Chiral Perturbation Theory Calculations Research Articles

REMARK ON PION SCATTERING LENGTHS

First it is shown that the tree amplitude for pion–pion scattering in the minimal linear sigma model has an exact expression which is proportional to a geometric series in the quantity [Formula: see text], where mB is the sigma mass which appears in the Lagrangian and is the only a priori unknown parameter in the model. This induces an infinite series for every predicted scattering length in which each term corresponds to a given order in the chiral perturbation theory counting. It is noted that, perhaps surprisingly, the pattern, though not the exact values, of chiral perturbation theory predictions for both the isotopic spin 0 and isotopic spin 2 s-wave pion–pion scattering lengths to orders p2, p4 and p6 seems to agree with this induced pattern. The values of the p8 terms are also given for comparison with a possible future chiral perturbation theory calculation. Further aspects of this approach and future directions are briefly discussed.

Up toNLO3heavy-baryon chiral perturbation theory calculation for theM1properties of three-nucleon systems

M1 properties, comprising magnetic moments and radiative capture of thermal neutron observables, are studied in two- and three-nucleon systems. We utilize meson exchange current derived up to N$^3$LO using heavy baryon chiral perturbation theory a la Weinberg. Calculations have been performed for several qualitatively different realistic nuclear Hamiltonians, which permits us to analyze model dependence of our results. Our results are found to be strongly correlated with the effective range parameters such as binding energies and the scattering lengths. Taking into account such correlation, the results are in good agreement with the experimental data with small model-dependence.

Manifestly-covariant p 3 calculation of nucleon Compton scattering

We compute the Compton scattering off the nucleons in the framework of manifestly covariant baryon chiral perturbation theory (BχPT). The results for observables differ substantially from the corresponding calculations in heavy-baryon chiral perturbation theory (HBχPT), most appreciably in the forward kinematics. We verify that the covariant p 3 result fulfills the forward-Compton-scattering sum rules. We also explore the effect of the Δ(1232) resonance at order p 4/Δ, with Δ ≈ 300 MeV, the resonance excitation energy. We find that the substantial effect of the Δ-excitation on the nucleon polarizabilities can naturally be accommodated in the manifestly covariant calculation.

Chiral condensate in neutron matter

A recent chiral perturbation theory calculation of the in-medium quark condensate 〈q¯q〉 is extended to the isospin-asymmetric case of pure neutron matter. In contrast to the behavior in isospin-symmetric nuclear matter we find only small deviations from the linear density approximation. This feature originates primarily from the reduced weight factors (e.g. 1/6 for the dominant contributions) of the 2π-exchange mechanisms in pure neutron matter. Our result suggests therefore that the tendencies for chiral symmetry restoration are actually favored in systems with large neutron excess (e.g. neutron stars). We also analyze the behavior of the density-dependent quark condensate 〈q¯q〉(ρn) in the chiral limit mπ→0.

Highlights and perspectives of the JLab spin physics program

Nucleon spin structure has been an active, exciting and intriguing subject of interest for the last three decades. Recent precision spin-structure data from Jefferson Lab have significantly advanced our knowledge of nucleon structure in the valence quark (high-x) region and improved our understanding of higher-twist effects, spin sum rules and quark-hadron duality. First, results of spin sum rules and polarizabilities in the low to intermediate Q 2 region are presented. Comparison with theoretical calculations, in particular with Chiral Perturbation Theory (ChPT) calculations, are discussed. Surprising disagreements of ChPT calculations with experimental results on the generalized spin polarizability, δ LT, were found. Then, precision measurements of the spin asymmetry, A 1, in the high-x region are presented. They provide crucial input for global fits to world data to extract polarized parton distribution functions. The up and down quark spin distributions in the nucleon were extracted. The results for Δd/d disagree with the leading-order pQCD prediction assuming hadron helicity conservation. Results of precision measurements of the g 2 structure function to study higher-twist effects are presented. The data indicate a significant higher-twist (twist-3 or higher) effect. The second moment of the spin structure functions and the twist-3 matrix element d 2 results were extracted. The high Q 2 result was compared with a Lattice QCD calculation. Results on the resonance spin-structure functions in the intermediate Q 2 range are presented, which, in combination with DIS data, enable a detailed study of quark-hadron duality in spin-structure functions. Finally, an experiment to study neutron transversity and transverse spin asymmetries is discussed. A future plan with the 12 GeV energy upgrade at JLab is briefly outlined.

