Pion Exchange Research Articles

Two-pion exchange as a leading-order contribution in chiral effective field theory

Pion exchange is the central ingredient to nucleon-nucleon interactions used in nuclear structure calculations, and one pion exchange (OPE) enters at leading order in chiral effective field theory. In the $^{2S+1}L_J={^1S_0}$ partial wave, however, OPE and a contact term needed for proper renormalization fail to produce the qualitative, and quantitative, features of the scattering phase shifts. Cutoff variation also revealed a surprisingly low breakdown momentum of about 330 MeV in this partial wave. Here we show that potentials consisting of OPE, two pion exchange (TPE), and a single contact address these problems and yield accurate and renormalization group (RG) invariant phase shifts in the $^1S_0$ partial wave. We demonstrate that a leading-order potential with TPE can be systematically improved by adding a contact quadratic in momenta. For momentum cutoffs below about 500 MeV, the removal of relevant physics from TPE loops needs to be compensated by additional contacts to keep RG invariance. Inclusion of the $\Delta$ isobar degree of freedom in the potential does not change the strong contributions of TPE.

Electromagnetic and strong isospin breaking in light meson masses

We study electromagnetic as well as strong isospin breaking effects in the isospin mass splittings of light pseudoscalar and vector mesons. To this end we employ a coupled system of quark Dyson-Schwinger and meson Bethe-Salpeter equations whose interaction kernels contain gluon, pion and photon exchange interactions. In bound states, QCD-induced isospin breaking is manifest on different levels. On the one hand, a different explicit up- and down-quark mass directly affects the propagators of the constituent quarks. On the other hand, it leads to different interaction kernels within the isospin multiplets. In addition, electromagnetic isospin breaking is induced via a photon exchange diagram. Using the kaon iso-doublet and the charged pion masses as input to determine the up, down and strange quark masses we find for the pion, kaon and rho meson mass splittings different patterns each. In particular, our results provide evidence that the effects from two sources of mass splittings, the different quark masses and the different quark charges, do not add up linearly.

A new effective potential for deuteron

We calculate for the first time the static properties of the deuteron, within the framework of supersymmetric quantum mechanics, analytically. A new effective potential and its partner are derived from a superpotential so that all parameters are fitted by the experimental data. An analytical expression is obtained for the deuteron wave function and contributions of the orthogonal 13 S 1 and 13 D 1 states are determined, explicitly. Compared to one pion exchange, the superpotential produces an electrostatic as well as two pion exchange terms for the potential. The saddle point radius of the potential and the maximum of the wave function are linearly proportional. In comparison with other methods, the approach presented in this paper is a new and extensible symmetry-based approach that, despite its straightforward calculations and explicit analytical expressions, provides a good explanation for two-body effective interactions such as two-nucleon systems and diatomic molecules.

Towards modeling cluster structure of 8Be with chiral interaction

Abstract How the nuclear force behaves in cluster states, in particular those consisting of the α clusters, has been investigated so far, but not yet elucidated. Today the chiral effective field theory (EFT) is established and it would shed new light on the microscopic understanding of the cluster states. We aim to address a possible source of the attraction in the cluster states of 8Be in view of the pion exchange. Namely, we investigate whether the two-pion-exchange interaction acts as a dominant attraction in the α + α system as predicted by a previous work. We describe theoretically the cluster structure of 8Be by the Brink model, for which the effective interaction is designed from the realistic nuclear force derived through the chiral EFT. The two-body matrix elements of the chiral interaction with the local-Gaussian bases are formulated within the approximation of the spin–isospin saturation forming an α particle. Introducing a global prefactor to the chiral interaction phenomenologically, the ground and low-lying excited states of 8Be, the scattering phase shift of the α–α system as well, are satisfactorily depicted. The attraction in the cluster states is found to be stemming from the two-pion-exchange contributions dominantly, along with nonnegligible short-range terms. The present work can be the foundation towards constructing realistic cluster models, by which the cluster states will be revealed microscopically in the next step.

