Two-pion Exchange Contributions Research Articles

Nucleon-nucleon scattering up to next-to-next-to-leading order in manifestly Lorentz-invariant chiral effective field theory: Peripheral phases

We study the nucleon-nucleon interaction up to next-to-next-to-leading order using time-ordered perturbation theory in the framework of manifestly Lorentz-invariant chiral effective field theory. We present the two-pion exchange contribution at one-loop level, which is consistent with the corresponding non-relativistic expressions in the large-nucleon-mass limit. Using the Born series truncated at one-loop order, we calculate the phase shifts and mixing angles of the partial waves with the angular momentum $l\geq 2$. Comparing with the results of non-relativistic formulation, we find an improved description of the phase shifts for some $D$ waves such as the $^3D_3$ one. For the other partial waves, both approaches show the globally similar results.

Nonperturbative two-pion exchange contributions to the nucleon-nucleon interaction in covariant baryon chiral perturbation theory

We calculate the nonperturbative two-pion exchange (TPE) contributions to the $NN$ interaction in covariant baryon chiral perturbation theory. We study how the nonperturbative resummation affects the $NN$ phase shifts for partial waves with $J \geq 3$ and $L \leq 6$. No significant differences are observed between the nonperturbative phase shifts and perturbative ones for most partial waves except for $^3D_3$, for which the nonperturbative resummation greatly improves the description of the phase shifts. However, a significant cutoff dependence is found for this partial wave and a reasonable description of the phase shifts can only be obtained with a particular cutoff. Furthermore, we compare the so-obtained nonperturbative phase shifts with those obtained in the heavy baryon chiral perturbation theory. We show that the contributions from relativistic nonperturbative TPE are more moderate than those from the nonrelativistic TPE obtained in the dimensional regularization scheme. A proper convergence pattern is observed for most of the partial waves studied except for $^3F_3$, $^3F_4$, and $^3H_6$, for which the subleading TPE contributions are a bit strong. We find that for $H$ and $I$ partial waves, the OPE alone can already describe the phase shifts reasonably well.

Two-pion exchange contributions to the nucleon-nucleon interaction in covariant baryon chiral perturbation theory

Employing the covariant baryon chiral perturbation theory, we calculate the leading and next-to-leading order two-pion exchange (TPE) contributions to $NN$ interaction up to order $O(p^3)$. We compare the so-obtained $NN$ phase shifts with $2\leq L\leq 6$ and mixing angles with $2\leq J\leq6$ with those obtained in the nonrelativistic baryon chiral perturbation theory, which allows us to check the relativistic corrections to the medium-range part of $NN$ interactions. We show that the contributions of relativistic TPE are more moderate than those of the nonrelativistic TPE. The relativistic corrections play an important role in F-waves especially the $^3\text{F}_2$ partial wave. Moreover, the relativistic results seem to converge faster than the nonrelativistic results in almost all the partial waves studied in the present work, consistent with the studies performed in the one-baryon sector.

Three-nucleon force in chiral effective field theory with explicit Δ (1232) degrees of freedom: Longest-range contributions at fourth order

We analyze the longest-range two-pion exchange contributions to the three-nucleon force at leading-loop order in the framework of heavy-baryon chiral effective field theory with explicit $\Delta(1232)$ degrees of freedom. All relevant low-energy constants which appear in the calculation are determined from pion-nucleon scattering. Comparing our results with the ones obtained in the $\Delta$-less theory at N4LO, we find effects of the $\Delta$ isobar for this particular topology to be rather well represented in terms of resonance saturation of various low-energy constants in the $\Delta$-less approach.