A parametrization of the baryon octet and decuplet masses

We construct a general parametrization of the baryon octet and decuplet mass operators including the three-body terms using the unit operator and the symmetry-breaking factors Md = diag(0, 1, 0) and Ms = diag(0, 0, 1) in conjunction with the spin operators. Our parametrization has the minimal number of operators needed to describe all the octet and decuplet masses. Investigating the likely size of the three-body terms, we find that contributions of the three-body hypercharge splittings are comparable to those from the one- and two-body isospin splittings and that contributions of the three-body isospin splitting operators are very small. We prove that, in dynamical calculations, one must go to three loops to get the three-body terms. We also find that the suggested hierarchy of sizes for terms in the most general expression for baryon masses that involve multiple factors of Md and/or Ms does not hold strictly for dynamical calculations in heavy-baryon chiral perturbation theory: terms of a given order in a meson loop expansion may appear both with the expected factors of Md and Ms, and with one factor more.

On the two-loop contributions to the pion mass

We derive a simplified representation for the pion mass to two loops in three-flavour chiral perturbation theory. For this purpose, we first determine the reduced expressions for the tensorial two-loop 2-point sunset integrals arising in chiral perturbation theory calculations. Making use of those relations, we obtain the expression for the pion mass in terms of the minimal set of master integrals. On the basis of known results for these, we arrive at an explicit analytic representation, up to the contribution from KKη intermediate states where a closed-form expression for the corresponding sunset integral is missing. However, the expansion of this function for a small pion mass leads to a simple representation which yields a very accurate approximation of this contribution. Finally, we also give a discussion of the numerical implications of our results.

Heavy-light semileptonic decays in staggered chiral perturbation theory

We calculate the form factors for the semileptonic decays of heavy-light pseudoscalar mesons in partially quenched staggered chiral perturbation theory ($\mathrm{S}\ensuremath{\chi}\mathrm{PT}$), working to leading order in $1/{m}_{Q}$, where ${m}_{Q}$ is the heavy-quark mass. We take the light meson in the final state to be a pseudoscalar corresponding to the exact chiral symmetry of staggered quarks. The treatment assumes the validity of the standard prescription for representing the staggered ``fourth-root trick'' within $\mathrm{S}\ensuremath{\chi}\mathrm{PT}$ by insertions of factors of $1/4$ for each sea-quark loop. Our calculation is based on an existing partially quenched continuum chiral perturbation theory calculation with degenerate sea quarks by Be\ifmmode \acute{c}\else \'{c}\fi{}irevi\ifmmode \acute{c}\else \'{c}\fi{}, Prelovsek, and Zupan, which we generalize to the staggered (and nondegenerate) case. As a byproduct, we obtain the continuum partially quenched results with nondegenerate sea quarks. We analyze the effects of nonleading chiral terms, and find a relation among the coefficients governing the analytic valence mass dependence at this order. Our results are useful in analyzing lattice computations of form factors $B\ensuremath{\rightarrow}\ensuremath{\pi}$ and $D\ensuremath{\rightarrow}K$, when the light quarks are simulated with the staggered action.

