Pion Properties Research Articles

A field theoretical model for QCD thermodynamics

The PNJL model is used to study the thermodynamics of Nf = 2 quark flavours interacting through chiral four-point couplings and propagating in a homogeneous Polyakov loop background. The input is fixed exclusively by selected pure-gauge lattice QCD results and by pion properties in vacuum. We present successful comparisons with lattice QCD thermodynamics expanded to finite chemical potential μq. Quark number susceptibilities at finite quark chemical potential are investigated. The validity of the Taylor expansion is then examined through a comparison between the full and truncated PNJL model calculations.

Polyakov loop, diquarks, and the two-flavor phase diagram

An updated version of the PNJL model is used to study the thermodynamics of ${N}_{f}=2$ quark flavors interacting through chiral four-point couplings and propagating in a homogeneous Polyakov loop background. Previous PNJL calculations are extended by introducing explicit diquark degrees of freedom and an improved effective potential for the Polyakov loop field. The mean field equations are treated under the aspect of accommodating group theoretical constraints and issues arising from the fermion sign problem. The input is fixed exclusively by selected pure-gauge lattice QCD results and by pion properties in vacuum. The resulting $(T,\ensuremath{\mu})$ phase diagram is studied with special emphasis on the critical point, its dependence on the quark mass and on Polyakov loop dynamics. We present successful comparisons with lattice QCD thermodynamics expanded to finite chemical potential $\ensuremath{\mu}$.

Phases of QCD, Polyakov Loop and Quasiparticles

QCD thermodynamics is studied using a model which combines Polyakov loop dynamics with spontaneous chiral symmetry breaking and its restoration (the PNJL model). The input is fixed entirely by pure-gauge lattice QCD results and by pion properties in vacuum. Successful comparisons with results from thermal lattice QCD are achieved, including extrapolations to finite quark chemical potential. The phase diagram and selected susceptibilties for two quark flavors (Nf = 2) are investigated with inclusion of diquark degrees of freedom.

