Hadronic Parity Violation in Few-Nucleon Systems

The weak interaction between quarks induces a parity-violating component in the interactions between nucleons, which is typically suppressed by a factor of ≈10-7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\approx 10^{-7}}$$\end{document} compared to the dominant parity-conserving part. Because of the short range of the weak interactions, it provides a unique probe of the strong dynamics that confine quarks into nucleons. An experimental program to map out this weak component of the nuclear force is underway at a number of facilities, including the Spallation Neutron Source at Oak Ridge National Laboratory. The corresponding observables are related to few-nucleon processes at very low energies, at which pionless effective field theory provides a reliable and model-independent theoretical approach to hadronic parity violation. Results in two- and three-nucleon systems, the role of parity-violating three-nucleon forces, and possible extensions to other few-nucleon systems are discussed.

erations. The DDH potential has been combined with a number of PC potentials; see, e.g., Refs. [1,24,29] for applications. More recently, starting with the work of Refs. [20,27], PV nucleon-nucleon interactions have been formulated and applied in the framework of effective field theory (EFT), with a comprehensive analysis of PV interactions in the so-called pionless and chiral EFTs performed in Ref. [35]. The advantage of the EFT approach is that it is model independent and that it provides a framework to consistently treat parity-conserving and parity-violating two-, three-, and few-nucleon interactions, as well as external currents.

Parity Violation in Pionless EFT
Most of the experimentally accessible processes involve very low energies well below the pion mass. At these energies, it is possible to formulate a so-called "pionless EFT" (EFT(π /)) solely in terms of nucleons as active degrees of freedom. All other hadrons, including pions, are integrated out and their contributions are taken into account through the values of the so-called low-energy couplings (LECs) of the nucleon contact terms. See, e.g., Refs. [3,4,23] for reviews. In the PV sector, the leading-order Lagrangian consists of five independent terms [15,22]; in the formalism including auxiliary dibaryon fields for NN S-wave states it takes the form [30] Ob with O some spin-isospin-operator, and I = diag(1, 1, −2). 1 The LECs g are unknown parameters in the EFT framework and have to be determined from a calculation in terms of the underlying standard model degrees of freedom or from comparison with experiment. Once they have been determined, the Lagrangian of Eq. (1) together with the corresponding Lagrangian in the PC sector can be used to predict PV observables. At present no theoretical determination of the LECs exists; therefore, the extraction from comparison with experimental data seems more feasible. This requires the consistent calculation of at least five PV observables within the pionless EFT framework. Results of an ongoing program with this goal are described in the following. While different conventions have been used in the individual calculations, all results presented here are adjusted to the conventions of Ref. [17].

Two-Nucleon Systems
The longitudinal asymmetry in the scattering of polarized nucleons off an unpolarized nucleon target is defined as where σ ± is the total cross section for the scattering of nucleons with helicity ±1. At leading order in EFT(π /) the asymmetries for → n n, → p p, and → n p scattering are given by [19,22,35] where M is the nucleon mass and p the nucleon momentum in the center-of-mass frame. Coulomb interactions are not considered in the pp result of Eq. (3). As shown in Ref. [22], they amount to a correction of about 3% or less at the energies and angular ranges that have been considered experimentally. The result for np scattering is related to another observable, the spin rotation angle in the transmission of a perpendicularly polarized neutron beam through a proton target. Up to next-to-leading order (NLO), the result for the rotation angle per unit length is [17] 1 ρ where ρ is the target density, γ t/s are the poles in the NN scattering amplitudes in the 3 S 1 and 1 S 0 channels, respectively, and Z t/s = 1 1−γ t,s r t/s , with r t/s the effective ranges in the corresponding channels. In addition to reactions involving only nucleons, hadronic parity violation can also be studied in processes involving external photons. Considerable experimental effort has been devoted to measuring the photon angular asymmetry A γ in polarized neutron capture, → n p → dγ , with a currently ongoing experiment at Oak Ridge's Spallation Neutron Source [14]. The angular asymmetry is defined through with θ the angle between the spin of the incoming neutron and the direction of the outgoing photon. In EFT(π /), A γ has been calculated in Refs. [26,30]. Adjusted to the conventions used here, the result at LO is where κ 1 is the anomalous nucleon isovector magnetic moment and a s the scattering length in the 1 S 0 channel. Another PV observable can be determined from the induced circular polarization in the capture of unpolarized neutrons, np → d → γ . The polarization is defined by where σ ± is the total cross section for outgoing photons with helicity ±1. The EFT(π /) result at threshold is [30] As seen by comparing Eqs. (6) and (8), the observables A γ and P γ are independent and provide complementary information on the PV couplings. Experimentally, it might be more feasible to measure the asymmetry A γ L in the inverse reaction, → γ d → np, which is equal to P γ for exactly reversed kinematics. A model calculation found strong dependence of A γ L on the choice of which PC two-nucleon model is used in the evaluation [28]. The measurement of A γ L is currently being considered as a flagship experiment for a possible upgrade of the HIGS facility at the Triangle Universities Nuclear Laboratory.