Sigma exchange in the nuclear force and effective field theory

In the phenomenological description of the nuclear interaction a crucial role is traditionally played by the exchange of a scalar I=0 meson, the sigma, of mass 500-600 MeV, which however is not seen clearly in the particle spectrum and which has a very ambiguous status in QCD. I show that a remarkably simple and reasonably controlled combination of ingredients can reproduce the features of this part of the nuclear force. The use of chiral perturbation theory calculations for two pion exchange supplemented by the Omnes function for pion rescattering suffices to reproduce the magnitude and shape of the exchange of a supposed $\sigma$ particle, even though no such particle is present in this calculation. I also show how these ingredients can describe the contact interaction that enters more modern descriptions of the internucleon interaction.

Erratum: Chiral corrections to kaon-nucleon scattering lengths [Phys. Rev. C 64, 045204 (2001)

We calculate the threshold T-matrices of kaon-nucleon and antikaon-nucleon scattering to one loop order in SU(3) heavy baryon chiral perturbation theory. To that order the complex-valued isospin-1 $\bar KN$ threshold T-matrix can be successfully predicted from the isospin-0 and 1 $KN$ threshold T-matrices. As expected perturbation theory fails to explain the isospin-0 $\bar KN$ threshold T-matrix which is completely dominated by the nearby subthreshold $\Lambda^*(1405)$-resonance. Cancelations of large terms of second and third chiral order are observed as they seem to be typical for SU(3) baryon chiral perturbation theory calculations. We also give the kaon and eta loop corrections to the $\pi N$ scattering lengths and we investigate $\pi\Lambda$ scattering to one-loop order. The second order s-wave low-energy constants are all of natural size and do not exceed 1 GeV$^{-1}$ in magnitude.

Finite volume corrections toππscattering

Lattice QCD studies of hadron-hadron interactions are performed by computing the energy levels of the system in a finite box. The shifts in energy levels proportional to inverse powers of the volume are related to scattering parameters in a model independent way. In addition, there are nonuniversal exponentially suppressed corrections that distort this relation. These terms are proportional to ${e}^{\ensuremath{-}{m}_{\ensuremath{\pi}}L}$ and become relevant as the chiral limit is approached. In this paper we report on a one-loop chiral perturbation theory calculation of the leading exponential corrections in the case of $I=2$ $\ensuremath{\pi}\ensuremath{\pi}$ scattering near threshold.

The pion and proton mass in finite volume

We calculate the finite volume effects for the pion and nucleon mass. For the pion mass we present the results of a full two-loop calculation in chiral perturbation theory. The outcome shows that the resummed version of the Luscher formula we presented in an earlier work does indeed give an excellent approximation to the two-loop result. In view of this result we apply the same resummed formula to the nucleon mass. In the nucleon sector the extension of the chiral expansion to higher quark masses appears to be more problematic and it is therefore more difficult to make reliable predictions for the size of the finite volume effects. We present some preliminary numerical estimates.

Minimal electromagnetic and mass difference corrections in πNscattering

Coulomb and mass difference corrections to low-energy pion-nucleon scattering are calculated and compared with previous work including potential models, dispersion relation methods and chiral perturbation theory calculations. Particular attention is paid to their role in testing isospin breaking.

Influence of quark boundary conditions on the pion mass in finite volume

We calculate the mass shift for the pion in a finite volume with renormalization group (RG) methods in the framework of the quark-mesons model. In particular, we investigate the importance of the quark effects on the pion mass. As in lattice gauge theory, the choice of quark boundary conditions has a noticeable effect on the pion mass shift in small volumes, in addition to the shift due to pion interactions. We compare our results to chiral perturbation theory calculations and find differences due to the fact that chiral perturbation theory only considers pion effects in the finite volume.

