Dynamical Nucleon-Pion System via Basis Light-Front Quantization
Weijie Du, Yang Li, Xingbo Zhao, James P. Vary
备注：
15 pages, 2 figures, Weijie Du, Yang Li, Xingbo Zhao and James P. Vary, "Dynamical Nucleon-Pion System via Basis Light-Front Quantization", Proceedings of the International Conference `Nuclear Theory in the Supercomputing Era - 2018' (NTSE-2018), Daejeon, South Korea, October 29 - November 2, 2018, eds. A. M. Shirokov and A. I. Mazur. Pacific National University, Khabarovsk, Russia, 2019, p. 92
We present the first application of the Basis Light-Front Quantization method to study a simple chiral model of the nucleon-pion system via an ab initio, non-perturbative, Hamiltonian approach. As a test problem, we consider the physical proton as the relativistic bound state of the nucleon-pion system. Based on the chiral model of the nucleon-pion system, we construct the mass-squared matrix of the system within our light-front basis representation. We obtain the proton's mass and the corresponding light-front wave function by solving the eigenvalue problem of the mass-squared matrix. With the resulting boost-invariant light-front wave function, we also compute the proton's parton distribution function.
Off-shell aspects of the one-pion-exchange potential and their relationship to different forms of the nonstatic (subleading-order) chiral two-pion-exchange nucleon-nucleon potential are discussed. Various types of off-shell behavior are categorized and numerous examples are given. Recently derived potentials based on chiral approaches are supplemented by a rather general form of the two-pion-exchange potential derived using old-fashioned methods. The latter is closely related to a general form of one-pion-exchange relativistic corrections and nonstatic two-pion-exchange three-nucleon forces de…
We develop a toy model of pion production in nucleon-nucleon collisions that reproduces some of the features of the chiral Lagrangian calculations. We calculate the production amplitude and examine some common approximations.
The scattering amplitude of the pion-nucleon elastic scattering is calculated by the lowest-order approximation of the perturbative expansion with the non-perturbative term. The self-energy of nucleon is determined so as to give the scattering parameters of the $S$-wave. It is shown that the scattering process mediated by the $σ$-meson plays a decisive role together with the pseudovector pion interaction.
This paper contains the results of an analysis of recent low-energy pion-nucleon scattering experiments. Obtained are phase shifts, the pion-nucleon coupling constant and an estimate of the Sigma term.
A new result for the nucleon sigma term from an updated pion-nucleon partial-wave and dispersion relation analysis of the Virginia Polytechnic Institute (now George Washington University) group is discussed. Using a method similar to that of Gasser, Leutwyler, Locher, and Sainio, we obtain Sigma=90 +/- 8 MeV (preliminary), in disagreement with the canonical result 64 +/- 8 MeV, but consistent with expectations based on new information on the $π$NN coupling constant, pionic atoms, and the Delta resonance width.
The leading contribution to the three-pion exchange nucleon-nucleon potential is calculated in the framework of chiral symmetry. It has pseudoscalar and axial components and is dominated by the former, which has a range of about 1.5 fm and tends to enhance the OPEP. The strength of this force does not depend on the pion mass and hence it survives in the chiral limit.
A microscopic model for the $N\bar N\toππ$ process is presented in the meson exchange framework, which in the pseudophysical region agrees with available quasiempirical information. The scalar ($σ$) and vector ($ρ$) piece of correlated two--pion exchange in the pion--nucleon interaction is then derived via dispersion integrals over the unitarity cut. Inherent ambiguities in the method and implications for the description of pion--nucleon scattering data are discussed.
The electromagnetic form factor of proton is studied following our previous calculation of the anomalous magnetic moment. The non-perturbative term is incorporated to correct the $Q^2$ dependence.
We compute the matrix elements for pion emission from the nucleon induced by twist-two light-cone operators in the kinematics where the pion is close to threshold. It is shown that the results of heavy baryon ChPT are in agreement with the results from the soft-pion theorem approach. Therefore the amplitude of soft-pion emission is directly related to the generalized parton distributions in the nucleon.
