7 pages, 3 figures, contribution to the International School of Nuclear Physics, 33rd Course: From Quarks and Gluons to Hadrons and Nuclei, Erice-Sicily: 16-24 September 2011
The latest experimental results in relativistic heavy-ion collisions show that the matter there produced requires transport coefficients because of the important collective properties found. We review the theoretical calculation of these transport coefficients in the hadron side at low temperatures by computing them in a gas composed of low energy pions. The interaction of these pions is taken from an effective chiral theory and further requiring scattering unitarity. The propagation of D and D* mesons in the thermalized pion gas is also studied in order to extract the heavy quark diffusion coefficients in the system.
Pion-pion scattering amplitude obtained from one-loop Chiral Perturbation Theory (ChPT) is crossing symmetric, however the corresponding partial wave amplitudes do not respect exact unitarity relation. There are different approaches to get unitarized partial wave amplitudes from ChPT. Here we consider the inverse amplitude method (IAM) that is often used to fit pion-pion phase shifts to experimental data, by adjusting free parameters. We measure the amount of crossing symmetry violation (CSV) in this case and we show that crossing symmetry is badly violated by the IAM unitarized ChPT amplitude…
The Vector Pion form factor below 1 GeV is analyzed using experimental data on its modulus, the P-wave pion pion phase shifts and dispersion relation. It is found that causality is satisfied. Using dispersion relation, terms proportional to s squared and s cubed are calculated using the experimental data, where s is the momentum transfer. They are much larger than the one-loop and two-loop Chiral Perturbation Theory calculations. Unitarized model calculations agree very well with dispersion relation results.
We calculate the pion-pion elastic scattering phase shifts for pion masses from the chiral limit to values of interest for lattice studies. At low energies, we use the standard Chiral Perturbation Theory expressions to one and two loops. In addition, we study the phase shifts mass dependence in the resonance region by means of dispersion theory in the form of unitarized Chiral Perturbation Theory and the Inverse Amplitude Method. We pay particular attention to the case when resonances are close to threshold, illustrating the different behavior between scalar and vector resonances. We also prov…
We report on our recent results in the study of the chiral extrapolation of the phase-shifts in elastic pion-pion scattering, using both standard and unitarized ChPT to one and two loops. In the standard ChPT approach, limited to low momenta, we study the S, P and D waves. Unitarization extends the analysis to energies of around 1 GeV, being compatible with standard ChPT at low energies for the S and P waves. We then compare with lattice results and find a good agreement of standard ChPT below p \sim 200 MeV for the I=2, J=0 and I=1, J=1 channels and up to p \sim 500 MeV for the I=2, J=2 chann…
Pion-pion scattering amplitude obtained from one-loop Chiral Perturbation Theory (ChPT) is crossing symmetric, however the corresponding partial-wave amplitudes do not respect exact unitarity relation. There are different approaches to get unitarized results from ChPT. Here we consider the inverse amplitude method (IAM) and, using the Roskies relations, we measure the amount of crossing symmetry violation when IAM is used in order to fit pion-pion phase-shifts to experimental data in the resonance region. We also show the unitarity violation of the crossing symmetric ChPT amplitude with its pa…
In a recent work on low energy pion-nucleon scattering, instead of using chiral perturbation theory (ChPT) amplitude, we started from a pion-nucleon {\it soft-pion} result and used elastic unitarity directly as a dynamical constraint to construct first-order unitarity corrected amplitudes. The resulting amplitudes are crossing symmetric but, as the ChPT ones, satisfy only approximate unitarity relation. In the present work, we reconsider our approach and we apply the inverse amplitude method (IAM) in order to access the energy resonance region. We present the resulting S- and P-wave phase shif…
We extend our previous analysis of the unitarized pion-nucleon scattering amplitude including up to fourth order terms in Heavy Baryon Chiral Perturbation Theory. We pay special attention to the stability of the generated Delta(1232 resonance, the convergence problems and the power counting of the chiral parameters.
By using unitarized two-loop Chiral Perturbation Theory partial waves to describe pion-pion scattering we find that the dominant component of the lightest scalar meson does not follow the q-qbar dependence on the number of colors that, in contrast, is obeyed by the lightest vectors. The method suggests that a subdominant q-qbar component of the f_0(600) possibly originates around 1 GeV.
