Chiral effective field theory analysis of hadronic parity violation in few-nucleon systems

Background: Weak interactions between quarks induce a parity-violating (PV) component in the nucleon-nucleon potential, whose effects are currently being studied in a number of experiments involving few-nucleon systems. In the present work, we reconsider the derivation of this PV component within a chiral effective field theory ($\chi$EFT) framework.

Purpose: The objectives of the present work are twofold. The first is to perform a detailed analysis of the PV nucleon-nucleon potential up to next-to-next-to-leading (N2LO) order in the chiral expansion, in particular, by determining the number of independent low-energy constants (LECs) at N2LO. The second objective is to investigate PV effects in a number of few-nucleon observables, including the $\vec{p}$-$p$ longitudinal asymmetry, the neutron spin rotation in $\vec{n}$-$p$ and $\vec{n}$-$d$ scattering, and the longitudinal asymmetry in the ${}^3$He${(\vec{n},p)}^3$H charge-exchange reaction.

Methods: The $\chi$EFT PV potential includes one-pion-exchange, two-pion-exchange, and contact terms as well as $1/M$ ($M$ being the nucleon mass) nonstatic corrections. Dimensional regularization is used to renormalize pion loops. The wave functions for the $A=2$–4 nuclei are obtained by using strong two- and three-body potentials also derived, for consistency, from $\chi$EFT. In the case of the $A=3$–4 systems, the wave functions are computed by expanding on a hyperspherical harmonics functions basis.

Results: We find that the PV potential at N2LO depends on six LECs: the pion-nucleon PV coupling constant $h_{\pi }^1$ and five parameters multiplying contact interactions. An estimate for the range of values of the various LECs is provided by using available experimental data, and these values are used to obtain predictions for the other PV observables.

Conclusions: The $\chi$EFT approach provides a very satisfactory framework to analyze PV effects in few-nucleon systems.

Figure 1

Time-ordered diagrams contributing to the PV potential (only a single time ordering is shown). Nucleons and pions are denoted by solid and dashed lines, respectively. The open (solid) circles represent PC (PV) vertices. Diagrams (h)–(u) are vertex corrections contributing to the OPE potential.

Figure 2

Contours of $h_{\pi }^1$ and $C$ values (in units of ${10}^{-7}$) corresponding to which ${\chi }^2=2$ for the $\vec{p}$-$p$ longitudinal asymmetry. The black solid (red dashed) contour is relative to $\Lambda =500$ (600) MeV.

Figure 3

Vertices entering the PV potential at $\mathrm{N}2\mathrm{LO}$. The solid (dashed) lines represent nucleons (pions). The open (solid) symbols denote PC (PV) vertices. In diagrams (3) and (4) the vertices are tadpole contributions from the $3\pi \mathit{NN}$ and $4\pi$ interaction Hamiltonians, respectively. The open square on a pion line as in diagram (5) represents a $\pi \pi$ vertex coming from the ${\ell }_3$ and ${\ell }_4$ terms in ${\mathcal{L}}_{\pi \pi }^{\left(4\right)}$ and the $\delta m_{\pi }^2$ term. The open square on a nucleon line as in diagram (6) represents insertions from the $c_1$ term in ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ and the $\delta M$ term.

Figure 4

Diagrams describing the $N\to N$ transition amplitude. The open (solid) circles represent contributions owing to PC (PV) $\pi \mathit{NN}$ vertices, while the open square represents the contribution from the $c_1$ term in ${\mathcal{L}}_{\pi N}^{\left(2\right)}$ and the $\delta M$ counterterm. The notation is as in Fig. 3.

Figure 5

Parts of a general diagram with the propagation of nucleons only.

Figure 6

Propagation of a pion in the intermediate state of a generic diagram. Only the pion line is shown.