The chiral two-pion exchange nucleon-nucleon interaction has nowadays a rather firm conceptual basis, but depends on low-energy constants, which may be extracted from fits to data. In order to facilitate this kind of application, we present here a parametrized version of our relativistic expansion of this component of the force to $O(q^4)$, performed recently.
Charge-independence breaking due to the pion-mass difference in the (chiral) two-pion-exchange nucleon-nucleon force is investigated. A general argument based on symmetries is presented that relates the charge-symmetric part of that force to the proton-proton case. The static potential linear in that mass difference is worked out as an explicit example by means of Feynman diagrams, and this confirms the general argument.
We give a full kinematical classification of all the tree-level two-pion photoproduction processes on the nucleon, which consists of seventeen diagrams. It suggests a method of analysis of two-pion data with little model bias.
We consider the relationship between P-wave pi-N scattering and the strength of the P-wave two-pion-exchange three-nucleon interaction (TPE3NI). We explain why effective theories that do not contain the delta resonance as an explicit degree of freedom tend to overestimate the strength of the TPE3NI. The overestimation can be remedied by higher-order terms in these ``delta-less'' theories, but such terms are not yet included in state-of-the-art chiral EFT calculations of the nuclear force. This suggests that these calculations can only predict the strength of the TPE3NI to an accuracy of +/-25%…
Using Dyson--Schwinger equations, the nucleon propagator is analyzed nonperturbatively in a field--theoretical model for the pion--nucleon interaction. Infinities are circumvented by using pion--nucleon form factors which define the physical scale. It is shown that the correct, finite, on--shell nucleon renormalization is important for the value of the mass--shift and the propagator. For physically acceptable forms of the pion--nucleon form factor the rainbow approximation together with renormalization is inconsistent. Going beyond the rainbow approximation, the full pion--nucleon vertex is mo…
A calculation of the pion-production operator up to next-to-next-to-leading order for s-wave pions is performed within chiral effective field theory. In the previous study [Phys. Rev. C 85, 054001 (2012)] we discussed the contribution of the pion-nucleon loops at the same order. Here we extend that study to include explicit Delta degrees of freedom and the 1/m_N^2 corrections to the pion-production amplitude. Using the power counting scheme where the Delta-nucleon mass difference is of the order of the characteristic momentum scale in the production process, we calculate all tree-level and loo…
We study the perturbative renormalizability of chiral two pion exchange in nucleon-nucleon scattering for p- and d-waves within the effective field theory approach. The one pion exchange potential is fully iterated at the leading order in the expansion, a choice generating a consistent and well-defined power counting that we explore in detail. The results show that perturbative chiral two pion exchange reproduces the data up to a center-of-mass momentum of k_cm ~300 MeV at NNLO and that the effective field theory expansion convergences up to k_cm ~ 350 MeV.
We outline the modified formulation of baryon chiral effective field theory for nucleon-nucleon scattering and discuss the issue of a possible power counting violation by the nucleon mass. We also present the results for the quark mass dependence of the $^1S_0$ and $^3S_1$ scattering lengths and the deuteron binding energy at leading order.
In the Yukawa model for nuclear forces, a simple relation between the charged and neutral pion-nucleon coupling constants is derived. The relation implies that the charged pion-nucleon coupling constant is larger than the neutral one since the np interaction is stronger than the pp interaction. The derived value of the charged pion-nucleon constant shows a very good agreement with one of the recent experimental values. The relative splitting between the charged and neutral pion-nucleon coupling constants is predicted to be practically the same as that between the charged and neutral pion masse…
The description of the three-nucleon system in terms of nucleon and $Δ$ degrees of freedom is extended to allow for explicit pion production (absorption) from single dynamic $Δ$ de-excitation (excitation) processes. This mechanism yields an energy dependent effective three-body hamiltonean. The Faddeev equations for the trinucleon bound state are solved with a force model that has already been tested in the two-nucleon system above pion-production threshold. The binding energy and other bound state properties are calculated. The contribution to the effective three-nucleon force arising from th…
Pion photoproduction amplitudes are calculated from a set of equations that have been derived by coupling an external photon to all places in a dressed pion-nucleon vertex. The calculation is consistent with gauge invariance, charge conservation, unitarity and covariance. To provide input to the photoproduction amplitude, a photon-nucleon vertex is calculated from a set of equations derived by complete attachment of photons to a dressed nucleon propagator. We check the accuracy of this vertex by extracting its nucleon electromagnetic form factors.
