Mesons and diquarks in the CFL phase of dense quark matter
D. Ebert, K. G. Klimenko
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3 pages, 2 figures; prepared for the XIII International Conference on Selected Problems of Modern Theoretical Physics (SPMTP08), Bogoliubov Laboratory of Theoretical Physics, Dubna, Russia, June 23-27, 2008
The spectrum of meson and diquark excitations of the color--flavor locked (CFL) phase of dense quark matter is considered in the framework of the Nambu -- Jona-Lasinio model. We have found that in this phase all Nambu--Goldstone bosons are realized as scalar and pseudoscalar diquarks. Other diquark excitations are resonances with mass value around 230 MeV. Mesons are stable particles in the CFL phase. Their masses vs chemical potential lie in the interval 300$÷$500 MeV.
Diquark excitations of dense quark matter are considered in the framework of the Nambu -- Jona-Lasinio model with three types of massless quarks in the presense of a quark number chemical potential $μ$. We investigate the effective action of meson- and diquark fields at sufficiently high values of $μ$, where the color--flavor locked (CFL) phase is realized, and prove the existence of NG-bosons in the sector of pseudoscalar diquarks. In the sector of scalar diquarks an additional NG-boson is found, corresponding to the spontaneous breaking of the U(1)$_B$ baryon symmetry in the CFL phase. Final…
The spectrum of meson and diquark excitations of dense quark matter is considered in the framework of the Nambu -- Jona-Lasinio model with three types of massless quarks in the presense of a quark number chemical potential $μ$. We investigate the effective action of meson- and diquark fields both at sufficiently large values of $μ>μ_c\approx 330$ MeV, where the color--flavor locked (CFL) phase is realized, and in the chirally broken phase of quark matter ($μ<μ_c$). In the last case all nine pseudoscalar mesons are Nambu -- Goldstone (NG) bosons, whereas the mass of the scalar meson nonet is tw…
The Ginzburg-Landau equations for magnetic and gluomagnetic gauge fields of the semi-superfluid strings in the color superconducting core of the neutron star with diquark condensate in the color-flavor-locking (CFL) phase are derived. The simultaneous coupling of the CFL diquark condensate to the magnetic and gluomagnetic gauge fields is taken into account. The asymptotic values of these string energies are defined from quantization conditions. It is shown that a lattice of semi-superfluid strings with minimal quantum of circulation emerges in the superconducting quark core during rotation of …
We use a Nambu-Jona Lasinio type model to investigate the phase diagram of dense quark matter under neutron star conditions in mean field approximation. The model contains selfconsistently determined quark masses and allows for diquark condensation in the scalar as well as in the pseudoscalar channel. The latter gives rise to the possibility of K^0 condensation in the CFL phase. In agreement with earlier studies we find that this CFLK^0 phase covers large regions of the phase diagram and that the predominant part of this phase is fully gapped. We show, however, that there exists a region at ve…
Using flow equations, we derive an effective quark-quark interaction and obtain the coupled set of gap equations for the condensates of the CFL phase of massless $N_f=3$ dense QCD. The formalism developed here enables one to consider more general case of nonzero $s$-quark mass.
We show that the color-flavor locked (CFL) phase of dense quark matter is unstable with respect to the formation of a Goldstone boson current in the vicinity of the point $m_s^2=2μΔ$. In this Goldstone boson current phase all components of the magnetic screening mass are real.
We investigate the color-singlet order parameters and the quark description of the Nambu-Goldstone (NG) bosons in the color-flavor locked (CFL) phase. We put emphasis on the NG boson (phason) called ``H'' associated with the $\mathrm{U_B(1)}$ symmetry breaking. We qualitatively argue the nature of H as the second sound in the hydrodynamic regime. We articulate, based on a diquark picture, how the structural change of the condensates and the associated NG bosons occurs continuously from hadronic to CFL quark matter if the quark-hadron continuity is realized. We sharpen the qualitative differenc…
At asymptotic high density and low temperature quarks form Cooper pairs in a color-flavor locked (CFL) configuration. The diquark condensates break spontaneously the baryon symmetry, and this fact makes the CFL phase also superfluid. At low temperatures the transport properties are dominated by the contribution of the superfluid phonon, the Goldstone boson associated to baryon symmetry breaking, in full analogy to what happens in superfluid He4. We discuss how to derive transport properties in the ultracold regime making use of an analogue model of gravity. We also review how this model can be…
We study the stability of the color-flavor locked (CFL) phase of dense quark matter with regard to the formation of a non-zero Goldstone boson current. We show that an instability appears in the vicinity of the point $μ_s=Δ$ which marks the appearance of gapless fermion modes in the CFL phase. Here, $μ_s=m_s^2/(2μ)$ is the shift in chemical potential due to the strange quark mass and $Δ$ is the gap in the chiral limit. We show that in the Goldstone boson current phase all components of the magnetic screening mass are real. In this work we do not take into account homogeneous kaon condensation.…
The approach of path integral hadronization, applied to a quark-diquark toy model, is used to derive an effective chiral meson-baryon Lagrangian. In this approach the Goldberger-Treiman relation, found at the quark-meson level, is re-established at the composite hadron level. For illu- strations, electromagnetic low energy characteristics of composite nucleons are estimated.