Coupled-channel approach to Tcc+ including three-body effects

A coupled-channel approach is applied to the charged tetraquark state ${T}_{cc}^{+}$ recently discovered by the LHCb Collaboration. The parameters of the interaction are fixed by a fit to the observed line shape in the three-body ${D}^{0}{D}^{0}{\ensuremath{\pi}}^{+}$ channel. Special attention is paid to the three-body dynamics in the ${T}_{cc}^{+}$ due to the finite life time of the ${D}^{*}$. An approach to the ${T}_{cc}^{+}$ is argued to be self-consistent only if both manifestations of the three-body dynamics, the pion exchange between the $D$ and ${D}^{*}$ mesons and the finite ${D}^{*}$ width, are taken into account simultaneously to ensure that three-body unitarity is preserved. This is especially important to precisely extract the pole position in the complex energy plane whose imaginary part is very sensitive to the details of the coupled-channel scheme employed. The ${D}^{0}{D}^{0}$ and ${D}^{0}{D}^{+}$ invariant mass distributions, predicted based on this analysis, are in good agreement with the LHCb data. The low-energy expansion of the ${D}^{*}D$ scattering amplitude is performed and the low-energy constants (the scattering length and effective range) are extracted. The compositeness parameter of the ${T}_{cc}^{+}$ is found to be close to unity, which implies that the ${T}_{cc}^{+}$ is a hadronic molecule generated by the interactions in the ${D}^{*+}{D}^{0}$ and ${D}^{*0}{D}^{+}$ channels. Employing heavy-quark spin symmetry, an isoscalar ${D}^{*}{D}^{*}$ molecular partner of the ${T}_{cc}^{+}$ with ${J}^{P}={1}^{+}$ is predicted under the assumption that the $D{D}^{*}\text{\ensuremath{-}}{D}^{*}{D}^{*}$ coupled-channel effects can be neglected.

Electric dipole moment of light nuclei in chiral effective field theory

CP-violating interactions at quark level generate CP-violating nuclear interactions and currents, which could be revealed by looking at the presence of a permanent nuclear electric dipole moment. Within the framework of chiral effective field theory, we discuss the derivation of the CP-violating nuclear potential up to next-to-next-to leading order (N2LO) and the preliminary results for the charge operator up to next-to leading order (NLO). Moreover, we introduce some renormalization argument which indicates that we need to promote the short-distance operator to the leading order (LO) in order to reabsorb the divergences generated by the one pion exchange. Finally, we present some selected numerical results for the electric dipole moments of 2H, 3He and 3H discussing the systematic errors introduced by the truncation of the chiral expansion.

Investigation of the Σ0 hyperon production in p(3.5GeV)+p collisions

The production mechanism of the Σ0 was investigated via the exclusive reaction pp → pK+Σ0 at a beam energy 3.5 GeV with the HADES detector. The preliminary value of the extracted total production cross section is found to be σ(pK+Σ0)[μb] = 17.7 ± 1.7(stat) ± 1.6(syst). The dynamics of the reaction pp → pK+Σ0 was investigated by studying the angular distributions in the center-of-mass, Gottfried-Jackson and helicity frames. The angular distributions in the center-of-mass frame supports the pion exchange mechanism. Furthermore, the helicity angular distributions are highly non-isotropic, which is a clear indication of resonant production component.

Experimental Tests and Interpretations of the X(2900)

We examine whether the LHCb vector udc¯s¯ state X(2900) can be interpreted as a kinematical cusp effect arising from D¯⁎K⁎ and D¯1K interactions. The production amplitude is modelled as a triangle diagram with hadronic final state interactions. A satisfactory fit to the Dalitz plot projection is obtained that leverages the singularities of the triangle diagram without the need for D¯K resonances. A somewhat better fit is obtained if the final state interactions are strong enough to generate resonances, although the evidence in favour of this scenario is not conclusive. We also make predictions for the production and decays of X(2900) states, and their possible charged partners, in B+ and B0 decays, considering a number of competing models for the states, including triangle diagrams mediated by quark exchange or pion exchange, and resonance scenarios including molecules and tetraquarks. Assuming only isospin symmetry and the dominance of colour-favoured weak decays, we find characteristic differences in the predictions of the different models. Future experimental studies can therefore discriminate among the competing interpretations for the states.