We study the dependence of the pion-pion scattering phase shifts on the light quark mass in both standard and unitarized SU(2) Chiral Perturbation Theory (ChPT) to one and two loops. We then use unitarized SU(3) ChPT to study the elastic f_0(600), kappa(800), rho(770) and K*(892) resonances. The quark masses are varied up to values of interest for lattice studies. We find a very soft dependence on the light quark mass of the pion-pion phase shifts at one loop and slightly stronger at two loops and a good agreement with lattice results. The SU(3) analysis shows that the properties of the rho(77…
A unitarized nonrelativistic meson model which is successful for the description of the heavy and light vector and pseudoscalar mesons yields, in its extension to the scalar mesons but for the same model parameters, a complete nonet below 1 GeV. In the unitarization scheme, real and virtual meson-meson decay channels are coupled to the quark-antiquark confinement channels. The flavor-dependent harmonic-oscillator confining potential itself has bound states epsilon(1.3 GeV), S(1.5 GeV), delta(1.3 GeV), kappa(1.4 GeV), similar to the results of other bound-state qqbar models. However, the full c…
By means of the Inverse Amplitude Method we unitarize the elastic pion-nucleon scattering amplitudes obtained from Heavy Baryon Chiral Perturbation Theory to O(q^3). Within this approach we can enlarge their applicability range and generate the Delta(1232) resonance. We can find a reasonable description of the pion nucleon phase shifts with (q^2) parameters in agreement with the resonance saturation hypothesis. However, the uncertainties in the analysis of the low energy data as well as the large number of chiral parameters, which can have strong correlations, allow us to obtain very good fits…
We show how the unitarized description of pion nucleon scattering within Heavy Baryon Chiral Perturbation Theory can be considerably improved, by a suitable reordering of the expansion over the nucleon mass. Within this framework, the $Δ$ resonance and its associated pole can be recovered from the chiral parameters obtained from low-energy determinations. In addition, we can obtain a good description of the six $S$ and $P$ wave phase shifts in terms of chiral parameters with a natural size and compatible with the Resonance Saturation Hypothesis.
We present our recent results for the rho and sigma mesons considered as resonances in pion-pion scattering in a thermal bath. We use chiral perturbation theory to fourth order in p for the low energy behaviour, then extend the analysis via the unitarization method of the Inverse Amplitude into the resonance region. The width of the rho broadens about twice the amount required by phase space considerations alone, its mass staying practically constant up to temperatures of order 150 MeV. The sigma meson behaves in accordance to chiral symmetry restoration expectations.
The first three coefficients of the Taylor's series expansion of the vector pion form factor as a function of the momentum transfer are evaluated using experimental data on the pion form factor and the P-wave pi pi phase shifts. The real part of the form factor as a function of energy is also calculated by dispersion relation. Comparisons these results with Chiral Perturbation Theory and unitarized models are given.
Using unitarized Chiral Perturbation Theory methods, we perform a detailed analysis of the $ππ$ scattering poles $f_0(600)$ and $ρ(770)$ behaviour when medium effects such as temperature or density drive the system towards Chiral Symmetry Restoration. In the analysis of real poles below threshold, we show that it is crucial to extend properly the unitarized amplitudes so that they match the perturbative Adler zeros. Our results do not show threshold enhancement effects at finite temperature in the $f_0(600)$ channel, which remains as a pole of broad nature. We also implement T=0 finite density…
We calculate the bulk viscosity of the massive pion gas within Unitarized Chiral Perturbation Theory. We obtain a low temperature peak arising from explicit conformal breaking due to the pion mass and another peak near the critical temperature, dominated by the conformal anomaly through gluon condensate terms. The correlation between bulk viscosity and conformal breaking supports a recent QCD proposal. We discuss the role of resonances, heavier states and large-$N_c$ counting.
We use low energy pion-pion phase-shifts in order to make distinction between the alternatives for the value of the quark-antiquark condensate $B_0$ in the chiral limit. We will consider the amplitude up to and including ${\cal O}(p^4)$ contributions within the Standard and Generalized Chiral Perturbation Theory frameworks. They are unitarized by means of Padé approximants in order to fit experimental phase-shifts in the resonance region. As the best fits correspond to $α= β= 1$, we conclude that pion-pion phase-shift analysis favors the standard ChPT scenario, which assumes just one, large le…
We study thermal properties of a large-N massless pion gas using a low-energy QCD approach given by an $O(N +1)/O(N)$ Nonlinear Sigma Model. We build diagrammatically the associated finite free energy to $O(TM^{3})$ in the pion mass expansion through an effective vertex that considers all the contributions coming from closed diagrams. Subsequently, we calculate finite order parameters such as the quark condensate and its respective derivative, i.e., the scalar susceptibility, in the chiral limit, along with their associated critical exponents. These results are compared with our previous unita…
The unitary isobar model MAID2007 has been developed to analyze the world data of pion photo- and electroproduction. The model contains both a common background and several resonance terms. The background is unitarized according to the K-matrix prescription, and the 13 four-star resonances with masses below 2 GeV are described by appropriately unitarized Breit-Wigner forms. The data have been analyzed by both single-energy and global fits, and the transverse and longitudinal helicity amplitudes have been extracted for the four-star resonances below 2 GeV. Because of its inherent simplicity, MA…
We report on our present work, where by means of the Inverse Amplitude Method we unitarize the elastic pion nucleon scattering amplitudes of Heavy Barion Chiral Perturbation Theory at O(q^3). We reproduce the scattering up to the inelastic thresholds including the Delta(1232) resonance. The fitted chiral constants are rather different from those obtained by fitting the extrapolated threshold parameters for the non-unitarized theory.