The nucleon-nucleon interaction arising from the exchange of three pions and the excitation of $Δ(1232)$-isobars in intermediate states is studied. Approximating the $Δ$-propagator by the inverse $Δ$N mass-splitting, analytical expressions are derived for the spectral-functions of the isoscalar and isovector central, spin-spin and tensor NN-potentials in momentum-space. A trans- lation of the spectral-functions into coordinate-space potentials reveals that the main effect of these specific exchange and excitation mechanisms is a repulsive isoscalar central NN-potential.
Our extraction of the pion-nucleon coupling constant from πp elastic scattering data is outlined. A partial wave analysis (T_π < 2100 MeV) is performed simultaneously with a fixed-t dispersion relation analysis (T_π < 800 MeV). The πNN coupling constant g^2/4πis searched to find the best fit. The result 13.73 \pm 0.01 \pm 0.07 (first error statistical, second systematic) is found to be insensitive to database changes and Coulomb barrier corrections. This value satisfies important elements of low energy QCD like the Goldberger-Treiman discrepancy, the Dashen-Weinstein sum rule, and chiral pertu…
We use the method of QCD sum rules in the presence of an external pion field to investigate isospin symmetry breakings in pion-nucleon couplings. Possible manifestations of such isospin symmetry breaking are examined in the context of low-energy nucleon-nucleon ($NN$) scattering. We discuss numerical results in relation to both the existing data and other theoretical predictions.
We have made a needed correction to our earlier paper (nucl-th/9511002), which contains a sign error. With this correction, the weak parity-violating pion-nucleon coupling is about 3 x 10^(-7).
We show how to renormalize chiral two pion exchange perturbatively if one pion exchange has already been fully iterated at leading order. This particular choice corresponds to the implementation of the counting proposal of Nogga, Timmermans and van Kolck at subleading orders. We illustrate why the perturbative treatment of the two pion exchange contributions is mandatory in order to avoid certain inconsistencies in Weinberg's counting. In addition, renormalizability implies modifications of the power counting which we explore for the particular case of the singlet channel.
There has been a discrepancy between values of the pion-nucleon sigma term extracted by two different methods for many years. Analysis of recent high precision pion-nucleon data has widened the gap between the two determinations. It is argued that the two extractions correspond to different quantities and that the difference between them can be understood and calculated.
In view of persisting misunderstanding about the determination of the pion-nucleon coupling constants in the Nijmegen multienergy partial-wave analyses of pp, np, and pbar-p scattering data, we present additional information which may clarify several points of discussion. We comment on several recent papers addressing the issue of the pion-nucleon coupling constant and criticizing the Nijmegen analyses.
We show that the tail of the chiral two-pion exchange nucleon-nucleon potential is proportional to the $πN$ scalar form factor and discuss how it can be translated into effective scalar meson interactions. We then construct a kernel for the process $NN\toπNN$, due to the exchange of two pions, which may be used in either three body forces or pion production in NN scattering. Our final expression involves a partial cancellation among three terms, due to chiral symmetry, but the net result is still important. We also find that, at large internucleon distances, the kernel has the same spatial dep…
Photocouplings for the nucleon resonances most relevant to total photoabsorption are calculated in a constituent quark model including gluon and pion exchange. The sensitivity of the results to the different ingredients of the model is discussed.