Heavy Hadronic Molecules Coupled with Multiquark States

We investigate the hidden-charm pentaquarks as a \(\varLambda _c{\bar{D}}^{(*)}-\varSigma _c^{(*)}{\bar{D}}^{(*)}\) hadronic molecule coupling to a compact five-quark state. The coupling to the compact state leads to an hadron short-range interaction generating an attraction. In addition, the one pion exchange potential (OPEP) as a long-range interaction is introduced by the Lagrangians satisfying the chiral and heavy quark spin symmetries. The OPEP has been known as a driving force to bind atomic nuclei, where the tensor term leading the coupled-channel effect generates a strong attraction. The mass degeneracy of heavy hadrons due to the heavy quark spin symmetry enhances the OPEP derived by the \(\pi D^{(*)}{\bar{D}}^{(*)}\) and \(\pi \varSigma _c^{(*)}\varSigma _c^{(*)}\) couplings. Introducing those interactions, we consistently explain the masses and widths of the \(P_c\) states reported by LHCb in 2019. The short range interaction has a dominant role to determine the energy-level structures, while the OPEP tensor term does the decay width.

Left-Right Symmetry and Leading Contributions to Neutrinoless Double Beta Decay.

We study the impact of the mixing (LR mixing) between the standard model W boson and its hypothetical, heavier right-handed parter W_{R} on the neutrinoless double beta decay (0νββ decay) rate. Our study is done in the minimal left-right symmetric model assuming a type-II dominance scenario with charge conjugation as the left-right symmetry. We then show that the 0νββ decay rate may be dominated by the contribution proportional to this LR mixing, which at the hadronic level induces the leading-order contribution to the interaction between two pions and two charged leptons. The resulting long-range pion exchange contribution can significantly enhance the decay rate compared to previously considered short-range contributions. Finally, we find that even if future cosmological experiments rule out the inverted hierarchy for neutrino masses, there are still good prospects for a positive signal in the next generation of 0νββ decay experiments.

Measurement of beam asymmetry for π−Δ++ photoproduction on the proton at Eγ=8.5GeV

We report a measurement of the $\pi^-$ photoproduction beam asymmetry for the reaction $\vec{\gamma} p \rightarrow \pi^- \Delta^{++}$ using data from the GlueX experiment in the photon beam energy range 8.2--8.8 GeV. The asymmetry $\Sigma$ is measured as a function of four-momentum transfer $t$ to the $\Delta^{++}$ and compared to phenomenological models. We find that $\Sigma$ varies as a function of $t$: negative at smaller values and positive at higher values of $|t|$. The reaction can be described theoretically by $t$-channel particle exchange requiring pseudoscalar, vector, and tensor intermediaries. In particular, this reaction requires charge exchange, allowing us to probe pion exchange and the significance of higher-order corrections to one-pion exchange at low momentum transfer. Constraining production mechanisms of conventional mesons may aid in the search for and study of unconventional mesons. This is the first measurement of the process at this energy.

Comparing different density-matrix expansions for long-range pion exchange

Empirical energy density functionals (EDFs) are generally successful in describing nuclear properties across the table of nuclides. But their limitations motivate using the density-matrix expansion (DME) to embed long-range pion interactions into a Skyrme functional. Recent results on the impact of the pion were both encouraging and puzzling, necessitating a careful re-examination of the DME implementation. Here we take the first steps, focusing on two-body scalar terms in the DME. Exchange energies with long-range one-pion contributions are well approximated by all DME implementations considered, with preference for variants that do not truncate at two derivatives in every EDF term. The use of the DME for chiral pion contributions is therefore supported by this investigation. For scalar-isovector energies it is important to treat neutrons and protons separately. The results are found to apply under broad conditions, although self-consistency is not yet tested.

Discriminating among interpretations for the X(2900) states

We make predictions for the production and decays of $X(2900)$ states, and their possible charged partners, in $B^+$ and $B^0$ decays, considering a number of competing models for the states, including triangle diagrams mediated by quark exchange or pion exchange, and resonance scenarios including molecules and tetraquarks. Assuming only isospin symmetry and the dominance of colour-favoured weak decays, we find characteristic differences in the predictions of the different models. Future experimental studies can therefore discriminate among the competing interpretations for the states.