Local chiral potentials with Δ -intermediate states and the structure of light nuclei

We present fully local versions of the minimally non-local nucleon-nucleon potentials constructed in a previous paper [M.\ Piarulli {\it et al.}, Phys.\ Rev.\ C {\bf 91}, 024003 (2015)], and use them in hypersperical-harmonics and quantum Monte Carlo calculations of ground and excited states of $^3$H, $^3$He, $^4$He, $^6$He, and $^6$Li nuclei. The long-range part of these local potentials includes one- and two-pion exchange contributions without and with $\Delta$-isobars in the intermediate states up to order $Q^3$ ($Q$ denotes generically the low momentum scale) in the chiral expansion, while the short-range part consists of contact interactions up to order $Q^4$. The low-energy constants multiplying these contact interactions are fitted to the 2013 Granada database in two different ranges of laboratory energies, either 0--125 MeV or 0--200 MeV, and to the deuteron binding energy and $nn$ singlet scattering length. Fits to these data are performed for three models characterized by long- and short-range cutoffs, $R_{\rm L}$ and $R_{\rm S}$ respectively, ranging from $(R_{\rm L},R_{\rm S})=(1.2,0.8)$ fm down to $(0.8,0.6)$ fm. The long-range (short-range) cutoff regularizes the one- and two-pion exchange (contact) part of the potential.

Minimally non-local nucleon-nucleon potentials with chiral two-pion exchange including Δ’s

A coordinate-space nucleon-nucleon potential is constructed in chiral effective field theory (χ EFT) retaining pions, nucleons and Δ-isobars as explicit degrees of freedom. The calculation of the potential is carried out by including one- and two-pionexchange contributions up to next-to-next-to-leading order (N2LO) and contact interactions up to next-to-next-to-next-to-leading order (N3LO). The low-energy constants multiplying these contact interactions are fitted to the 2013 Granada database in the laboratory-energy range 0–300 MeV.

Precision Nucleon-Nucleon Potential at Fifth Order in the Chiral Expansion.

We present a nucleon-nucleon potential at fifth order in chiral effective field theory. We find a substantial improvement in the description of nucleon-nucleon phase shifts as compared to the fourth-order results utilizing a coordinate-space regularization. This provides clear evidence of the corresponding two-pion exchange contributions with all low-energy constants being determined from pion-nucleon scattering. The fifth-order corrections to nucleon-nucleon observables appear to be of a natural size, which confirms the good convergence of the chiral expansion for nuclear forces. Furthermore, the obtained results provide strong support for the novel way of quantifying the theoretical uncertainty due to the truncation of the chiral expansion proposed by the authors. Our work opens up new perspectives for precision ab initio calculations in few- and many-nucleon systems and is especially relevant for ongoing efforts towards a quantitative understanding of the structure of the three-nucleon force in the framework of chiral effective field theory.

Electron-deuteron scattering based on the Chiral Effective Field Theory

Based on the Chiral Effective Field Theory (ChEFT) dynamical picture of the two-pion exchange (TPE) contributions to the nuclear current operator which appear at higher order chiral expansions were considered. Their role in the electron-deuteron scattering reactions was studied and chiral predictions were compared with those obtained in the conventional framework. Results for cross section and various polarization observables are presented. The bound and scattering states were calculated with five different chiral nucleon-nucleon (NN) potentials which leads to the so-called theoretical uncertainty bands for the predicted results.

Chiral three-nucleon force at N4LO: Longest-range contributions

We derive the sub-subleading two-pion exchange contributions to the three-nucleon force which appear at next-to-next-to-next-to-next-to-leading order in chiral effective field theory. In order to determine the low-energy constants, a complete analysis of pion-nucleon scattering at the subleading-loop order in the heavy-baryon expansion is carried out utilizing the power counting scheme employed in the derivation of the nuclear forces. We discuss the convergence of the chiral expansion for this particular three-nucleon force topology and give the values of the low-energy constants which provide the most realistic description of the three-nucleon force when the chiral expansion is truncated at next-to-next-to-leading order.