We present recent results on finite temperature electromagnetic form factors and the electrical conductivity in a pion gas. The standard Chiral Perturbation Theory power counting needs to be modified for transport coefficients. We pay special attention to unitarity and to possible applications for dilepton and photon production.
We review or main results concerning the transport coefficients of a light meson gas, in particular we focus on the case of a massive pion gas. Leading order results according to the chiral power-counting are presented for the DC electrical conductivity, thermal conductivity, shear viscosity, and bulk viscosity. We also comment on the possible correlation between the bulk viscosity and the trace anomaly in QCD, as well as the relation between unitarity and a minimum of the quotient $η/s$ near the phase transition.
We study the bulk viscosity of a pion gas in unitarized Chiral Perturbation Theory at low and moderate temperatures, below any phase transition to a quark-gluon plasma phase. We argue that inelastic processes are irrelevant and exponentially suppressed at low temperatures. Since the system falls out of chemical equilibrium upon expansion,a pion chemical potential must be introduced, so we extend the existing theory to include it. We control the zero modes of the collision operator and Landau's conditions of fit when solving the Boltzmann equation with the elastic collision kernel. The dependen…
We study deuteron production using no-coalescence hydrodynamic + transport simulations of central AuAu collisions at $\sqrt{s_{NN}} = 7 - 200$ GeV. Deuterons are sampled thermally at the transition from hydrodynamics to transport, and interact in transport dominantly via $πp n \leftrightarrow πd$ reactions. The measured proton, Lambda, and deuteron transverse momentum spectra and yields are reproduced well for all collision energies considered. We further provide a possible explanation for the measured minimum in the energy dependence of the coalescence parameter, $B_2(\sqrt{s_{NN}})$ as well …
We compute the transport coefficients (drag and momentum diffusion) of the low-lying heavy baryons $Λ_c$ and $Λ_b$ in a medium of light mesons formed at the later stages of high-energy heavy-ion collisions. We employ the Fokker-Planck approach to obtain the transport coefficients from unitarized baryon-meson interactions based on effective field theories that respect chiral and heavy-quark symmetries. We provide the transport coefficients as a function of temperature and heavy-baryon momentum, and analyze the applicability of certain nonrelativistic estimates. Moreover we compare our outcome f…
During the expansion of a heavy ion collision, the system passes close to the $O(4)$ critical point of QCD, and thus the fluctuations of the order parameter $(σ, \vecπ)$ are expected to be enhanced. Our goal is to compute how these enhanced fluctuations modify the transport coefficients of QCD near the pseudo-critical point. We also make a phenomenological estimate for how chiral fluctuations could effect the momentum spectrum of soft pions. We first formulate the appropriate stochastic hydrodynamic equations close to the $O(4)$ critical point. Then, working in mean field, we determine the cor…
The electromagnetic response of an interacting system of pions has been studied at finite temperature. The corresponding transport parameter i.e. electrical conductivity has been estimated by solving the relativistic transport equation in presence of a finite electric field employing the Chapman-Enskog technique where the collision term has been treated in the relaxation time approximation. The scattering amplitudes of charged pions modeled by $ρ$ and $σ$ meson exchange using an effective Lagrangian have been obtained at finite temperature by introducing self-energy corrections in the thermal …
In the late stage of the evolution of a pion system in high-energy heavy-ion collisions when pions undergo multiple scatterings, the quantum transport of the interfering pair of identical pions plays an important role in determining the characteristics of the Hanbury-Brown-Twiss (HBT) interference. We study the quantum transport of the interfering pair using the path-integral method, in which the evolution of the bulk matter is described by relativistic hydrodynamics while the paths of the two interfering pions by test particles following the fluid positions and velocity fields. We investigate…
The bulk and shear viscosities of a pion gas is obtained by solving the relativistic transport equation in the Chapman-Enskog approximation. In-medium effects are introduced in the $ππ$ cross-section through one-loop self-energies in the propagator of the exchanged $ρ$ and $σ$ mesons. The effect of early chemical freeze-out in heavy ion collisions is implemented through a temperature dependent pion chemical potential. These are found to affect the temperature dependence of the bulk and shear viscosities significantly.