Anatoly Ivanovich Larkin

Anatoly Ivanovich Larkin—“Tolya” to his friends and colleagues—died of heart failure on 4 August 2005 in Aspen, Colorado. He was a universally recognized leader in condensed matter theory and a celebrated teacher of several generations of theorists. Until the very end, he was remarkably productive and remained a desirable collaborator for many younger colleagues. His recent publications have the same depth and originality as the first paper he published 47 years ago.Born on 14 October 1932 in the small town of Kolomna in central Russia, Larkin went to Moscow to study physics at the Institute for Physical Engineering. He was taught by Igor Tamm, Mikhail Leontovich, Isaac Pomeranchuk, and Arkadii Migdal, and after two years of research under Andrei Sakharov’s guidance, he joined Migdal’s group at the Kurchatov Institute. In 1966 he moved to the newly organized L. D. Landau Institute for Theoretical Physics in Chernogolovka, near Moscow, which became his home for the next 25 years. In 1995 he joined the University of Minnesota as a professor of theoretical physics, but he returned to Chernogolovka every summer.Larkin’s research career began at the time when young theorists were still expected to study all branches of theoretical physics. In his first papers, Larkin applied the newly developed Matsubara diagrammatics to the calculation of the thermodynamics of plasmas and the energy losses of fast particles passing through them. In a series of joint publications with Migdal, Larkin used the Landau Fermi-liquid theory to describe the properties of atomic nuclei.Following the development of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity in 1957, Larkin began his studies of that subject, which was to become his lifelong passion. Although he made a significant contribution to particle and nuclear physics and to the theories of phase transitions, magnetism, and disorder, it was superconductivity to which he kept returning and in which his impact was truly outstanding. Among his greatest achievements are the prediction of the fluctuation-driven enhancement of conductivity above the critical temperature T c, the theory of the Josephson effect in superconductor–normal metal–superconductor junctions and in superconducting point contacts, the prediction of pairing with finite momentum, a semiclassical theory of nonequilibrium phenomena including nonlinear flux flow, the theory of collective creep, and thermally activated flux dynamics in the high-T c materials. In 1961 Valentin Vaks and Larkin suggested that the properties of soft pions could be explained by a BCS-like spontaneous breaking of chiral symmetry. That concept proved to be an important step toward the creation of the standard model of particle physics. In a 1969 paper, Larkin solved the problem of the singularity of thermodynamic functions at the type II phase transition in uniaxial ferroelectrics. It was the first successful application of the renormalization group to condensed matter theory. The paper was significant in the development of the theory of critical phenomena. Larkin also made a great impact on the physics of one-dimensional metals. His calculations of the Green’s functions of fermions and the correlation functions of density and spin density laid the foundations of the modern theory of the field. His further contributions included applying the theory of collective pinning to the dynamics of charge density waves, and developing the theory of the thermal activation of Frölich’s conductivity. In the theory of disordered conductors, Larkin and his collaborators initiated several new research directions: weak localization, mesoscopics, and quantum chaos. Larkin was also known as a great teacher, one who could work with students of different personalities. Lev Aslamazov, Vladimir Fileov, Lev Ioffe, Vadim Geshkenbein, Victor Galitski, Chushun Tian, and the four of us all did PhDs (or equivalent works) under his supervision. Yuri Dreyzin, Konstantin Matveev, and Valery Rupasov were his students as undergraduates. Working with Larkin created a unique feeling of joy, which attracted many different collaborators to him.Larkin received many formal signs of recognition: He was elected a full member of the Russian Academy of Sciences (1991), and was a recipient of numerous prizes, including the Fritz London Memorial Award (1990), the Hewlett-Packard Europhysics Prize (1993), and the Lars Onsager Prize (2002). He was greatly respected by his colleagues, who admired him for his talent, modesty, and tireless work.His death in the Colorado mountains was entirely unexpected. He was still hard at work on several ambitious projects. The coming years will doubtless see the publication of new papers under the familiar name of A. I. Larkin. Anatoly Ivanovich Larkin JOHN LINNPPT|High resolution© 2006 American Institute of Physics.

Chiral symmetry restoration and pion properties in a q-deformed NJL model

We review the implementation of a q-deformed fermionic algebra in the Nambu-Jona-Lasinio model (NJL). The gap equations obtained from a deformed condensate as well as from the deformation of the NJL Hamiltonian are discussed. The effect of both temperature and deformation in the chiral symmetry restoration process as well as in the pion properties is studied.

Residual interaction effects on deeply bound pionic states in Sn and Pb isotopes

We have studied the residual interaction effects theoretically on the deeply bound pionic states in Pb and Sn isotopes. We need to evaluate the residual interaction effects carefully in order to deduce the nuclear medium effects for pion properties, which are believed to provide valuable information on nuclear chiral dynamics. The s- and p-wave $\pi-N$ interactions are used for the pion-nucleon residual interactions. We show that the complex energy shifts are around [(10-20)+i(2-7)]keV for 1s states in Sn, which should be taken into account in the analyses of the high precision data of deeply bound pionic $1s$ states in Sn isotopes.

Twisted mass chiral perturbation theory at next-to-leading order

We study the properties of pions in twisted mass lattice QCD (with two degenerate flavors) using chiral perturbation theory (ChPT). We work to next-to-leading order (NLO) in a power counting scheme in which m_q ~ a \Lambda_QCD^2, with m_q the physical quark mass and a the lattice spacing. We argue that automatic O(a) improvement of physical quantities at maximal twist, which has been demonstrated in general if m_q >> a \Lambda_QCD^2, holds even if m_q ~ a \Lambda_QCD^2, as long as one uses an appropriate non-perturbative definition of the twist angle. We demonstrate this with explicit calculations, for arbitrary twist angle, of all pionic quantities that involve no more than a single pion in the initial and final states: masses, decay constants, form factors and condensates, as well as the differences between alternate definitions of twist angle. We also calculate the axial and pseudoscalar form factors of the pion, quantities which violate flavor and parity, and which vanish in the continuum limit. These are of interest because they are not automatically O(a) improved at maximal twist. They allow a determination of the unknown low energy constants introduced by discretization errors, and provide tests of the accuracy of ChPT at NLO. We extend our results into the regime where m_q ~ a^2 \Lambda_QCD^3, and argue in favor of a recent proposal that automatic O(a) improvement at maximal twist remains valid in this regime.

