Forbidden beta decay form factors

[ajdin]

Homework Statement

As a part of the EFT course, we were introduced to some recent applications of the EFT to beta decays. The idea is to consider the non-relativistic decomposition and Lee-Yang lagrangian to derive the non-relativistic version of it and to carry on with the computation of the differential decay width. At the leading order, the relevant hadronic matrix elements (known as Fermi and Gamow-Teller matrix elements) are given as:

$M_F \sim <p| \psi_{p}^{\dagger} \psi_{n} |n> = 2m_N j_+ \delta_{j_3 + 1,j_3'} \delta_{J,J'} \delta_{J_z,J_z'},$

and

$<j,j_3',J',J_z'|\psi^{\dagger}_p \sigma^3 \psi_n |j,j_3,J,J_z> = 2m_N j_+ r \delta_{j_3 + 1, j_3'} \delta_{J_z',J_z} C^{J,J_z \otimes 1,0}_{J',J_z},$

$<j,j_3',J',J_z'|\psi^{\dagger}_p \frac{\sigma^+}{\sqrt{2}} \psi_n |j,j_3,J,J_z> = -2m_N j_+ r \delta_{j_3+1,j_3'}\delta_{J_z', J_z+1}C^{J,J_z \otimes 1,1}_{J',J_z+1},$

$<j,j_3',J',J_z'|\psi^{\dagger}_p \frac{\sigma^-}{\sqrt{2}} \psi_n |j,j_3,J,J_z> = 2m_N j_+ r \delta_{j_3+1,j_3'}\delta_{J_z', J_z-1}C^{J,J_z \otimes 1,-1}_{J',J_z-1}$.

These are the matrix elements at the leading order. Now, in order to describe the forbidden beta decays in this formalism, it turns out it is useful to consider following matrix elements

$M_1 \sim <\mathcal{N}'| \psi_p^{\dagger} (\vec{\sigma}\cdot \vec{\nabla}) \psi_n |\mathcal{N}> ,$

$M_2 \sim <\mathcal{N}'| \psi_p^{\dagger} \sigma^i (\vec{\sigma}\cdot \vec{\nabla}) \psi_n |\mathcal{N}>.$

The main question is how to parametrize these matrix elements.

Relevant Equations

$\frac{1}{\sqrt{2}} \psi_p^{\dagger} \sigma^3 \psi_n |J,J_z> \sim \sum_{ \tilde{J} = J-1}^{J+1} |\tilde{J},J_z > C_{\tilde{J},J_z}^{J,J_z \otimes 1,0}$

$\frac{1}{\sqrt{2}} \psi_p^{\dagger} \sigma^{\pm} \psi_n |J,J_z> \sim \mp \sum_{ \tilde{J} = J-1}^{J+1} |\tilde{J},J_z \pm 1 > C_{\tilde{J},J_z \pm 1}^{J,J_z \otimes 1,\pm 1}$

My idea was to consider first the structure of the matrix element and to see if there are any possible constraints that we could use for parametrization. If I am not mistaken, we are dealing with the hadronic decay governed by QCD which conserves parity. Since we have a derivative operator inside the matrix element, which is parity odd, we need to have a parity odd nuclei transition in order to have this requirement satisfied. It turns out that for $L = 1$ (first forbidden decay), we have $\Delta J = 0, 1, 2$ and $\Delta \pi = (-1)^L = -1$, which is in agreement with the previous statement. Also since we have the derivative operator, I think that momentum should explicitly appear on the right hand side of the equation. Unfortunately, I didn't make much more progress in parametrizing the transition beyond these remarks. One thing I tried is to convert the product $\vec{\sigma} \cdot \vec{\nabla}$ into something we have seen for the Fermi and Gamow-Teller transitions. Using $\sigma^{\pm} = \sigma^1 \pm i\sigma^2$ we get:
$$ \vec{\sigma}\cdot \vec{\nabla} = \frac{\sigma^+}{2} (\nabla_1 - i\nabla_2) + \frac{\sigma^-}{2} (\nabla_1 + i\nabla_2) + \sigma^3 \nabla_3,$$
and $M_1$ would then become:
$$ M_1 \sim <\mathcal{N}'| \psi_p^{\dagger} (\vec{\sigma}\cdot \vec{\nabla}) \psi_n |\mathcal{N}> = <\mathcal{N}'| \psi_p^{\dagger} \frac{\sigma^+}{2} (\nabla_1 - i\nabla_2) \psi_n |\mathcal{N}> + <\mathcal{N}'| \psi_p^{\dagger} \frac{\sigma^-}{2} (\nabla_1 + i\nabla_2) \psi_n |\mathcal{N}> + <\mathcal{N}'| \psi_p^{\dagger} \sigma^3 \nabla_3 \psi_n |\mathcal{N}> .$$
The nablas acting on the neutron field would give the momentum component and the terms in $M_1$ could maybe be written as:
$$<\mathcal{N}'| \psi_p^{\dagger} \frac{\sigma^+}{2} (\nabla_1 - i\nabla_2) \psi_n |\mathcal{N}> \sim \frac{1}{2} [f_1(p_1^2) p_{n1} - if_2(p_2^2) p_{n2} ] <\mathcal{N}'|\psi_p^{\dagger} \sigma^+ \psi_n |\mathcal{N}>$$
$$<\mathcal{N}'| \psi_p^{\dagger} \frac{\sigma^-}{2} (\nabla_1 + i\nabla_2) \psi_n |\mathcal{N}> \sim \frac{1}{2} [f_1(p_1^2) p_{n1} +if_2(p_2^2) p_{n2} ] <\mathcal{N}'|\psi_p^{\dagger} \sigma^- \psi_n |\mathcal{N}>$$
$$<\mathcal{N}'| \psi_p^{\dagger} \sigma^3 \nabla_3 \psi_n |\mathcal{N}> \sim f_3(p_3^2) p_{n3} <\mathcal{N}'|\psi_p^{\dagger} \sigma^3 \psi_n |\mathcal{N}>.$$
I would like to hear your opinion and ideas about this problem. Thank you very much for your help!

 

[Nate810]

I think that the approach you are taking is a very good one. In order to parametrize the matrix element, it is important to consider the structure of the matrix element and any possible constraints that it may have due to conservation laws or symmetry principles. In this case, the fact that the derivative operator is parity-odd means that we must have a parity-odd nuclear transition in order for the matrix element to be nonzero. This leads to the constraints you mentioned, namely that for L=1 (first forbidden decay), we have $\Delta J=0,1,2$ and $\Delta \pi = (-1)^L = -1$. In addition, you made some progress in attempting to convert the product $\vec{\sigma}\cdot \vec{\nabla}$ into something similar to the Fermi and Gamow-Teller transitions. I think this is a good idea, since these matrix elements are well known and can provide insight into the structure of the matrix element we are dealing with. It makes sense to express the matrix element as a sum of terms that contain the momentum of the neutron as well as the spin components of the proton and neutron. However, it is not clear to me what the functions $f_1(p_1^2)$, $f_2(p_2^2)$, and $f_3(p_3^2)$ represent. Are these just general functions that parametrize the matrix element? If so, it would be helpful to know what assumptions were made about these functions in order to derive the expression for the matrix element. Overall, I think you are on the right track with this problem and I am looking forward to hearing more about your progress. Good luck!