This is a conference mostly devoted to diquarks. Although an exotic meson can in principle be composed of a diquark and an antidiquark, such an exotic is unlikely to be experimentally observable in the near future. The reason, according to the model, is that diquark-antidiquark exotics have masses well above the threshold for decay into two mesons, and are likely to have widths too large to make them observable. A possible exception is a $ u d \bar b \bar b$ exotic, but it will be a long time before such a state can be observed even if it is stable against strong decay.
I demonstrate how confinement in Coulomb-gauge QCD makes quark-quark states of the color anti-triplet (diquarks) move out of the physical spectrum. Mesons as color singlet quark-antiquark states, on the other hand have finite masses and for highly excited states in the meson spectra effective restoration of chiral symmetry can be observed.
Diquark-antidiquark states are expected to exist as a natural complement of mesons and baryons. Although they were predicted long ago, and some candidates were found experimentally, none has, as yet, been reliably identified. We suggest that the search for the so-called $C(1480)$-meson in reactions such as photoproduction $γN\rightarrowϕπN$ and $K N \rightarrow ϕπΛ$ should provide a decisive way to settle this issue. Estimates of the cross sections are given using present experimental information on the C-meson and assuming its diquark-antidiquark structure. Sizable cross sections are predicte…
The spectrum of meson and diquark excitations of dense cold quark matter is investigated in the framework of a Nambu--Jona-Lasinio type model for light quarks of two flavors. It was found out that a first order phase transition occurs when the chemical potential $μ$ exceeds the critical value $μ_c=350$ MeV. Above $μ_c$, the diquark condensate $\vev{qq}$ forms, breaking the color symmetry of strong interaction. The masses of $π$- and $σ$-mesons are shown to grow with the chemical potential $μ$ in the color-superconducting phase, but the mesons themselves become almost stable particles due to th…
We argue that in the color-flavor-locking (CFL) superconducting phase classical soliton solutions can exist, whose excitations should be interpreted as states formed by a quark (or an antiquark) and condensed diquarks. This finding extends the picture of quark-hadron-continuity showing the existence of a region, intermediate between the CFL and the hypernuclear phase, where chiral solitons and Nambu Goldstone bosons can exist. We derive an expression of the soliton mass in terms of the QCD coupling, $g_s$, and the Nambu Goldstone boson parameters. From the quark-hadron continuity we can draw a…
Unified analysis of screening masses for vector and axial-vector mesons and their diquark partners in the Contact Interaction model
SECIHTI-Mesoamerican Centre for Theoretical Physics, Universidad Autónoma de Chiapas（墨西哥自治大学中美洲理论物理中心）；Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Sinaloa（锡那罗亚自治大学理学院）；Facultad de Ciencias de la Tierra y el Espacio, Universidad Autónoma de Sinaloa（锡那罗亚自治大学地球与空间科学学院）
The earlier suggested and developed idea of quark-hadron duality, underlying "bremsstrahlung-weighted" sum rules for total or polarized photon interaction cross sections, is applied to the description of excitation of light scalar mesons in gamma-gamma interactions. The emphasis is put on the discussion of a role of the scalar diquark cluster degrees of freedom in the radiative formation of light scalar mesons.
Exotic mesons with hidden strange ($s\bar s$) and heavy quark pairs ($Q\bar Q$), where $Q=c,b$, are considered as diquark-antidiquark systems, $(Qs)(\bar Q\bar s)$. Taking into account that these states can recombinate into two-meson ones, we study the interplay of these states in terms of the dispersion relation D-function technique. The classification of exotic mesons is discussed, coefficients for decay modes are given, predictions for new states are presented. The nonet structure for $((Qq)(\bar Q \bar q)$, ($(Qs)(\bar Q\bar s$)), ($(Qq)(\bar Q\bar s))$- states ($q=u,d$) is suggested.