Neutral pion decay in dense Skyrmion matter

We study the density dependence of the decay ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ using the Skyrme Lagrangian to describe simultaneously both the matter background and mesonic fluctuations. The classical ground state configuration has different chiral properties depending on the Skyrmion density, which is reflected in the physical properties of pion fluctuating on top of the classical background. This leads to large suppression at high density of both photo-production from the neutral pion and the reverse process. The effective charges of ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$ are also discussed in the same framework.

In-medium chiral perturbation theory

We report on how to tackle the problem of establishing a chiral effective field theory in nuclear matter with explicit pion fields and in the presence of external sources (Ann. Phys. 297. 27 (2002)). We have made use of the results of J.A. Oller (Phys. Rev. C 65, 025204 (2002)) where the generating functional for the in-medium chiral SU (2) x SU (2) Lagrangian has been derived. Within this approach we develop the so-called standard power counting rules for the calculation of in-medium pion properties if the residual nucleon energies are of the order of the pion mass. In addition, for the case of vanishing residual nucleon energies. a modified scheme (non-standard counting) is introduced. For both schemes the pertinent scales where the chiral expansions have to break down are established as well. We have performed a systematic analysis of n-point in-medium Green functions up to and including next-to-leading order when the standard rules apply. These include the in-medium contributions to quark condensates, pion propagators, pion masses and couplings of the axial-vector, vector and pseudoscalar currents to pions.

Sliding vacua in dense skyrmion matter

In continuation of our systematic effort to understand hadronic matter at high density, we study dense skyrmion matter and its chiral phase structure in an effective field theory implemented with the trace anomaly of QCD applicable in the large N c limit. By incorporating a dilaton field χ associated with broken conformal symmetry of QCD into the simplest form of skyrmion Lagrangian, we simulate the effect of “sliding vacua” influenced by the presence of matter and obtain what could correspond to the “intrinsic dependence” on the background of the system, i.e., matter density or temperature, that results when a generic chiral effective field theory of strong interactions is matched to QCD at a matching scale near the chiral scale Λ χ ∼4 πf π ∼1 GeV. The properties of the Goldstone pions and the dilaton scalar near the chiral phase transition are studied by looking at the pertinent excitations of given quantum numbers on top of a skyrmion matter and their behavior in the vicinity of the phase transition from Goldstone mode to Wigner mode characterized by the changeover from the FCC crystal to the half-skyrmion CC crystal. We recover from the model certain features that are connected to Brown–Rho scaling and that suggest how to give a precise meaning to the latter in the framework of an effective field theory that is matched to QCD.

In-medium chiral perturbation theory

This talk will report about a systematical implementation of a chiral effective field theory in nuclear matter with explicit pion fields and in the presence of external sources[1]. Within the generating functional approach of Ref.[2] the so-called standard power counting rules for the calculation of in-medium pion properties are developed that apply if the residual nucleon energies are of the order of the pion mass. In addition, for the case of vanishing residual nucleon energies, a modified scheme (non-standard counting) is introduced. For both schemes the pertinent scales where the chiral expansions have to break down are established as well. We report about a systematic analysis of n-point in-medium Green functions up to and including next-to-leading order when the standard rules apply. These include the in-medium contributions to quark condensates, pion propagators, pion masses and couplings of the axial-vector, vector and pseudoscalar currents to pions.