Nuclear energy density functional from chiral two-nucleon aaand three-nucleon interactions

An improved density-matrix expansion is used to calculate the nuclear energy density functional from chiral two- and three-nucleon interactions. The two-body interaction comprises long-range one- and two-pion exchange contributions and a set of contact terms contributing up to fourth power in momenta. In addition we employ the leading-order chiral three-nucleon interaction with its parameters c E , c D and c 1,3,4 fixed in calculations of nuclear few-body systems. With this input the nuclear energy density functional is derived to first order in the two- and three-nucleon interaction. We find that the strength functions F ∇(ρ) and F so (ρ) of the surface and spin-orbit terms compare in the relevant density range reasonably with the results of phenomenological Skyrme forces. However, an improved description requires (at least) the treatment of the two-body interaction to second order. This observation is in line with the deficiencies in the nuclear matter equations of state $$\bar E(\rho )$$ that remain in the Hartree-Fock approximation with low-momentum two- and three-nucleon interactions.

Signatures of the chiral two-pion exchange electromagnetic currents in theH2andHe3photodisintegration reactions

The recently derived long-range two-pion exchange (TPE) contributions to the nuclear current operator which appear at next-to-leading order (NLO) of the chiral expansion are used to describe electromagnetic processes. We study their role in the photodisintegration of 2H and 3He and compare our predictions with experimental data. The bound and scattering states are calculated using five different parametrizations of the chiral next-to-next-to-leading order (N2LO) nucleon-nucleon (NN) potential which allows us to estimate the theoretical uncertainty at a given order in the chiral expansion. For some observables the results are very close to the predictions based on the AV18 NN potential and the current operator (partly) consistent with this force. In the most cases, the addition of long-range TPE currents improved the description of the experimental data.

Two-pion exchange electromagnetic current in chiral effective field theory using the method of unitary transformation

Using the method of unitary transformation in combination with chiral effective field theory we derive the two-pion exchange contributions to the two-nucleon electromagnetic current. A formal definition of the current operator in this scheme and the power counting is presented. We discuss the implications of additional unitary transformations that have to be present to ensure the renormalizability of the one-pion exchange current. Further, we give explicit and compact results for the current in coordinate-space.

Two-pion exchange contribution to proton-proton scattering at medium energies

The contribution of the box and crossed two-pion-exchange diagrams to proton-proton scattering at ${90}_{\mathrm{c}.\mathrm{m}.}^{○}$ is calculated in the laboratory momentum range up to 12 GeV/$c$. Relativistic form factors related to the nucleon and pion size and representing the pion source distribution are included at each vertex of the pion-nucleon interaction. These form factors depend on the four-momenta of the exchanged pions and scattering nucleons. Feynman-diagram amplitudes calculated without form factors are checked against those derived from dispersion relations. In this comparison, it was observed that a very short-range part of the crossed diagram, neglected in dispersion-relation calculations of the two-pion-exchange nucleon-nucleon potential, gives a sizable contribution. In the Feynman-diagram calculation with form factors the agreement with measured spin-separated cross sections, as well as amplitudes in the lower part of the energy range considered, is better for pion-nucleon pseudovector vis \`a vis pseudoscalar coupling although potentially important corrections for inelastic effects are lacking. While strengths of the box and crossed diagrams are comparable for laboratory momenta below 2 GeV/$c$, the crossed diagram dominates for larger momenta, largely due to the kinematics of the crossed diagram allowing a smaller momentum transfer in the nucleon center of mass. An important contribution arises from the principal-value part of the integrals which is non-zero when form factors are included. The imaginary part of the amplitude arising from the form factor remains to be understood. It seems that the importance of the exchange of color singlets may extend higher in energy than expected.

Two-pion exchange contributions to the relativisticNNkernel: Peripheral scattering

The relativistic one-boson-exchange model for NN scattering is extended by including two-pion exchange (TPE) contributions in the kernel. We develop the formalism for the evaluation of the TPE diagrams within the relativistic quasipotential approach. The peripheral partial waves in elastic NN scattering are studied within this model. The TPE interactions contain a strongly attractive isoscalar-scalar component which requires a low value of the cutoff parameter: {lambda}=650-800 MeV. With this prescription, the peripheral waves can be reasonably described.

