Two-nucleon Operator Research Articles

In-medium k-body reduction of n-body operators

The computational cost of ab initio nuclear structure calculations is rendered particularly acute by the presence of (at least) three-nucleon interactions. This feature becomes especially critical now that many-body methods aim at extending their reach beyond mid-mass nuclei. Consequently, state-of-the-art ab initio calculations are typically performed while approximating three-nucleon interactions in terms of effective, i.e. system-dependent, zero-, one- and two-nucleon operators. While straightforward in doubly closed-shell nuclei, existing approximation methods based on normal-ordering techniques involve either two- and three-body density matrices or a symmetry-breaking one-body density matrix in open-shell systems. In order to avoid such complications, a simple, flexible, universal and accurate approximation technique involving the convolution of the initial operator with a sole symmetry-invariant one-body matrix is presently formulated and tested numerically. Employed with a low-resolution Hamiltonian, the novel approximation method is shown to induce errors below $2-3\%$ across a large range of nuclei, observables and many-body methods.

Normal-ordered k-body approximation in particle-number-breaking theories

The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schr\"odinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schr\"odinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle number conservation is broken (and potentially restored), a particle-number-conserving normal-ordered k-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed.

A neutrinoless double beta decay master formula from effective field theory

We present a master formula describing the neutrinoless-double-beta decay (0νββ) rate induced by lepton-number-violating (LNV) operators up to dimension nine in the Standard Model Effective Field Theory. We provide an end-to-end framework connecting the possibly very high LNV scale to the nuclear scale, through a chain of effective field theories. Starting at the electroweak scale, we integrate out the heavy Standard Model degrees of freedom and we match to an SU(3)c ⊗ U(1)em effective theory. After evolving the resulting effective Lagrangian to the QCD scale, we use chiral perturbation theory to derive the lepton-number-violating chiral Lagrangian. The chiral Lagrangian is used to derive the two-nucleon 0νββ transition operators to leading order in the chiral power counting. Based on renormalization arguments we show that in various cases short-range two-nucleon operators need to be enhanced to leading order. We show that all required nuclear matrix elements can be taken from existing calculations. Our final result is a master formula that describes the 0νββ rate in terms of phase-space factors, nuclear matrix elements, hadronic low-energy constants, QCD evolution factors, and high-energy LNV Wilson coefficients, including all the interference terms. Our master formula can be easily matched to any model where LNV originates at energy scales above the electroweak scale. As an explicit example, we match our formula to the minimal left-right-symmetric model in which contributions of operators of different dimension compete, and we discuss the resulting phenomenology.

Muon capture on the deuteron and3He

We plan to investigate the role of meson exchange currents in the description of the μ − + d → νμ + n + n and μ − + 3 He → νμ + 3 H reactions. They both are treated as the decay of the corresponding muonic atoms, with the muon initially on the lowest K shell. The muon binding energy in these atoms can be safely neglected and in the initial state we deal essentially with the deuteron (or 3 He) and muon at rest. These two reactions are interesting for several reasons. First of all, they offer a testing ground for the nuclear wave functions, which for any nucleon-nucleon (NN) and three-nucleon (3N) forces can be constructed for such light systems with great accuracy. In these reactions few-nucleon weak current operators are an important dynamical ingredient. In the current operators apart from the relatively well known single nucleon contributions, two-nucleon parts (generated by various meson exchanges) play an important role. Their details are not well known and several models should be considered. We present our formalism for dealing with these reactions and a simple method for partial wave decomposition of the two-nucleon operators. The crucial nuclear matrix elements of the corresponding weak current operators will be calculated in the momentum space and using partial wave decomposition. The effect of meson exchanges will be investigated in the energy spectrum of the emitted neutrinos (in the deuteron case) and in the total decay rates for the two reactions. We will employ various models of NN and 3N forces, such as the Bonn B or chiral NNLO potentials. Our results with the single nucleon currents look already very promising and we hope for the improvement in the description of the experimental data, when dominant two-nucleon current operators are included in our framework.