In-medium Chiral Perturbation Theory beyond the Mean-Field Approximation

An explicit expression for the generating functional of two-flavor low-energy QCD with external sources in the presence of nonvanishing nucleon densities was derived recently (J. A. Oller, Phys. Rev. C65 (2002) 025204). Within this approach we derive power counting rules for the calculation of in-medium pion properties. We develop the so-called standard rules for residual nucleon energies of the order of the pion mass and a modified scheme (nonstandard counting) for vanishing residual nucleon energies. We also establish the different scales for the range of applicability of this perturbative expansion, which are 6πfπ≃0.7 GeV for standard and 6π2fπ2/2mN≃0.27 GeV for nonstandard counting, respectively. We have performed a systematic analysis of n-point in-medium Green functions up to and including next-to-leading order when the standard rules apply. These include the in-medium contributions to quark condensates, pion propagators, pion masses, and couplings of the axial-vector, vector, and pseudoscalar currents to pions. In particular, we find a mass shift for negatively charged pions in heavy nuclei, ΔMπ−=(18±m 5) MeV, that agrees with recent determinations from deeply bound pionic 207Pb. We have also established the absence of in-medium renormalization in the π0→γγ decay amplitude up to the same order. The study of ππ scattering requires the use of the nonstandard counting and the calculation is done at leading order. Even at that order we establish new contributions not considered so far. We also point toward further possible improvements of this scheme and touch upon its relation to more conventional many-body approaches.

Mesonic fluctuations in a nonlocal Nambu–Jona-Lasinio model

The effects of meson fluctuations are studied in a nonlocal generalization of the Nambu–Jona-Lasinio model, by including terms of next-to-leading order (NLO) in 1/ N c . In the model with only scalar and pseudoscalar interactions NLO contributions to the quark condensate are found to be very small. This is a result of cancellation between virtual mesons and Fock terms, which occurs for the parameter sets of most interest. In the quark self-energy, similar cancellations arise in the tadpole diagrams, although not in other NLO pieces which contribute at the ∼25% level. The effects on pion properties are also found to be small. NLO contributions from real ππ intermediate states increase the sigma-meson mass by ∼30%. In an extended model with vector and axial interactions, there are indications that NLO effects could be larger.

Resonating mean-field theoretical description of [formula omitted] and [formula omitted] mesons by the Nambu–Jona-Lasinio model

In order to show the advantage of the resonating (Res-) mean-field theory for a fermion system with large quantum fluctuations over the usual mean-field theory, we have applied it successfully to the Nambu–Jona-Lasinio model to describe precisely phenomena associated with pionic excitation. We apply it further to describe more accurately phenomena which are associate simultaneously with pionic and sigma-mesonic excitations. A state with large quantum fluctuations is approximated by a superposition of three Dirac seas composed of constituent quark masses, i.e., nonorthogonal Slater determinants (S-dets) with different correlation structures. We make a direct optimization of the Res-mean-field ground-state energy with respect to the Res-mean-field parameters, the constituent quark masses. Then the Res-mean-field ground and excited states generated with the S-dets explain most of mass spectra and associated properties of pion and sigma meson.

Erratum: Enhanced chiral logarithms in partially quenched QCD [Phys. Rev. D 56, 7052 (1997)

I discuss the properties of pions in ``partially quenched'' theories, i.e. those in which the valence and sea quark masses, $m_V$ and $m_S$, are different. I point out that for lattice fermions which retain some chiral symmetry on the lattice, e.g. staggered fermions, the leading order prediction of the chiral expansion is that the mass of the pion depends only on $m_V$, and is independent of $m_S$. This surprising result is shown to receive corrections from loop effects which are of relative size $m_S \ln m_V$, and which thus diverge when the valence quark mass vanishes. Using partially quenched chiral perturbation theory, I calculate the full one-loop correction to the mass and decay constant of pions composed of two non-degenerate quarks, and suggest various combinations for which the prediction is independent of the unknown coefficients of the analytic terms in the chiral Lagrangian. These results can also be tested with Wilson fermions if one uses a non-perturbative definition of the quark mass.

