing to include any experimental lifetime measurements. To do this, you need to calculate the decay width (\Gamma) for every possible decay channel (which is a standard QFT sort of calculation). Then, the branching to channel, i, is
B_i = \frac{\Gamma_i}{\sum_j \Gamma_j}

I will add that these sorts of calculations get a bit more complicated when any of the particles involved is hadronic, as the corrections involved with the hadronic bound states are not something that can be dealt with analytically.

 

[lpetrich]

Experimentally determined branching ratio:
B_{\tau} = \frac{\tau_{\tau}}{\tau_{\mu}} \left( \frac{m_{\tau}^5}{m_{\mu}^5} \right) = 0.1797

You should look at my calculation. You handled only tau -> one flavor decays, while I handled tau -> muon, tau -> electron, and tau -> quarks/hadrons. You have to consider every decay mode that you can, though you may end up dismissing many of them as too slow to contribute much. Like tau -> muon + neutrinos + lots of photons. Each photon multiplies the decay rate by (alpha), which is about 1/137, and decays with many photons are thus VERY slow.

Using experimentally determined lifetime values to predict a theoretical lifetime value seems to be circular logic 'petitio principii' to me, and still does not seem to prove 'ab initio' or first theoretical principle, and does not seem to rely on basic and established laws of nature without additional assumptions or special models.
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Experiment is the ultimate source of the parameter values. However, an important test of a theory is whether or not

(Number of parameters in it) < (Number of observational and experimental results that it purportedly explains)

To a first approximation, the stronger the inequality the better.

 

[Orion1]

Are these equations closer to a QFT calculation?

QFT branching channel:
B_i = \frac{\Gamma_i}{\sum_j \Gamma_j}

Integration via substitution:
\tau_{i} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{i}^5} B_{i} \Delta_{m} \Delta_{W} \Delta_{\gamma} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{i}^5} \left( \frac{\Gamma_i}{\sum_j \Gamma_j} \right) \Delta_{m} \Delta_{W} \Delta_{\gamma}

QFT branching channel lifetime:
\boxed{\tau_{i} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{i}^5} \left( \frac{\Gamma_i}{\sum_j \Gamma_j} \right) \Delta_{m} \Delta_{W} \Delta_{\gamma}}

And what about the correction terms of \Delta_{m} \Delta_{W} \Delta_{\gamma} with respect to both i and j?
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Reference:
http://en.wikipedia.org/wiki/Quantum_field_theory"