S matrix and decaying particles

In summary, the S matrix formalism can't predict that a new particle is created at the time of the interaction. However, by starting with a neutron in the in state, you get the equivalent decay rate.
  • #1
pscplaton
6
1
Hi All,

The S-matrix is defined as the inner product of the in- and out-states, as in Eq. (3.2.1) in Weinberg's QFT vol 1:
[itex]S_{βα}=(Ψ−β,Ψ+α)[/itex]

[itex]\Psi_{±α}[/itex] are the eigenstates of the full Hamiltonian with a non-zero interaction term.

Can [itex]\alpha[/itex] describes a neutron ? Since it is not stable, it is not an eigenstate of the full Hamiltonian, so it should not... But then how can you compute the decay rate of the neutron within the S matrix formalism ?

I am also wondering how to take neutrinon into account. Supposing [itex]\alpha[/itex] can describe a neutron,
the neutron may decays as: n → p + [itex]e^−[/itex] + [itex]\overline{\nu}_e[/itex]. But [itex]\overline{\nu}_e[/itex] is not an eigenstate of the full Hamiltonian ; I would rather use the mass eigenstates [itex]\nu_1, \nu_2, \nu_3[/itex]. But then it means that the S matrix formalism is unable to predict that it is really a [itex]\nu_e[/itex] that is created at the time of the interaction, which could lead to false predictions... What do you think ?
 
Physics news on Phys.org
  • #2
If you have access to Peskin-Schröder, there is a discussion on unstable particles in chapter 7 in relation to the discussion of the optical theorem.

When it regards neutrinos, you are right. The states should really be the mass eigenstates. However, they remain coherent over pretty large distances due to the very small mass splittings, resulting in neutrino oscillations. If you do not measure the neutrino, you can simply do the total decay rate as a sum of the decay rates into the different mass eigenstates. The result will be essentially ##\Gamma = (|U_{e1}|^2+|U_{e2}|^2+|U_{e3}|^2)\Gamma_0##, where ##\Gamma_0## is the decay rate you would have to one neutrino state if there was no mixing and ##U## is the PMNS matrix. Since the PMNS (as far as we know) is unitary, the sum of the squared PMNS matrix elements equals 1.
 
  • #3
Strictly speaking, only stable particles are allowed in the in and out states. The neutron then appears as a resonance in scattering of a neutrino, electron, and proton. The decay rate is identified with the imaginary part of the location of the pole in the scattering amplitude.

Happily, it turns out that this decay rate is equivalent to what you get by starting with a neutron in the in state, even though you are not really allowed to do this.

Srednicki's text also has a discussion of this.
 
  • #4
Thanks for your replies. I will have a look to these references
 

What is the S matrix?

The S matrix, also known as the scattering matrix, is a mathematical tool used in quantum mechanics to describe the interactions between particles. It is a matrix of complex numbers that relates the initial and final states of a system undergoing a scattering process.

How is the S matrix used in particle physics?

The S matrix is used to calculate the probability amplitudes for particles to interact and scatter off of each other. It can also be used to determine the properties of particles, such as their mass and spin, by analyzing the patterns in the scattering data.

What are decaying particles?

Decaying particles are unstable particles that have a finite lifetime and eventually break down into other particles. This process is known as particle decay and can be observed in high-energy collisions or in natural radioactive processes.

How does the S matrix describe decaying particles?

The S matrix takes into account the decaying nature of particles by including a complex phase factor in the calculation of probability amplitudes. This factor accounts for the decay rate of the particle and allows for the prediction of the final state of the decaying particle and its decay products.

What is the relationship between the S matrix and unitarity?

Unitarity is a fundamental principle in quantum mechanics that states that the total probability of all possible outcomes of a process must equal one. The S matrix satisfies this principle, as it conserves probability and ensures that the sum of all possible outcomes is always equal to one.

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
1
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
5
Views
4K
  • High Energy, Nuclear, Particle Physics
2
Replies
49
Views
9K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Quantum Physics
Replies
4
Views
4K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Quantum Physics
Replies
1
Views
2K
Back
Top