Pion Properties as a Goldstone Boson in the Dual Ginzburg–Landau Theory**The project supported in part by National Natural Science Foundation of China, the Doctoral Program Foundation of Institution of Higher Education, and the State Education Commission of China

By solving the Schwinger–Dyson equation for quark self-energy and Bethe–Salpeter equation for pseudo-scalar vertex, we investigate the static pion properties in the dual Ginzburg–Landau theory that includes a description of confinement. Pion mass formula and the Gell–Mann–Oakes–Renner relation are shown. The pion mass, resulting from the. current quark mass, and pion decay constant are calculated. Our results show that the dynamical effect of light quarks is unimportant for the mass of pion.

Pion properties in the 1/ Nc-corrected NJL model

We investigate the effect of mesonic fluctuations on the pion propagator in the Nambu–Jona-Lasinio (NJL) model by explicitly taking into account 1/ N c -corrections. Because of the non-renormalizability of the model we have to regularize the meson loops with an independent cutoff parameter Λ M . Whereas for moderate values of Λ M the pion properties change only quantitatively we encounter strong instabilities for larger values of Λ M .

In-medium dependence and Coulomb effects of the pion production in heavy ion collisions

The properties of the high energy pions observed in heavy ion collisions, in particular in the system Au on Au at 1 GeV/nucleon are investigated. The reaction dynamics is described within the Quantum Molecular Dynamics (QMD) approach. It is shown that high energy pions freeze out early and originate from the hot, compressed matter. N*-resonances are found to give an important contribution toward the high energy tail of the pion. Further the role of in-medium effects in the description of charged pion yields and spectra is investigated using a microscopic potential derived from the Brueckner G-matrix which is obtained with the Reid soft-core potential. It is seen that the high energy part of the spectra is relatively more suppressed due to in-medium effects as compared to the low energy part. A comparison to experiments further demonstrates that the present calculations describe reasonably well the neutral (TAPS) and charged (FOPI) pion spectra. The observed energy dependence of the π−/π+ ratio, i.e. deviations from the isobar model prediction, is due to Coulomb effects and again indicate that high energy pions probe the hot and dense phase of the reaction. These findings are confirmed independently by a simple phase space analysis.

Enhanced chiral logarithms in partially quenched QCD

I discuss the properties of pions in {open_quotes}partially quenched{close_quotes} theories, i.e., those in which the valence and sea quark masses m{sub V} and m{sub S} are different. I point out that for lattice fermions which retain some chiral symmetry on the lattice, e.g., staggered fermions, the leading order prediction of the chiral expansion is that the mass of the pion depends only on m{sub V}, and is independent of m{sub S}. This surprising result is shown to receive corrections from loop effects which are of relative size m{sub S}lnm{sub V}, and which thus diverge when the valence quark mass vanishes. Using partially quenched chiral perturbation theory, I calculate the full one-loop correction to the mass and decay constant of pions composed of two nondegenerate quarks, and suggest various combinations for which the prediction is independent of the unknown coefficients of the analytic terms in the chiral Lagrangian. These results can also be tested with Wilson fermions if one uses a nonperturbative definition of the quark mass. {copyright} {ital 1997} {ital The American Physical Society}

Measurements of pion properties and production amplitudes: Probing confinement scale QCD

Two basic characteristics of QCD at low energies are the appearance of the chiral anomaly (e.g. responsible for the π 0 → γγ decay) and the prediction of Goldstone Bosons due to the spontaneous breaking of chiral symmetry. It is therefore important to accurately measure the properties, low energy interactions, and the production and decay rates of the approximate Goldstone Bosons (π, η, K) which are found in nature. These experimental data will provide a test for any theory of the strong interactions at low energies. Here a subset of such experiments which can be performed with low energy electromagnetic accelerators and e + e − colliders will be examined. The present status of the near threshold γp → πN reaction with real and virtual photons is reviewed since there has been considerable recent progress in the study of this reaction. The status of the radiative lifetime and transition form factors of the π 0, η and η′ mesons are briefly reviewed and some possible future work is suggested.