We investigate the effect of the medium on the thermal conductivity of a pion gas out of chemical equilibrium by solving the relativistic transport equation in the Chapman-Enskog and relaxation time approximations. Using an effective model for the pi-pi cross-section involving rho and sigma meson exchange, medium effects are incorporated through thermal one-loop self-energies. The temperature dependence of the thermal conductivity is observed to be significantly affected.
We analyze the evolution of hydrodynamic fluctuations for QCD matter below $T_c$ in the chiral limit, where the pions (the Goldstone modes) must be treated as additional non-abelian superfluid degrees of freedom, reflecting the broken $SU_L(2) \times SU_R(2)$ symmetry of the theory. In the presence of a finite pion mass $m_π$, the hydrodynamic theory is ordinary hydrodynamics at long distances, and superfluid-like at short distances. The presence of the superfluid degrees of freedom then gives specific contributions to the bulk viscosity, the shear viscosity, and diffusion coefficients of the …
Updated Hadron List for Transport Simulations of Heavy-Ion Collisions
Institute for Theoretical Physics, Goethe University（法兰克福歌德大学理论物理研究所）；Frankfurt Institute for Advanced Studies（法兰克福高等研究院）；Helmholtz Research Academy Hesse for FAIR (HFHF), GSI Helmholtz Center（黑森亥姆霍兹FAIR研究学院（HFHF），GSI亥姆霍兹中心）；Center for Theoretical Physics – a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院莱因韦伯理论物理中心）；Pennsylvania State University（宾夕法尼亚州立大学）；Physics Department, University of Houston（休斯顿大学物理系）；GSI Helmholtzzentrum für Schwerionenforschung（GSI亥姆霍兹重离子研究中心）
Transport and hydrodynamical models used to describe the expansion stage of a heavy-ion collision at the CERN SPS give different dilepton spectrum even if they are tuned to reproduce the observed hadron spectra. To understand the origin of this difference we compare the dilepton emission from transport and hydrodynamical models using similar initial states in both models. We find that the requirement of pion number conservation in a hydrodynamical model does not change the dilepton emission. Also the mass distribution from the transport model indicates faster cooling and longer lifetime of the…
Hadronic interactions are crucial for the dynamical description of heavy-ion reactions at low collision energies and in the late dilute stages at high collision energies. In particular, the properties and decay channels of resonances are an essential ingredient of hadronic transport approaches. The HADES collaboration measured particle production in collisions of pions with carbon and tungsten nuclei at $E_\text{kin} = 1.7\,\text{GeV}$. Such reactions are of high interest, because they allow to probe the properties of baryonic resonances produced in a much cleaner environment than the usual nu…
Based on a Relativistic Boltzmann-Uehling- Uhlenbeck transport model, proton and $^{3}$He induced reactions on $^{197}$Au target at beam energies of 2.8, 5, 10 and 16.587 GeV/nucleon are studied. It is found that compared with proton induced reactions, $^{3}$He induced reactions give larger cross sections of pion production, about 5 times those of the proton induced reactions. And more importantly, pion production from $^{3}$He induced reaction is more inclined to low-angle emission. Neutrino production via positively charged pion is also discussed accordingly.
Nonequilibrium Phenomenology of Identified Particle Spectra in Heavy-Ion Collisions at LHC Energies
Institute of Theoretical Physics, University of Wrocław（弗罗茨瓦夫大学理论物理研究所）；Center for Advanced Systems Understanding (CASUS)（高级系统理解中心）；Helmholtz-Zentrum Dresden-Rossendorf (HZDR)（德累斯顿-罗斯多夫亥姆霍兹中心）；Institut für Kernphysik, Goethe Universität Frankfurt am Main（法兰克福歌德大学核物理研究所）；Institute of Physics, University of Rostock（罗斯托克大学物理研究所）
Within a multi-phase transport (AMPT) model with the string melting mechanism and imported initial electric quadrupole moment, the difference between the elliptic flow of positive and negative pions is calculated. The slope parameter $r$ of the linear dependence of $Δv_{2}=v_{2}(π^{-})-v_{2}(π^{+})$ on $A_{ch}=(N^{+}-N^{-})/(N^{+}+N^{-})$ is yielded owing to the parton cascade, which converts the initial electric quadrupole distribution into the final charge-dependent elliptic flow. The slope parameter $r$ is found to be increased by the hadronization given by the coalescence, and decreased by…
