Decay width

The decay width is a particularly in the nuclear and particle physics used measured variable , from which the life of short-lived particles states ( resonances ) can be determined. “Width” refers to the half- width of the relevant maximum ( peak ) in the graphically displayed excitation function ( cross section as a function of the center of gravity energy ). The width thus directly shows the energy uncertainty of the state. The smaller the (decay) width of a peak, the longer the lifetime of the corresponding particle state and vice versa. According to quantum mechanics , an unstable state cannot have any sharply defined energy; only states that do not change can be eigenstates of the Hamilton operator and thus have fixed energies. ${\displaystyle \Gamma }$

The decay width has the dimension of an energy and is z. B. specified in electron volts .

Decay constant and lifetime

According to the law of decay, there is a number N of identical, unstable particles

${\displaystyle -{\frac {\mathrm {d} N}{\mathrm {d} t}}=-{\dot {N}}=\lambda \cdot N,}$

so that at time t :

${\displaystyle N(t)=N_{0}\cdot \mathrm {e} ^{-\lambda t}.}$

With

• the number of particles at the time ${\displaystyle N_{0}}$ ${\displaystyle t=0}$
• the decay constant  λ (decay probability per unit of time). The dimension of the decay constant is therefore that of an inverse time, the usual unit  s ⁻¹ . The reciprocal of the decay constant is the lifetime : ${\displaystyle \tau }$ ${\displaystyle \tau ={\frac {1}{\lambda }}}$

Total decay width

The total decay width of a resonance, i.e. H. of a short-lived particle can be determined by plotting the measured cross-section over the center of gravity energy ( excitation function ) and adapting a Breit-Wigner curve to the measured values using the least squares method ( compensation calculation ). ${\displaystyle \Gamma }$

According to one of the forms of the energy-time uncertainty relation , as it emerges from the scattering theory on a potential well, the measured energy uncertainty is proportional to the decay constant, i.e. inversely proportional to the lifetime: ${\displaystyle \Gamma }$

${\displaystyle {\begin{aligned}\Gamma &=\hbar \cdot \lambda \\&=\hbar \cdot {\frac {1}{\tau }}\end{aligned}}}$

with the reduced Planck's constant . ${\displaystyle \hbar }$

To put it clearly: the energy of a state is the more precisely defined (or its energy uncertainty the smaller) the longer its lifespan.

The mean lifetime of a particle state can therefore be determined from the measurement of its total decay width. 1 MeV corresponds to a lifetime of 6.58 · 10 ⁻²²  seconds, i.e. a half-life of 4.56 · 10 ⁻²²  seconds.

Partial decay width

In elementary particle physics, mostly only the decay of a single particle into different final states with different decay products is of interest. Here, too, there is a blurring of the energy released corresponding to the service life.

Since most unstable particles can decay into different final states, one defines a partial width for each decay channel  i . The sum of all partial widths is the total decay width described above : ${\displaystyle \Gamma _{i}}$

${\displaystyle \Gamma =\Gamma _{\mathrm {tot} }=\sum _{i=1}^{n}\Gamma _{i}}$.

The same applies to them as above

${\displaystyle \Gamma ={\frac {\hbar }{\tau }}}$

or in natural units ( ) ${\displaystyle \hbar =c=1}$

${\displaystyle \Gamma ={\frac {1}{\tau }}}$

All possible decay channels and their partial widths must also be taken into account in the theoretical calculation of the service life.

For the sake of clarity, a partial width is sometimes expressed by the reciprocal of the relevant partial decay constant , i.e. by a partial mean life . But this is a fictitious , unobservable quantity; the real decay always takes place according to the total decay constant.

Branching ratio

The branching ratio (English: branching fraction or branching ratio )

${\displaystyle B_{i}={\frac {\Gamma _{i}}{\Gamma }}}$

describes the probability with which a particle state decays into a certain final state.

For example, the positively charged pion ( ) decays into a positively charged muon and the associated muon neutrino in 99.9877 percent of all cases , and into a positron and an electron neutrino in only 0.0123 percent of cases . In addition, there are other decay channels, which, however, occur even more rarely with branching ratios in the order of magnitude of 10 ⁻⁴ … 10 ⁻⁹ . ${\displaystyle \pi ^{+}}$

Calculation of the decay width

The partial decay widths can be calculated from the matrix element of the scatter matrix of the corresponding process and the phase space factor according to Fermi's golden rule . One finds in the rest system of the decaying particle

${\displaystyle \mathrm {d} \Gamma ={\frac {1}{2m}}|{\mathcal {M}}|^{2}\mathrm {d} \Pi _{\mathrm {LIPS} }}$

With

• the mass of the particle , ${\displaystyle m}$
• the matrix element and ${\displaystyle {\mathcal {M}}}$
• the Lorentz invariant phase space element

${\displaystyle \mathrm {d} \Pi _{\mathrm {LIPS} }=\prod _{j}{\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{(2\pi )^{3}}}{\frac {1}{2E_{j}}}(2\pi )^{4}\delta ^{(4)}\left(\sum _{j}p_{j}^{\mu }\right),}$

where the product runs over all particles in the final state and are their energies and their momentum. ${\displaystyle j}$ ${\displaystyle E_{j}}$ ${\displaystyle p_{j}}$

literature

• J. Bleck-Neuhaus: Elementary Particles . 2nd edition, Springer 2012, ISBN 978-3-642-32578-6

Individual evidence

1. ↑ John M. Blatt, Viktor Weisskopf : Theoretische Kernphysik . 1st edition. BGTeubner, Leipzig 1959, p. 354 ff .
2. ^ Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 61-62 (English).