We present a relativistic calculation of pion scattering, pion photoproduction and Compton scattering on the nucleon in the energy region of the \Del-resonance (upto 450 MeV photon lab energy), in a unified framework which obeys the unitarity constraint. It is found that the recent data on the cross section for nucleon Compton scattering determine accurately the parameters of the electromagnetic nucleon--\Del\ coupling. The calculated pion-photoproduction partial-wave amplitudes agree well with the recent Arndt analysis.
We have developed a model for the N N --> N N pi pi reaction and evaluated cross sections for the different charged channels. The low energy part of those channels where the pions can be in an isospin zero state is dominated by N* excitation, driven by an isoscalar source followed by the decay N* --> N (pi pi, T=0, s-wave). At higher energies, and in channels where the pions are not in T=0, Delta excitation mechanisms become relevant.
A general expression for the $π-N$ and $π-Δ$ scattering lengths is derived in the framework of a microscopic calculation. Annihilation, negative energy wave-functions and spontaneous chiral symmetry are included consistently. The point-like limit is used to calculate the scattering lengths.
We study the effect of isospin projections on a two source picture of coherent pions coming from nucleon-antinucleon annihilation. We show that important Hanbury-Brown Twiss correlations among like charge pion pairs compared with unlike arise from these isospin projections.
We summarize results obtained in our studies of the pion-nucleon coupling constant. Several different techniques have been applied to pi-N and NN elastic scattering data, and the existing database for single-pion photoproduction. The most reliable determination comes from pi-N elastic scattering. The sensitivity in this reaction was found to be greater, by at least a factor of 3, when compared with analyses of NN elastic scattering or single-pion photoproduction.
We develop a Lorenz- and gauge-invariant dynamical model for pion electroproduction in the resonance region. The model is based on solving of the Salpeter (instantaneous) equation for the pion-nucleon interaction with a hadron-exchange potential. We find that the one-particle-exchange kernel of the Salpeter equation for pion electroproduction develops an unphysical singularity for a finite value of $Q^{2}$. We analyse two methods of dealing with this problem. Results of our model are compared with recent single-polarization data for pion electroproduction.
We study the perturbative renormalizability of chiral two pion exchange for the singlet and triplet channels within effective field theory, provided that the one pion exchange piece of the interaction has been fully iterated. We determine the number of counterterms/subtractions needed in order to obtain finite results when the cut-off is removed, resulting in three counterterms for the singlet channel and six for the triplet. The results show that chiral two pion exchange can be treated perturbatively up to a center-of-mass momentum of k ~ 200-300 MeV in the singlet channel and k ~ 300-400 in …
The procedure for deriving the Ward-Takahashi identity is applied to the pseudovector coupling pion-nucleon system in which the isovector and axial-vector current does not conserved. The relation is given in closed form although it is more complicated than that of the quantum electrodynamics. The application to the propagator has revealed the interesting results.
We review recent progress in our understanding of elastic and inelastic pion reactions on two nucleon systems from the point of view of effective field theory. The discussion includes pion-deuteron scattering, gamma d to pi-nn, and NN to NNpi. At the end some remarks are made on strangeness production reactions.
We investigate p-wave pion production in nucleon-nucleon collisions up to next-to-next-to-leading order in chiral effective field theory. In particular, we show that it is possible to describe simultaneously the p-wave amplitudes in the $pn\to ppπ^-$, $pp\to pnπ^+$, $pp\to dπ^+$ channels by adjusting a single low-energy constant accompanying the short-range operator which is available at this order. This study provides a non-trivial test of the applicability of chiral effective field theory to reactions of the type $NN\to NNπ$.
