Nucleon current from quark current

Question

For free neutron decay we consider the effective current-current Lagrangian of quarks and leptons:

\begin{equation} \mathcal{L}=\frac{G_F\cos{\theta_c}}{2\sqrt{2}}[\overline{u}\gamma^\mu(1-\gamma^5)d][\overline{e}\gamma_\mu(1-\gamma^5)\nu_e], \tag{1}\label{Lang} \end{equation} where $u$, $d$, $e$, $\nu_e$ are the quantum field operators of up-quark, down-quark, electron and electron-neutrino respectively. Obviously the tree-level hadronic current $\overline{u}\gamma^\mu(1-\gamma^5)d$ has to be converted to nucleon current. The most general nucleon current written in terms of form-factors is given by, \begin{align} \overline{u}\gamma^\mu(1-\gamma^5)d \rightarrow \overline{\Psi}\tau^+\Big[g_V(q^2)\gamma^\mu+ig_M(q^2)\sigma^{\mu\nu}q_\nu-g_A(q^2)\gamma^\mu\gamma^5+g_P(q^2)q^\mu\gamma^5 \Big]\Psi,\tag{2}\label{nucl-curr} \end{align} where $\Psi^T= (\psi_p, \psi_n)$ is the iso-doublet nucleon field in terms protons and neutrons. $\tau^+$ is raising operator in isospin space. The various momentum dependent form-factors $g_V$, $g_M$, $g_A$ and $g_P$ are defined as the vector, weak magnetism, axial-vector and induced pseudoscalar, respectively. The four-momentum vector $q^\mu=p_f^\mu-p^\mu_i$ is the momentum transfer between initial and final nucleon.

My question is, how do we get the nucleon current from the basic quark current? Can anyone suggest a pedagogical reference of how to get \eqref{nucl-curr}? All I have found is heuristic arguments of how to construct a Lorentz-vector like current from available Lorentz vectors $\gamma^\mu$, $q^\mu$ etc using general symmetry considerations. I know the treatment is quite involved, but any suggestions would be appreciated.

Answer 1

A good question, meaning, sadly, that there is no good answer. The answer is (properly) either in hyper-technical lattice gauge theory, so not pedagogical. Or, else books and reviews of the 1970s, after the triumph of the quark model and the acceptance in principle of QCD, but before the industry of correction computation/estimation got its start: the golden age of hadronization modeling.

As you indicate, in the absence of good quark wave functions for nucleons, all one has is symmetry: Lorentz, parity, CP, flavor (isospin), etc... and systematic tabulation of matrix elements between hadron states of short distance QCD operators, correlating them among dozens and dozens of processes, with a few models/aids such as PCAC, the Goldberger-Treiman relation, etc. Shining examples of these are two books now sometimes sniffed at as dated, which you might, or might not, find in your library:

• Weak Interactions of Leptons and Quarks, by Eugene D. Commins & Philip H. Bucksbaum (Cambridge University Press, 1983) ISBN-13 : 978-1423446637

• Elementary Particles and Their Currents, by Jeremy Bernstein (W. H. Freeman and Co., 1968) ASIN : B0000COC6N

and old QFT texts from the early 80s, expediting the cultural transition to the SM. Typically, neutral current and neutrino scattering reviews, by necessity, had to review the stuff. In any case, what was part of the street slang of that generation is rapidly getting lost, and I am hard pressed to find systematic dictionaries for it.

Today, the absolutely best starting point is

• Dynamics of the Standard Model, by Donoghue, Golowich & Holstein, (Cambridge University Press; 2nd edition, 2014) ISBN-13: 978-0521768672