Pions and Contacts at N4LO: Some details on the chiral nuclear force

In this work we have performed a detailed study of chiral nuclear forces at N4LO approximation applied to selected channels of the neutron–proton (np) scattering. The idea is to analyse the different contributions to the nucleon–nucleon (NN) potential by separating the part coming from the exchange of pions and the one coming from the contact interactions. We consider two state-of-the-art chiral interactions at N4LO which are constructed using different regularization procedures: the non-local Idaho–Salamanca force and the semi-local interaction from the Bochum group. In order to compare the two types of regularization we consider both interactions with a 500 MeV cutoff and to analyse the cutoff dependence we select the Bochum potential with three different cutoff values: 500, 450 and 400 MeV. Our results show that the balance between pion exchanges and contact interactions depends strongly on the regularization procedure. The non-local angle-independent regularization of both components of the interaction implemented in the Idaho–Salamanca potential makes the contact terms to be present at large distances while the local regularization of the pion exchanges in the Bochum potential restricts the contact interactions to small distances. Also, the value of the cutoff affects the strength of the potential but the interplay between pion exchanges and contact terms remains qualitatively the same.

How to reveal the nature of three or more pentaquark states

Within the chiral unitary approach and with the constraints of heavy quark spin symmetry, we study the coupled channel interactions of ${\bar D}^{(*)}\Sigma_c^{(*)}$ channels, close to whose thresholds three pentaquark-like $P_c$ states have been reported by the LHCb Collaboration. In the present work, we take into account the contributions of pion exchanges via box diagrams to the interaction potentials, and therefore lift the degeneracy in the masses of ${\bar D}^*\Sigma_c^{(*)}$ spin multiplets. Fitting the $J/\psi p$ invariant mass distributions in the $\Lambda_b^0 \to J/\psi K^- p$ decay, we find that the LHCb pentaquark states can not be reproduced in the direct $J/\psi p$ production in the $\Lambda_b^0$ decay, and can only be indirectly produced in the final state interactions of the $\Lambda_b^0$ decay products, ${\bar D}^*\Sigma_c^{(*)}$, which further supports the nature of these states as $\bar{D}\Sigma_c$ molecules. Based on the fit results obtained, we study the partial decay widths/branching ratios to other decay channels, $\bar{D}^* \Lambda_c$, $\bar{D} \Lambda_c$, and $\eta_c N$, and the corresponding invariant mass distributions. The resonances with $J^P=\frac{1}{2}^-$, $P_c(4312)$, $P_c(4440)$ and the one of $\bar{D}^* \Sigma_c^*$ around 4500 MeV, have large partial decay width into $\eta_c N$, and thus, can be easily seen in the $\eta_c N$ invariant mass distributions. By contrast, the states with $J^P=\frac{3}{2}^-$, $P_c(4457)$, the (predicted) narrow $P_c(4380)$ and the bound state of $\bar{D}^* \Sigma_c^*$ with a mass of about 4520 MeV, do not decay into $\eta_c N$. Therefore, the $\eta_c N$ channel should be studied in future to provide further insights into the nature of these states, especially that of the $P_c(4440)$ and $P_c(4457)$.

Convergence of nuclear effective field theory with perturbative pions

The classic paper by Fleming, Mehen and Stewart cast doubts on the convergence of spin-triplet nucleon-nucleon partial wave scattering amplitudes when following the proposal of Kaplan, Savage and Wise to construct nuclear effective field theory around the unitary fermion limit with perturbative pion exchange. FMS identified the subclass of iterated one-pion exchange potential graphs as the cause of this poor convergence, which they showed persisted in the chiral limit. Theoretical tools are developed here to compute these Feynman graphs analytically to high order in all angular momentum channels simultaneously, examining the amplitudes computed to seven loops in the $L=J$ channels, and three loops in the coupled $L=J\pm1$ channels. One finds that there is nothing pathological about the perturbative expansion of a $1/r^3$ potential in general, and that the expansion converges satisfactorily in all partial waves except those with the lowest angular momentum, particularly the ${}^3P_0$ and the coupled ${}^3S_1-{}^3D_1$ channels. The results corroborate work by Birse, which suggests possible avenues to explore for improving the range of validity of the EFT expansion.