We investigate the influence of the real part of the in-medium pion optical potential on the pion dynamics in intermediate energy heavy ion reactions at 1 GeV/A. For different models, i.e. a phenomenological model and the $Δ$--hole model, a pionic potential is extracted from the dispersion relation and used in Quantum Molecular Dynamics calculations. In addition with the inelastic scattering processes we thus take care of both, real and imaginary part of the pion optical potential. A strong influence of the real pionic potential on the pion in-plane flow is observed. In general such a potentia…
A unified picture of nucleon antinucleon annihilation into pions emerges from a classical description of the pion wave produced in annihilation and the subsequent quantization of that wave as a coherent state. When the constraints of energy-momentum and iso-spin conservation are imposed on the coherent state, the pion number distribution and charge ratios are found to be in excellent agreement with experiment.
A covariant model of elastic pion-nucleon scattering based on the Bethe-Salpeter equation is presented. The kernel consists of s- and u-channel nucleon and delta poles, along with rho and sigma exchange in the t-channel. A good fit is obtained to the s- and p-wave phase shifts up to the two-pion production threshold.
By introducing electromagnetic formfactors in the spacelike region we extend an effective, gauge invariant Lagrangian model considering Born terms, baryon resonances up to 1.7 GeV and vector meson contributions (Rho, Omega) to calculate electroproduction of Pi-mesons. This model forms the basis for predictions of pion induced dilepton production on the nucleon. Therefore, the implemented formfactors are constructed in such a way that their analytic continuation into the timelike region is possible. It is shown that the seagull term and the N*(1520) resonance play a dominant role in the dilepto…
Nuclear processes involving momenta much below the mass of the pion may be described by an effective field theory in which the pions do not appear as explicit degrees of freedom. The effects of the pion and all other virtual hadrons are reproduced by the coefficients of gauge-invariant local operators involving the nucleon field. Nucleon-nucleon scattering phase shift data constrains many of the coefficients that appear in the effective Lagrangean but at some order in the expansion coefficients enter that must be constrained by other observables. We compute several observables in the two-nucle…
A status report is given for the current theoretical understanding of pion production close to threshold. In the first part of the talk predictions of a meson exchange model are compared to recent polarized data for the reactions $pp \to ppπ^0$ as well as $pp\to pnπ^+$ revealing, that the former reaction is badly described whereas the predictions for the latter turn out to be consistent with the data. Recent progress in the application of chiral perturbation theory allows to understand this difference.
We report on recent results obtained by the above collaboration on the collision processes involving three nucleons, where we pay particular attention on the dynamical role of the pion. After discussing the case at intermediate energies, where real pions can be produced and detected, we have considered the case at lower energies, where the pions being exchanged are virtual. The study has revealed the presence of some new pion-exchange mechanisms, which leads to a new three-nucleon force of tensor structure. Recently, the effect of this tensor three-nucleon force to the spin observables for neu…
We derive the leading one-loop contribution to the one-pion exchange and short-range two-nucleon electromagnetic current operator in the framework of chiral effective field theory. The derivation is carried out using the method of unitary transformation. Explicit results for the current and charge densities are given in momentum and coordinate space.
We use a recently improved version of the chiral nucleon-nucleon potential at next-to-next-to-leading order to calculate the $^1S_0$ pairing gap in isospin-symmetric nuclear matter. The pairing potential consists of the long-range one- and two-pion exchange terms and two short-distance NN-contact couplings. We find that the inclusion of the two-pion exchange at next-to-next-to-leading order reduces substantially the cut-off dependence of the $^1S_0$ pairing gap determined by solving a regularised BCS equation. Our results are close to those obtained with the universal low-momentum nucleon-nucl…
We show on two examples, namely a calculation for charge symmetry breaking in pn->d pi^0 that allows one to extract the quark mass difference induced part of the proton--neutron mass difference and a high precision calculation for pion--deuteron scattering and its implications for the value of the charged pion--nucleon coupling constant, how QCD tests can be performed from low energy hadronic observables.
