Weak Interaction Hamiltonian V-A Structure Derivation

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SUMMARY

The discussion focuses on deriving the V-A structure of the weak interaction Hamiltonian as presented in the Feynman & Gell-Mann 1958 paper. The key conclusion is that only the vector (V) and axial vector (A) contributions survive due to the properties of the left chiral projection operator, \(P_L\). The participant encounters difficulties in achieving the expected V-A structure after simplifying the Hamiltonian, leading to confusion about the derivation process. The main issue lies in the algebraic manipulation of the terms involving \(C_V\) and \(C_A\).

PREREQUISITES
  • Understanding of quantum field theory concepts, particularly weak interactions.
  • Familiarity with the properties of chiral projection operators, specifically \(P_L\) and \(P_R\).
  • Knowledge of Dirac matrices and their algebraic properties.
  • Experience with particle physics literature, particularly the Feynman & Gell-Mann 1958 paper.
NEXT STEPS
  • Review the derivation of the weak interaction Hamiltonian in the context of quantum electrodynamics.
  • Study the implications of chiral symmetry breaking in particle physics.
  • Examine the role of the axial vector current in weak interactions.
  • Explore advanced topics in quantum field theory, focusing on the application of the V-A structure in particle decay processes.
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This discussion is beneficial for theoretical physicists, graduate students in particle physics, and researchers focusing on weak interactions and quantum field theory derivations.

deuteron
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the Hamiltonian I end up with is very different than the usual V-A structure, and I don't know why
Hi! I am trying to come to the V-A structure of the weak interaction Hamiltonian, but I am having some issues with it.
In Feynman & Gell-Mann 1958 paper, they argue that each particle field must be projected onto its left chiral component, which means:

$$H=\displaystyle\sum_{i=S,V,T,A,P} C_i (\overline{\psi_n } O_i \psi_p )(\overline{\psi_\nu} O_i \psi_e) \to \displaystyle\sum_i C_i ([\overline{P_L\psi_n}] O_i [P_L\psi_p]) ([\overline{P_L\psi_\nu}] O_i[P_L\psi_e])$$

and then, they argue that since ##P_L^2 = P_L##, and ##P_RP_L=0##,
$$S:\quad P_RP_L=0,
\quad P: \quad P_R\gamma_5 P_L = -P_RP_L = 0,
\quad T:\quad P_R\sigma_{\mu\nu} P_L = \sigma_{\mu\nu} P_RP_L = 0,
\quad V:\quad P_R\gamma_\mu P_L = \gamma_\mu P_LP_L=\gamma_\mu P_L,
\quad A:\quad P_R\gamma_\mu \gamma_5 P_L = \gamma_\mu P_L\gamma_5P_L = -\gamma_\mu P_LP_L=-\gamma_\mu P_L$$
so, only the V and A contributions must survive.

And this is where I start to have a problem, because when I only take the V and A contributions, I get:
$$H =C_V ( \overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e) + C_A(\overline{\psi_p}(-\gamma_\mu P_L) \psi_n)(\overline{\psi_\nu} (-\gamma_\mu P_L) \psi_e)
\\ = C_V ( \overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e) + C_A(\overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e)
\\ = (C_V+C_A) (\overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e)
\\ = \frac {C_V+C_A}4 ( \overline{\psi_p} \gamma_\mu (1-\gamma_5) \psi_n)(\overline{\psi_\nu} \gamma_\mu (1-\gamma_5) \psi_e)
\\ = \frac{C_V+C_A}4 [ \overline{\psi_p} \gamma_\mu \psi_n - \overline{\psi_p}\gamma_\mu \gamma_5 \psi_n ] [ \overline{\psi_\nu} \gamma_\mu \psi_e - \overline{\psi_\nu} \gamma_\mu\gamma_5 \psi_e]
\\ = \frac{C_V+C_A}4 [(\overline{\psi_p}\gamma_\mu \psi_n )(\overline{\psi_\nu} \gamma_\mu \psi_e ) -(\overline{\psi_p}\gamma_\mu \psi_n ) ( \overline{\psi_\nu} \gamma_\mu\gamma_5 \psi_e) + (\overline{\psi_p}\gamma_\mu \gamma_5 \psi_n) ( \overline{\psi_\nu} \gamma_\mu \gamma_5 \psi_e)-(\overline{\psi_p} \gamma_\mu \gamma_5 \psi_n )(\overline{\psi_\nu} \gamma_\mu \psi_e) ]$$

And this does not have the V-A structure, or the term with the ##\lambda## whatsoever. My question is: at which step am I doing something wrong?

Thanks for reading so far :)
 
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