Chiral effective theory predictions for deuteron form factor ratios at low Q2

We use chiral effective theory (χET) to predict the deuteron form factor ratio GC/GQ as well as ratios of deuteron to nucleon form factors. These ratios are calculated to next-to-next-to-leading order. At this order the chiral expansion for the NN isoscalar charge operator (including consistently calculated 1/M corrections) is a parameter-free prediction of the effective theory. Use of this operator in conjunction with NLO and NNLO χET wavefunctions produces results that are consistent with extant experimental data for Q2 < 0.35 GeV2. These wavefunctions predict a deuteron quadrupole moment GQ(Q2 = 0) = 0.278–0.282 fm2—with the variation arising from short-distance contributions to this quantity. The variation is of the same size as the discrepancy between the theoretical result and the experimental value. This motivates the renormalization of GQ via a two-nucleon operator that couples to quadrupole photons. After that renormalization we obtain a robust prediction for the shape of GC/GQ at Q2 < 0.3 GeV2. This allows us to make precise, model-independent predictions for the values of this ratio that will be measured at the lower end of the kinematic range explored at BLAST. We also present results for the ratio GC/GM.

Deuteron matrix elements in chiral effective theory at leading order

We consider matrix elements of two-nucleon operators that arise in chiral effective theories of the two-nucleon system. Generically, the short-distance piece of these operators scales as 1 / r n , with r the relative separation of the two nucleons. We show that, when evaluated between the leading-order wave functions obtained in this effective theory, these two-nucleon operators are independent of the cutoff used to renormalize the two-body problem for n = 1 and 2. However, for n ⩾ 3 general arguments about the short-distance behavior of the leading-order deuteron wave function show that the matrix element will diverge.

Inequalities for low-energy symmetric nuclear matter

Using effective field theory we prove inequalities for the correlations of two-nucleon operators in low-energy symmetric nuclear matter. For physical values of operator coefficients in the effective Lagrangian, the S = 1, I = 0 channel correlations must have the lowest energy and longest correlation length in the two-nucleon sector. This result is valid at nonzero density and temperature.

On the reactions [formula omitted

The total cross section for π +d → pp and the angular distribution and asymmetry parameters for the inverse reaction p p → π + d have been calculated with different models for the pion absorption (and production) operator and several sets of nuclear wave functions. The absorption operator includes both single-nucleon and two-nucleon operators. Fair agreement can be achieved with all empirical parameters at low energies, but significant discrepancies occur at high energies. The results are very sensitive to the model for the nucleon-nucleon interaction used to generate the wave functions.

Nucleon isobars in nuclear structure

The example quoted previously was that of the thermal radiative capture n+p + d+y. In the Chemtob-Rho (1) approach used by Prof. Blin-Stoyle the exchange current correction was evaluated by first deriving a two-nucleon operator. This operator was then sandwiched between two-nucleon wavefunctions to give the final correction. However? a closer examination of the result showed that the A contribution to the two-body operator was important. This term can be illustrated as in Fig. l(a) and contributes about 2.4 % to the total matrix element. The pair and pionic terms shown in Figs. l(b) and l(c) contribute a further 3.8 % and

On the determination of the nuclear pair correlation function from the high energy photo-effect

In accordance with Levinger's predictions, the absorption of a high energy photon frequently results in the simultaneous ejection of a neutron and proton. The present paper provides a theoretical analysis of this direct reaction which starts from first principles and is based on the following assumptions: (i) the impulse and closure approximations are valid, (ii) the photo-nuclear interaction is a sum of two-nucleon operators, and (iii) the pair correlation function is given by ∂ s ( r 1, r 2)|g( r 1 − r 2)| 2 where ∂ s is the shell model correlation function and g( x) contains those correlations directly due to nuclear forces. The cross section is then shown to be proportional to the probability for finding two nucleons with a certain total momentum in a relative S-state, the contributions from P-states being demonstrably small. The further assumptions required to relate the nuclear cross section to the one for photodissociation of deuterium are examined, while final state interactions are treated with an optical model. The theory is compared with the present data and found to be in good agreement. The type of further information required for ascertaining the validity of assumptions (ii) and (iii) is discussed, and a method for acquiring it suggested.