We present the first application of the Basis Light-Front Quantization method to a simple chiral model of the nucleon-pion system as a relativistic bound state for the physical proton. The light-front mass-squared matrix of the nucleon-pion system is obtained within a truncated basis. The mass and the corresponding light-front wave function (LFWF) of the proton are computed by numerical diagonalization of the resulting mass-squared matrix. With the boost invariant LFWF, we calculate the probability density distribution of the pion's longitudinal momentum fraction and the Dirac form factor of t…
We analyse the interplay between nucleon-nucleon potentials deduced from chiral perturbation theory and a coarse grained representation of the short distance interactions by delta-shells potentials below a certain cut-off distance. While we find that the number of parameters is greatly reduced when Chiral Two Pion Exchange contributions are included we also observe that discerning the necessity of improvements on the interaction requires a detailed analysis of all error sources. Our points are best illustrated by computing deuteron static properties as well as electromagnetic form factors afte…
In most simulations of nonrelativistic nuclear systems, the wave functions found solving the many-body Schrödinger equations describe the quantum-mechanical amplitudes of the nucleonic degrees of freedom. In those simulations the pionic contributions are encoded in nuclear potentials and electroweak currents, and they determine the low-momentum behavior. In this work we present an alternative quantum Monte Carlo formalism in which both relativistic pions and nonrelativistic nucleons are explicitly included in the quantum-mechanical states of the system. We report the renormalization of the nuc…
Production of p-wave pions in nucleon-nucleon collisions is studied according to an improved power counting that embodies the constraints of chiral symmetry. Contributions from the first two non-vanishing orders are calculated. We find reasonable convergence and agreement with data for a spin-triplet cross section in $pp\to ppπ^0$,with no free parameters. Agreement with existing data for a spin-singlet cross section in $pp\to pnπ^+$ constrains a short-range operator shown recently to contribute significantly to the three-nucleon potential.
Understanding pion production in nucleon-nucleon collisions near threshold has been a challenge for the last decades. In particular, the reaction pp -> pppi0 is highly sensitive to short-range mechanisms, because isospin conservation suppresses the otherwise dominant pion exchange term. However, the relative importance of the various reaction processes has been very difficult to establish. After reviewing the state-of-the-art of the theoretical approaches, we address the validity of the distorted-wave Born approximation (DWBA) through its link to the time-ordered perturbation theory (TOPT) dia…
We undertake a simultaneous partial wave analysis to proton-proton and neutron-proton scattering data below pion production threshold up to LAB energies of 350MeV. We represent the interaction as a sum of delta-shells in configuration space below 3fm and a charge dependent one pion exchange potential above 3fm together with electromagnetic effects. We obtain chi^2 = 2813|pp + 3985|np with a total of N = 2747|pp + 3691|np data obtained till 2013 and a total number of 46 fitting parameters yielding chi^2 /d.o.f = 1.06. Special attention is paid to estimate the errors of the phenomenological inte…
A model is developed for the hadronic and electromagnetic interaction in the two-nucleon system above pion threshold in the framework of meson, nucleon and $Δ$ degrees of freedom. It is based on time-ordered perturbation theory and includes full meson retardation in potentials and exchange currents as well as loop contributions to the nucleonic one-body current. Results for some selected processes are discussed.
We examine the effect of isospin-violating meson-nucleon coupling constants on low-energy pion-nucleon scattering. We compute the couplings in the context of a nonrelativistic quark model. The difference between the up and down constituent masses induces a coupling of the neutral pion to the proton that is slightly larger than the corresponding one for the neutron. This difference generates a large isospin-violating correction---proportional to the isospin-even contribution arising from the nucleon Born terms---to the charge-exchange ($π^{-}p \rightarrow π^{0}n$) amplitude. In contrast to the …
We compute pion-nucleon scattering at tree-level within a gauged linear sigma model which contains the nucleon and its chiral partner. Such an investigation in principle allows to make definite predictions as to whether the main contribution to the nucleon mass comes from the chiral condensate or from the mixing with its chiral partner. We find that there seems to be no set of model parameters that allows for a simultaneous description of all experimentally measured scattering lengths and range parameters. This indicates the need to improve the dynamical ingredients of the model.