*REVISED VERSION* Two quark propagators with different analytic structure are employed in on and off mass-shell Bethe-Salpeter type equations for the pion and scalar diquark form-factors. One of the quark propagators has been calculated with the inclusion of a trivial quark-gluon vertex and, as a consequence, contains a complex conjugate pair of logarithmic branch points. The other quark propagator includes a non-trivial vertex ansatz and is entire, with an essential singularity at infinity. The different quark propagators lead to qualitatively different diquark form-factors.
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Masses of the ground state light tetraquarks are dynamically calculated in the framework of the relativistic diquark-antidiquark picture. The internal structure of the diquark is taken into account by calculating the form factor of the diquark-gluon interaction in terms of the overlap integral of the diquark wave functions. It is found that scalar mesons with masses below 1 GeV: f_0(600) (σ), K^*_0(800) (κ), f_0(980) and a_0(980) agree well with the light tetraquark interpretation.
We investigate both (pseudo)scalar mesons and diquarks in the presence of external magnetic field in the framework of the two-flavored Nambu--Jona-Lasinio (NJL) model, where mesons and diquarks are constructed by infinite sum of quark-loop chains by using random phase approximation. The polarization function of the quark-loop is calculated to the leading order of $1/N_c$ expansion by taking the quark propagator in the Landau level representation. We systematically investigate the masses behaviors of scalar $σ$ meson, neutral and charged pions as well as the scalar diquarks, with respect to the…
We investigate the meson, baryon and anti-baryon productions in πp and Kp collisions in terms of a constituent cascade model which includes charm flavour. The constituent diquark in an incident baryon has a tendency to get a large momentum fraction, resulting in considerable difference of subenergies for particle productions from quark-diquark and quark-antiquark chains in a meson-baryon collision. As a consequence, the leading particle effects on D meson productions are expected to be quite different in π+ and π- beams at present accelerator energies.
In this article, we take the point of view that the $0^{++}$ nonet mesons below $1 GeV$ are diquark-antidiquark states $(qq)_{\bar{3}}(\bar{q}\bar{q})_3$, and devote to determine their masses in the framework of the QCD sum rules approach with the interpolating currents constructed from scalar-scalar type and pseudoscalar-pseudoscalar type diquark pairs respectively. The numerical results indicate that the $0^{++}$ nonet mesons may have two possible diquark-antidiquark substructures.
We compare two frequently discussed competing structures for a stable $\bar b \bar b u d$ tetraquark with quantum numbers $I(J^P) = 0(1^+)$ by considering a meson-meson as well as a diquark-antidiquark creation operator. We treat the heavy antiquarks as static with fixed positions and find diquark-antidiquark dominance for $\bar b \bar b$ separations $r < 0.2 \, \text{fm}$, while for $r > 0.5 \, \text{fm}$ the system essentially corresponds to a pair of $B$ mesons. For the meson-meson to diquark-antidiquark ratio of the tetraquark we obtain around $58\%/42\%$.
We study the spectroscopy of the states which defy conventional $c\bar{c}$ charmonium and $b\bar{b}$ bottomonium interpretation respectively, and are termed as exotic states. In August 2003 a state X(3872), was discovered [K. Abe\textit{\ et al.} (Belle Collaboration), Phys. Rev. Lett. \textbf{91}, 262001 (2003) [arXiv:hep-ex/0309032]] and in December 2007 a state Y(10890) was reported [K. F. Chen \textit{et. al.} [Belle Collaboration], Phys. Rev. Lett. \textbf{100}, 112001 (2008) [arXiv:0710.2577]. One possible interpretation of such exotic state is that they are tetraquark (diquark-antiquark…
The spectrum of meson and diquark excitations in cold color-superconducting (2SC) quark matter is investigated under local color and electric neutrality constraints with $β$-equilibrium. A 2-flavored Nambu--Jona-Lasinio type model including a baryon $μ_B$, color $μ_8$, and electric $μ_Q$ chemical potentials is used. Two relations between coupling constants $H$ and $G$ in the diquark- and quark-antiquark channels, correspondingly, are treated, $H=3G/4$ and $H=G$. At $H=3G/4$ the gapless- and at $H=G$ the gapped neutral color superconductivity is realized. It is shown that color and electrical n…
In the Nambu--Jona-Lasinio (NJL-) model baryons are considered either as quark-diquark-composites or as soliton configurations of mesons. The construction of a hybrid model, which possesses a dynamical interplay between both pictures, is presented. The Bethe-Salpeter equation for diquarks and the Faddeev equation for diquark-quark bound states in the background of a soliton configuration is solved.
