Cheng-Dashen Point Research Articles

High-Precision Determination of the Pion-Nucleon σ Term from Roy-Steiner Equations.

We present a determination of the pion-nucleon (πN) σ term σ_{πN} based on the Cheng-Dashen low-energy theorem (LET), taking advantage of the recent high-precision data from pionic atoms to pin down the πN scattering lengths as well as of constraints from analyticity, unitarity, and crossing symmetry in the form of Roy-Steiner equations to perform the extrapolation to the Cheng-Dashen point in a reliable manner. With isospin-violating corrections included both in the scattering lengths and the LET, we obtain σ_{πN}=(59.1±1.9±3.0) MeV=(59.1±3.5) MeV, where the first error refers to uncertainties in the πN amplitude and the second to the LET. Consequences for the scalar nucleon couplings relevant for the direct detection of dark matter are discussed.

New evaluation of theπNΣterm

A new evaluation of the {pi}N{sigma} term is presented that incorporates recent s-channel phase shifts and t-channel {pi}{pi} phase shifts. We also introduce analyticity-based extrapolation techniques that, along with standard dispersion relation methods, produce a more reliable extrapolation to the Cheng-Dashen point. A recent George Washington University (GWU) phase-shift analysis leads to a {sigma} term of 81{+-}6 MeV.

THE PION-NUCLEON SIGMA TERM FROM UPDATED T-CHANNEL INPUT

Dispersion relations along interior hyperbolas and a set of hyperbolas passing through the Cheng-Dashen point are used to calculate the pion-nucleon sigma term. The t-channel input is updated using the recent GWU partial wave solution and ππ phase shifts from calculations based on Roy equations. Obtained values for the sigma term are still within the error bars of the previous Karlsruhe result.

Analysis of the pion-kaon sigma term and related topics

We calculate the one-loop contributions to the difference \Delta_{\pi K} between the isoscalar on-shell pion-kaon scattering amplitude at the Cheng-Dashen point and the scalar form factor \Gamma_K (2M_\pi^2) in the framework of three flavor chiral perturbation theory. These corrections turn out to be small. This is further sharpened by treating the kaons as heavy fields (two flavor chiral perturbation theory). We also analyze the two-loop corrections to the kaon scalar form factor based on a dispersive technique. We find that these corrections are smaller than in the comparable case of the scalar form factor of the pion. This is related to the weaker final state interactions in the pion-kaon channel.

Low energy analysis of πN→πN

We derive a representation for the pion nucleon scattering amplitude that is valid to the fourth order of the chiral expansion. To obtain the correct analytic structure of the singularities in the low energy region, we have performed the calculation in a relativistic framework (infrared regularization). The result can be written in terms of functions of a single variable. We study the corresponding dispersion relations and discuss the problems encountered in the straightforward nonrelativistic expansion of the infrared singularities. As an application, we evaluate the corrections to the Goldberger-Treiman relation and to the low energy theorem that relates the value of the amplitude at the Cheng-Dashen point to the \sigma-term. While chiral symmetry does govern the behaviour of the amplitude in the vicinity of this point, the representation for the scattering amplitude is not accurate enough to use it for an extrapolation of the experimental data to the subthreshold region. We propose to perform this extrapolation on the basis of a set of integral equations that interrelate the lowest partial waves and are analogous to the Roy equations for \pi\pi scattering.

Virtual photons in baryon chiral perturbation theory

We construct the general Lagrangian for relativistic and heavy baryon chiral perturbation theory with virtual photons to fourth order. We work out the electromagnetic and strong isospin violating contributions to the nucleon self-energy, the nucleon mass and the scalar form factor of the nucleon. Electromagnetic effects for the shift to the Cheng-Dashen point can be as large as 2 MeV We also discuss the corrections to Weinberg's prediction for the scattering length difference a( π 0 p) − a( π 0 n) and show that they are small.

Generalized heavy baryon chiral perturbation theory

Standard SU(2) Heavy Baryon Chiral Perturbation Theory is extended in order to include the case of small or even vanishing quark condensate. The effective lagrangian is given to ${\cal O}(p^2)$ in its most general form and to ${\cal O}(p^3)$ in the scalar sector. A method is developed to efficiently construct the relativistic baryonic effective lagrangian for chiral SU(2) to all orders in the chiral expansion. As a first application, mass- and wave-function renormalization as well as the scalar form factor of the nucleon is calculated to ${\cal O}(p^3)$. The result is compared to a dispersive analysis of the nucleon scalar form factor adopted to the case of a small quark condensate. In this latter analysis, the shift of the scalar form factor between the Cheng-Dashen point and zero momentum transfer is found to be enhanced over the result assuming strong quark condensation by up to a factor of two, with substantial deviations starting to be visible for $r=m_s/\hat{m}\lesssim 12$.