Chiral2πexchange at fourth order and peripheralNNscattering

We calculate the impact of the complete set of two-pion exchange contributions at chiral order four (also known as next-to-next-to-next-to-leading order, N3LO) on peripheral partial waves of nucleon-nucleon scattering. Our calculations are based upon the analytical studies by Kaiser. It turns out that the contribution of order four is substantially smaller than the one of order three, indicating convergence of the chiral expansion. We compare the prediction from chiral pion-exchange with the corresponding one from conventional meson-theory as represented by the Bonn Full Model and find, in general, good agreement. Our calculations provide a sound basis for investigating the issue whether the low-energy constants determined from pi-N lead to reasonable predictions for NN.

Accurate nucleon–nucleon potential based upon chiral perturbation theory

We present an accurate nucleon-nucleon ($NN$) potential based upon chiral effective Lagrangians. The model includes one- and two-pion exchange contributions up to chiral order three. We show that a quantitative fit of the $NN$ $D$-wave phase shifts requires contact terms (which represent the short range force) of order four. Within this framework, the $NN$ phase shifts below 300 MeV lab. energy and the properties of the deuteron are reproduced with high-precision. This chiral $NN$ potential represents a reliable starting point for testing the chiral effective field theory approach in exact few-nucleon and microscopic nuclear many-body calculations. An important implication of the present work is that the chiral $2\pi$ exchange at order four is of crucial interest for future chiral $NN$ potential development.

Peripheral nucleon-nucleon phase shifts and chiral symmetry

Within the one-loop approximation of baryon chiral perturbation theory we calculate all one-pion and two-pion exchange contributions to the nucleon-nucleon interaction. In fact we construct the elastic NN-scattering amplitude up to and including third order in small momenta. The phase shifts with orbital angular momentum L ≥2 and the mixing angles with J ≥ 2 are given parameter free and thus allow for a detailed test of chiral symmetry in the two-nucleon system. We find that for the D-waves the 2π-exchange corrections are too large as compared with empirical phase shifts, signaling the increasing importance of shorter range effects in lower partial waves. For higher partial waves, especially for G-waves, the model-independent 2π-exchange corrections bring the chiral prediction close to empirical NN phase shifts. We propose to use the chiral NN phase shifts with L ≥ 3 as input in a future phase-shift analysis. Furthermore, we compute the irreducible two-pion exchange NN potentials in coordinate space. They turn out to be of van der Waals type, with exponential screening of two-pion mass range.

Two-pion exchange contributions to nuclear charge asymmetry.

In a nonrelativistic formalism we derive a charge-symmetry-breaking (CSB) two-pion exchange interaction ${\mathit{V}}_{\mathit{nn}}$(2\ensuremath{\pi})-${\mathit{V}}_{\mathit{pp}}$(2\ensuremath{\pi}), which includes both CSB vertex corrections and the CSB effect of propagators arising from the mass differences of the intermediate baryons in different charge states. While the former are small, the latter gives a contribution of the sign and scale comparable to the experimental difference in the effective range parameters and the binding energy difference between $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$. \textcopyright{} 1996 The American Physical Society.

Charge symmetry breaking two-pion exchange.

Two-pion exchange (TPE) contribution to the charge symmetry breaking class IV neutron-proton interaction is examined in a potential and coupled channels approach. Based on nonrelativistic \ensuremath{\pi}NN and \ensuremath{\pi}N\ensuremath{\Delta} vertices, a TPE interaction is treated in two ways, as a potential or as a part calculable by the coupled channels method plus a residual potential interaction. A practical parametrization of the TPE potentials is given, which can also be used in the case of class III charge symmetry breaking (CSB) forces as well as for charge symmetric interactions. The results show that below 300 MeV the TPE contribution to CSB in elastic np scattering is insignificant, whereas at higher energies it should not be neglected.

Contribution of charge symmetry breaking forces to energy differences in mirror nuclei

Contributions to energy differences in mirror nuclei and to the scattering length differences a pp − a nn and a nn − a np are evaluated from class III and class IV type charge symmetry breaking (CSB) potentials. We consider ϱ 0-ω mixing π 0-ν mixing, one- and two-pion exchange contributions. Potentials which are consistent with these scattering length differences and with recent CSB p-n elastic scattering data are found to contribute about 150–250 keV to the energy differences in mirror nuclei.