Three-Nucleon Systems
One of the advantages of the EFT approach is the ability to estimate the relative strengths of two-and threenucleon (3N) interactions based on power counting. In the PC sector, naive application of the power counting rules predicts that the leading 3N interactions are suppressed and start to contribute at next-to-next-to-leading order (NNLO). However, it was shown that for the case of nd scattering in the 2 S 1 2 channel a 3N interaction is needed at LO to remove cutoff dependence in the results for the scattering amplitude [5,6]. In the PV sector, 3N interactions are similarly predicted to first appear at NNLO. However, the unexpected "promotion" of the 3N terms in the PC sector raises the question whether PV 3N interactions have to be considered at lower orders than naively expected. Reference [16] showed by analyzing N d scattering that no PV 3N interaction terms are required at LO and NLO for the renormalization of the scattering amplitude. This implies that up to an estimated accuracy of ≈ 10% even three-and other few-nucleon observables can be analyzed in terms of PV two-nucleon interactions, and no additional terms have to be taken into account.
The theoretical considerations of Ref. [16] are confirmed by the calculation of PV nd scattering in Refs. [17,32]. The results for the scattering amplitude can be related to the spin rotation angle of polarized neutrons in a deuteron target. The rotation angle per unit length up to NLO is given by [17] 1 ρ including theoretical error estimates. Reference [32] also contains a LO result for the longitudinal asymmetry in → n d scattering.

Other Few-Body Systems
A number of further few-body systems provide the opportunity to study hadronic parity violation. Measurement of a PV angular asymmetry in the charge-exchange reaction → n + 3 He → 3 H + p is planned at the SNS in the near future [8]. Model and hybrid calculations of the asymmetry exist [33,18], but a consistent EFT calculation has not been performed to date. Experimental results exist for the longitudinal asymmetry in → p 4 He scattering [21] and n 4 He spin rotation [31]. Corresponding model calculations can be found in Refs. [25,13] and [2,12], respectively. Consistent EFT calculations of these observables are desirable, and the necessary few-body techniques continue to be developed, putting this goal within reach.

Conclusions and Outlook
Because of its origin in the interplay of short-range weak and strong interactions at low energy, hadronic parity violation offers a unique probe of nonperturbative QCD. The use of EFT(π /) provides a unified framework to treat PC and PV interactions as well as external currents on the same footing. In addition, the EFT power counting provides a method to estimate theoretical errors. Up to NLO, PV NN interactions in EFT(π /) can be parameterized in terms of five independent couplings, corresponding to five S-P wave transitions in the two-nucleon system. These couplings are not determined within the EFT, but can be determined from comparison with experimental results. The presented results form part of a comprehensive study of PV observables in few-nucleon systems that will allow a consistent and model-independent analysis and interpretation of hadronic parity violation. A first preliminary determination of a PV pion-nucleon coupling from lattice QCD can be found in Ref. [34]. This presents an important step in the determination of the PV couplings in terms of the underlying framework of the standard model. Additional EFT(π /) calculations of PV observables in fewnucleon systems will further constrain the unknown couplings and will help in improving our understanding of hadronic parity violation.