Isospin violation in pion–nucleon scattering

Abstract We construct the complete effective chiral pion–nucleon Lagrangian in the presence of virtual photons to one loop. As an application, we consider strong and electromagnetic isospin violation for scattering of neutral pions off nucleons. We show that for the scattering lengths these isospin violating terms are of the same size as the purely hadronic ones. We also analyze isospin-violating effects for the σ -term. These can be as large as 10% for the absolute value but are negligible for the shift to the Cheng–Dashen point.

On the analysis of the pion-nucleon σ-term: The size of the remainder at the Cheng-Dashen point

We calculate the one-loop contributions of order M π 4 to the difference Δ R between the on-shell pion-nucleon scattering amplitude D +(0, 2M π 2) at the Cheng-Dashen point ν = 0, t = 2 M π 2 and the scalar form factor σ(2 M π 2) in the framework of heavy baryon chiral perturbation theory. We proof that to this order Δ R contains no chiral logarithms and therefore it vanishes simply as M π 4 in the chiral limit. Numerically, we find as an upper limit Δ R ⋍ 2 MeV .

Low energy (T π

Recent experimental results of pion nucleon interactions below pion energies of about 300 MeV are reviewed. The Karlsruhe-Helsinki phases from the dispersion analysis of 1979 (KH80) are used for comparison as a ‘standard’ because they respect Mandelstam analyticity and unitary and, hence, allow a unique extrapolation to the threshold and to the Cheng-Dashen point which is crucial for a determination of the σ-term of pion nucleon scattering.

Importance of the meson cloud to hadron structure

We present a review of our recent results on the role of the mesonic cloud in the structure of hadrons in both soft and hard kinematical regimes. We compute the pion and nucleon form factors of the scalar operator uu + dd within a meson exchange model. Our results agree with recent dispersion relation analyses near the Cheng-Dashen point but show some deviation at higher energies. In particular, we confirm the observation that the strong ππ interaction gives rise to a scalar square radius of the nucleon of 1.5 fm 2 and a 15 MeV contribution to the pion nucleon sigma term. Some aspects of the meson cloud around the nucleon for deep-inelastic lepton scattering are studied in the framework of the Sullivan formalism. We present a simple two-phase model of the nucleon. Renormalization of the valence quark distribution due to the mesonic cloud is taken into account explicitly. We study the dependence of different quantities on the cut-off parameter of the form factor, the role of different mesons in deep-inelastic scattering, and SU(2) F symmetry breaking in the nucleon sea in connection to the Gottfried Sum Rule. It is possible to obtain agreement with the CCFR data using relatively hard meson-N-N form factors. The E615 data on ( u + d)/2 − s restrict the cut-off parameter in the dipole form factor to about 1.2 GeV. For this value the cut-off parameter we get the largest violation of the Gottfried Sum Rule, about half of that observed by NMC. Mesonic models predict violation of the SU (2) symmetry in the nucleon sea which seems to be necessary to explain the violation of the Gottfried Sum Rule. Since up to now there is no consensus concerning the explanation of the NMC effect, one has to study the role the d/ u asymmetry may play in other processes. Here we study the effect of the asymmetry for the Drell-Yan processes. We find that careful analysis of the dilepton production in the p-p and p-n collisions should shed new light on the interesting problem of the asymmetry. We discuss also the role of the asymmetry in proton-nucleus Drell-Yan processes. We find that moderate asymmetries concentrated in the small- x region are not excluded by the E772 experimental data.

Explicit chiral symmetry breaking in the Nambu-Jona-Lasinio model

We consider a chirally symmetric bosonization of the SU(2) Nambu-Jona-Lasinio model within the Pauli-Villars regularization scheme. Special attention is paid to the way in which chiral symmetry is broken explicitly. The parameters of the model are fixed in the light of chiral perturbation theory by performing a covariant derivative expansion in the presence of external fields. As a by-product we obtain the corresponding low-energy parameters and pion radii as well as some threshold parameters for pion-pion scattering. The nucleon is obtained in terms of the solitonic solutions of the action in the sector with baryon number equal to one. It is found that for a constituent quark mass M ∼ 350 MeV most of the calculated vacuum and pion properties agree reasonably well with the experimental ones and coincide with the region where localized solitons with the right size exist. For this value, however, the scalar and vector pion radii turn out to be very small. A unique determination of the sigma term is proposed, obtaining a value of σ(0) = 41.3 MeV. The scalar nucleon form factor is evaluated in the Breit frame. The extrapolation to the Cheng-Dashen point leads to σ(2 m 2) − σ(0) = 7.4 MeV.