Perturbative chiral nucleon–nucleon potential for the 3P0 partial wave

We study perturbativeness of chiral nuclear forces in the 3P0 channel. In previous works, the focus has been on the one-pion exchange, and the applicable window of perturbative pion exchanges has been shown to span from the threshold to center-of-mass momentum k ≃ 180 MeV. We will examine, instead, whether the cancellation of short- and long-range parts can sufficiently soften the 3P0 chiral force to make it more amenable to perturbation theory. The result is encouraging, as the combined 3P0 force is shown to be perturbative up to k ≃ 280 MeV, covering many nuclear-structure calculations.

The Problem of Renormalization of Chiral Nuclear Forces

Ever since quantum field theory was first applied to the derivation of nuclear forces in the mid-20th century, the renormalization of pion exchange with realistic couplings has presented a challenge. The implementation of effective field theories (EFTs) in the 1990s promised a solution to this problem but unexpected obstacles were encountered. The response of the nuclear community has been to focus on chiral potentials with regulators chosen to produce a good description of data. Meanwhile, a successful EFT without explicit pion exchange --- Pionless EFT --- has been formulated where renormalization is achieved order by order in a systematic expansion of low-energy nuclear observables. I describe how lessons from Pionless EFT are being applied to the construction of a properly renormalized Chiral EFT.

Nucleon axial structure from lattice QCD

We present a new analysis method that allows one to understand and model excited state contributions in observables that are dominated by a pion pole. We apply this method to extract axial and (induced) pseudoscalar nucleon isovector form factors, which satisfy the constraints due to the partial conservation of the axial current up to expected discretization effects. Effective field theory predicts that the leading contribution to the (induced) pseudoscalar form factor originates from an exchange of a virtual pion, and thus exhibits pion pole dominance. Using our new method, we can recover this behavior directly from lattice data. The numerical analysis is based on a large set of ensembles generated by the CLS effort, including physical pion masses, large volumes (with up to 963 × 192 sites and Lmπ = 6.4), and lattice spacings down to 0.039 fm, which allows us to take all the relevant limits. We find that some observables are much more sensitive to the choice of parametrization of the form factors than others. On the one hand, the z-expansion leads to significantly smaller values for the axial dipole mass than the dipole ansatz ( {M}_A^{z-mathit{exp}} = 1.02(10) GeV versus {M}_A^{mathrm{dipole}} = 1.31(8) GeV). On the other hand, we find that the result for the induced pseudoscalar coupling at the muon capture point is almost independent of the choice of parametrization ( {g}_P^{ast z-mathit{exp}} = 8.68(45) and {g}_P^{ast mathrm{dipole}} = 8.30(24)), and is in good agreement with both, chiral perturbation theory predictions and experimental measurement via ordinary muon capture. We also determine the axial coupling constant gA.

One pion exchange and the quantum numbers of the Pc(4440) and Pc(4457) pentaquarks

The LHCb collaboration has recently discovered three pentaquark-like states---the ${P}_{c}(4312)$, ${P}_{c}(4440)$ and ${P}_{c}(4457)$---close to the $\overline{D}{\mathrm{\ensuremath{\Sigma}}}_{c}$ and the ${\overline{D}}^{*}{\mathrm{\ensuremath{\Sigma}}}_{c}$ meson-baryon thresholds. The standard interpretation is that they are heavy antimeson-baryon molecules. Their quantum numbers have not been determined yet, which implies two possibilities for the ${P}_{c}(4440)$ and ${P}_{c}(4457)$: ${J}^{P}={\frac{1}{2}}^{\ensuremath{-}}$ and ${J}^{P}={\frac{3}{2}}^{\ensuremath{-}}$. The preferred interpretation within a contact-range effective field theory is that the ${P}_{c}(4440)$ is the ${J}^{P}={\frac{1}{2}}^{\ensuremath{-}}$ molecule, while the ${P}_{c}(4457)$ is the ${J}^{P}={\frac{3}{2}}^{\ensuremath{-}}$ one. Here we show that when the one pion exchange potential between the heavy antimeson and heavy baryon is taken into account, this conclusion changes, with the contrary identification being as likely as the original one. The identification is however cutoff dependent, which suggests that improvements of the present description (e.g., the inclusion of subleading order corrections, like two-pion exchanges) are necessary in order to disambiguate the spectroscopy of the molecular pentaquarks.