Interpretation and relevance of current matrix element

[center o bass]

In reading about the axial anomaly I stumbles across a matrix element on the form

$$\langle p,k|j^{\mu 53}|0\rangle. $$

and I have seen similar matrix elements turn up other places. But when matrix elements (correlation functions) of the form

$$ \langle 0 |T \phi(x) \phi(y)|0\rangle $$

have the interpretation of the probability amplitude of the field to create a particle and propagate between x and y, as well as the relevance of being essential in computation of the S-matrix elements; I have not read about the relevance, nor the interpretation of current matrix elements like the one above.

I can guess that their interpretation is something like the amplitude for a current creating particles.. but where are they relevant in terms of calculating physical quantities?

 

[andrien]

Have not you seen such things in say qed.Interaction in this case is -ieψ^-γ_μψA^μ which is also written as j_μA^μ,now if you want to calculate a matrix element then you have to take this interaction between initial and final states(first order).Now A_μ can create or destroy a photon while the current element if written in terms of it's positive and negative frequency part can destroy electron and positron,create positrons and electrons and also can represent scattering.

 

[The_Duck]

Here is one case. When a current is involved in an interaction term in the Lagrangian, you would like to know the amplitude for that current to create/annihilate the various states in the theory, because that tells you the amplitude for those states to participate in the interaction.

An important example is in the weak interaction. Below the W boson mass, the weak interaction can be approximated by a Lagrangian with four-fermion interactions like

$(\bar{e} \gamma_\mu (1 - \gamma_5) \nu_e)(\bar{u} \gamma^\mu (1 - \gamma_5) d)$.

Note that this interaction is a product of two currents. This interaction term will produce pion decay by letting a $\bar{u} d$ quark-antiquark pair turn into an electron and an electron antineutrino. (In the full theory this would be mediated by a W boson, but at low energies we can integrate out the W and we get the above four-fermion interaction).

The amplitude for this decay will be proportional to the matrix element

$\langle e, \bar{\nu} | (\bar{e} \gamma_\mu (1 - \gamma_5) \nu_e)(\bar{u} \gamma^\mu (1 - \gamma_5) d) | \pi \rangle$

where we have supplied appropriate initial and final states. If we make a certain approximation we can simplify this matrix element as the product of two matrix elements of simpler operators:

$\langle e, \bar{\nu} | (\bar{e} \gamma_\mu (1 - \gamma_5) \nu_e)|0 \rangle \langle 0 |(\bar{u} \gamma^\mu (1 - \gamma_5) d) | \pi \rangle$

This is the amplitude for the rightmost current to annihilate the pion initial state, leaving the vacuum, multiplied by the amplitude for the leftmost current to create the electron and antineutrino from the vacuum. This is only an approximation to the original amplitude, because the initial and final states could for instance exchange a photon, and then the amplitude would not factorize in this neat way. But such effects are down by factors of coupling constants and loop factors, so this factorization into a product the matrix elements of two currents is a good approximation.

In this case it is easy to evaluate the leftmost matrix element. The rightmost matrix element depends on the complicated strong-interaction physics that produces the pion bound state, so it is hard to calculate. But it is necessary to compute this matrix element in order to predict the pion decay rate.