The scalar form factor of the pion, kaon and nucleon

We compute the pion and nucleon form factors of the scalar operator u u + d d within a meson-exchange model. Our results agree with recent dispersion-relation analyses near the Cheng-Dashen point but show some deviation at higher energies. In particular, we confirm the observation that the strong ππ interaction gives rise to a scalar square radius of the nucleon of 1.5 fm 2 and a 15 MeV contribution to the sigma term σ(2 m π 2).

Nucleons with chiral loops

This article extends the analysis of the low-energy structure of Green functions in QCD to processes with one external nucleon. We calculate various nucleon matrix elements of quark currents and determine the elastic πN scattering amplitude T πN for off-shell pions to one loop. As an application of our representation of T πN we evaluate the difference Δ R between the amplitude D at the Cheng-Dashen point and the scalar form factor σ(2 M π 2 ). The obtained result is a factor of 50 too small to explain the difference between the value of σ extracted from elastic πN scattering and the value obtained from the baryon mass spectrum.

A new interpretation of the πNσ-term

An explicit relation between the σ-term, the S-wave πN scattering length a (+) and the range parameter b (+) is established. The range term is shown to be the dominant feature. The important P-wave background terms in the extrapolation of the physical amplitude to the Cheng-Dashen point are shown to be very small if a path v 2+2 mv B ≅0, i.e., with cos θ≅0, is chosen. This avoids the compensating cancellations between large numbers typical of other procedures. Correction terms are also shown to be small using physical arguments. The present relation is in a convenient form for phenomenological applications.

A new determination of the πN sigma term using hyperbolic dispersion relations in the (ν2,t) plane

Dispersion relations in the (ν2,t) plane along hyperbolas are used in order to extrapolate the invariant isospin-even πN amplitude D+(ν2,t) to the Cheng-Dashen point, ν=0, t=2μ. The fluctuation of the results obtained with different hyperbolas gives a realistic estimate of the errors, except for errors of the partial wave solution and of the ππ $$ - N\bar N$$ amplitudes assumed at t < 4μ2 —If our ππ $$ - N\bar N$$ partial waves are used, which are based on the ππs-wave scattering length a 0 0 =0.28 μ-1, the result for the sigma term is 64±8 MeV, in agreement with earlier determinations.—The discrepancy with the theoretical prediction σπN≈ 30 MeV is smaller by only 8 MeV, if our $$ - N\bar N$$ amplitudes are modified in such a way that the threshold behaviour of the ππs-wave agrees with Weinberg's prediction a 0 0 =0.16 μ-1. Further progress depends on new accurate experimental π±p scattering data in the Coulomb interference region at low energies.

Determinations of the πN σ term and s-wave scattering lengths

Determinations of the σ term and s-wave scattering lengths for pion-nucleon scattering are made using interior dispersion relations. The result for the σ term is obtained by extrapolating to the Cheng-Dashen point in only one variable. It is completely independent of the pion-nucleon coupling constant and insensitive to d and f waves in the resonance region. The use of interior dispersion relations allows a series of s-wave scattering length determinations to be made. It is observed that data at intermediate scattering angles gives more consistent results than data near the forward or backward directions, lending further support to the σ term calculation which utilizes only interior angle information.

Comment on theπNσTerm

The sum rule for the on-shell pion-nucleon scattering amplitude at the Cheng-Dashen (CD) point $\ensuremath{\nu}={\ensuremath{\nu}}_{B}=0$ has been recalculated. Instead of the value 1.7${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$ as obtained by CD, we find $T(0, 0)\ensuremath{\simeq}1.1{\ensuremath{\mu}}^{\ensuremath{-}1} \mathrm{and} 1.3{\ensuremath{\mu}}^{\ensuremath{-}1}$, when the new and old Lovelace phase shifts are used, respectively.

Implications for theΔNπInteraction andσTerm from Low-EnergyπNScattering

A model for low-energy $\ensuremath{\pi}N$ scattering consisting of $N$, $\ensuremath{\rho}$, $\ensuremath{\Delta}$, and $\ensuremath{\sigma}$ exchanges together with current-algebra constraints is presented. We derive the $\ensuremath{\Delta}$ contributions to the scattering amplitude using a general $\ensuremath{\Delta}$ propagator and $\ensuremath{\pi}N\ensuremath{\Delta}$ interaction. We show that the data are consistent with the simplest structure for both the $\ensuremath{\Delta}$ propagator and the interaction, A $\ensuremath{\sigma}$ term of 42 \ifmmode\pm\else\textpm\fi{} 10 MeV at the Cheng-Dashen point is found, and predictions of the $\ensuremath{\sigma}$ term at $t=0$, the $\ensuremath{\pi}\mathrm{NN}$ coupling constant, and the isospin-even $s$-wave scattering length are made.