Pion-pion scattering amplitude

We obtain reliable $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitudes consistent with experimental data, both at low and high energies, and fulfilling appropriate analyticity properties. We do this by first fitting experimental low energy (${s}^{1/2}\ensuremath{\le}1.42\text{ }\mathrm{GeV}$) phase shifts and inelasticities with expressions that incorporate analyticity and unitarity. In particular, for the S wave with isospin 0, we discuss in detail several sets of experimental data. This provides low energy partial wave amplitudes that summarize the known experimental information. Then, we impose Regge behavior as follows from factorization and experimental data for the imaginary parts of the scattering amplitudes at higher energy, and check fulfillment of dispersion relations up to 0.925 GeV. This allows us to improve our fits. The ensuing $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitudes are then shown to verify dispersion relations up to 1.42 GeV, as well as $s\ensuremath{-}t\ensuremath{-}u$ crossing sum rules and other consistency conditions. The improved parametrizations therefore provide a reliable representation of pion-pion amplitudes with which one can test chiral perturbation theory calculations, pionium decays, or use as input for $CP$-violating $K$ decays. In this respect, we find $[{a}_{0}^{(0)}\ensuremath{-}{a}_{0}^{(2)}{]}^{2}=(0.077\ifmmode\pm\else\textpm\fi{}0.008){M}_{\ensuremath{\pi}}^{\ensuremath{-}2}$ and ${\ensuremath{\delta}}_{0}^{(0)}({m}_{K}^{2})\ensuremath{-}{\ensuremath{\delta}}_{0}^{(2)}({m}_{K}^{2})=52.9\ifmmode\pm\else\textpm\fi{}{1.6}^{\mathrm{o}}$.

Finite volume effects in chiral perturbation theory

There has recently been an intense activity in the study of finite volume effects by means of chiral perturbation theory. In this contribution I review recent work in this field both for the epsilon-- (Mpi L \lesssim 1) and the p--regime (Mpi L \gg 1). For the latter I emphasize the importance of going beyond leading order calculations in chiral perturbation theory and the usefulness of asymptotic formulae a la L"uscher used in combination with CHPT.

Finite volume effects using lattice chiral perturbation theory

Lattice regularization is used to perform chiral perturbation theory calculations in finite volume. The lattice spacing is chosen small enough to be irrelevant, and numerical results are obtained from simple summations.

Light quark mass ratio from the Dalitz plot ofη→π+π−π0decay

High statistics Dalitz-plot distribution of $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ decay obtained recently by KLOE collaboration is fitted to the results of corresponding theoretical calculations in Chiral Perturbation Theory (ChPT) with unitarity corrections taken into account. The quark mass ratio $Q=\sqrt{[{m}_{s}^{2}\ensuremath{-}({m}_{d}+{m}_{u}{)}^{2}/4]/({m}_{d}^{2}\ensuremath{-}{m}_{u}^{2})}$ can be obtained from this analysis. We get $Q=22.8\ifmmode\pm\else\textpm\fi{}0.4$ which differs from the value ${Q}_{DT}=24.2$ that follows from Dashen's theorem and agrees with recently calculated electromagnetic kaon mass difference.

Measurement of the Generalized Forward Spin Polarizabilities of the Neutron

The generalized forward spin polarizabilities gamma(0) and delta(LT) of the neutron have been extracted for the first time in a Q2 range from 0.1 to 0.9 GeV2. Since gamma(0) is sensitive to nucleon resonances and delta(LT) is insensitive to the Delta resonance, it is expected that the pair of forward spin polarizabilities should provide benchmark tests of the current understanding of the chiral dynamics of QCD. The new results on delta(LT) show significant disagreement with chiral perturbation theory calculations, while the data for gamma(0) at low Q2 are in good agreement with a next-to-leading-order relativistic baryon chiral perturbation theory calculation. The data show good agreement with the phenomenological MAID model.

Relative branching ratio of the η→π0γγ decay channel

The η→π0γγ rare decay was measured at the AGS with the Crystal Ball photon spectrometer and its relative branching ratio was extracted to be B1=(8.3±2.8±1.4)×10−4, based on the analysis of 3×107 detected η mesons. This leads to a lower partial width for this eta channel than past measurements and is in line with chiral perturbation theory calculations.