In this article, we take the point of view that the scalar mesons $ f_0(980)$ and $a_0(980)$ are diquark-antidiquark states $(qq)_{\bar{3}}(\bar{q}\bar{q})_3$, and devote to determine their masses in the framework of the QCD sum rules approach with the interpolating currents constructed from scalar-scalar type and pseudoscalar-pseudoscalar type diquark pairs respectively. The numerical results indicate that the scalar mesons $ f_0(980)$ and $a_0(980)$ may have two possible diquark-antidiquark substructures.
A quark-meson-diquark plasma is considered within the PNJL model for dynamical chiral symmetry breaking and restoration in quark matter. Based on a generalized Beth-Uhlenbeck approach to mesons and diquarks we present the thermodynamics of this system including the Mott dissociation of mesons and diquarks at finite temperature. A striking result is the suppression of the diquark abundance below the chiral restoration temperature by the coupling to the Polyakov loop, because of their color degree of freedom. This is understood in close analogy to the suppression of quark distributions by the sa…
In recent years, the existence of a hadronically stable $\bar{b} \bar{b} u d$ tetraquark with quantum numbers $I(J^P) = 0(1^+)$ was confirmed by first principles lattice QCD computations. In this work we use lattice QCD to compare two frequently discussed competing structures for this tetraquark by considering meson-meson as well as diquark-antidiquark creation operators. We use the static-light approximation, where the two $\bar{b}$ quarks are assumed to be infinitely heavy with frozen positions, while the light $u$ and $d$ quarks are fully relativistic. By minimizing effective energies and b…
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We study the fine structure in the spectrum of known and predicted negative-parity hidden-charm exotic meson states, which comprise the lowest $P$-wave multiplet in the dynamical diquark model. Starting with a form previously shown to successfully describe the $S$-wave states, we develop a 5-parameter Hamiltonian that includes spin-orbit and tensor terms. After discussing the experimental status of the observed $J^{PC} = 1^{--}$ states $Y$ with respect to masses and decay modes (classified by eigenvalues of heavy-quark spin), we note a number of inconsistencies between measurements from differ…
We calculate the decay rate of light scalar mesons, using a diquark chiral effective theory, recently proposed to describe exotic hadrons. In the effective theory the light scalar mesons are postulated to be bound states of diquark and anti-diquark. We find our results are in good agreement with experimental data. The axial couplings of diquarks with pions and kaons are found to be quite small and the perturbation is reliable. It shows that the diquark picture captures the correct physics of light scalar mesons.
We study the structure of the nuclear force by using a path-integral hadronization approach in a chiral quark-diquark model. After the construction of the chiral quark-diquark model, we hadronize it to obtain a meson-baryon Lagrangian. The effective meson-baryon Lagrangian incorporates chiral symmetry and the composite description of the mesons and baryons. Using the effective meson-baryon Lagrangian we investigate the structure of the nuclear force in the simple case of neglecting the axial-vector diquark. It is shown that the meson-baryon Lagrangian contains two kinds of the nuclear force; t…
We study the effect of a non-zero strange quark mass on the color-flavor-locked (CFL) phase of high density quark matter. We have previously shown that for a strange quark mass $m_s\sim m_u^{1/3}Δ^{2/3}$ the CFL state becomes unstable toward the formation of a neutral kaon condensate. Recently, several authors discovered that for $m_s\sim (2Δp_F)^{1/2}$ the CFL state contains gapless fermions, and that the gapless modes lead to an instability in current-current correlation functions. Using an effective theory of the CFL state we demonstrate that this instability is resolved by the formation of…
We investigate mesons and diquarks as resonant states above chiral critical temperature $T_c$ in flavor SU(2) Nambu--Jona-Lasinio model. For each kind of resonance, we solve the pole equation for the resonant mass in the complex energy plane, and find an ultimate temperature where the pole starts to disappear. The phase diagram including these limit temperatures in $T-μ$ plane is obtained. The maximum limit temperature at $μ=0$ is approximately two times $T_c$.
We present an update on the phase diagram of two-color QCD from a chiral effective model approach based on a quark-meson-diquark model using the Functional Renormalization Group (FRG). We discuss the impact of perturbative UV contributions, the inclusion of gauge field dynamics via a phenomenological Polyakov loop potential, and the impact of matter backcoupling on the gauge sector. The corresponding phase diagram including these effects is found to be in qualitative agreement with recent lattice investigations.
In this article, we study the structures of the pseudoscalar mesons $π$, $K$ and the scalar diquarks $U^a$, $D^a$, $S^a$ in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with the confining effective potential. The $u$, $d$, $s$ quarks have small current masses, and the renormalization is very large, the mass poles in the timelike region are absent which implements confinement naturally. The Bethe-Salpeter wavefunctions of the pseudoscalar mesons $π$, $K$ and the scalar diquarks $U^a$, $D^a$, $S^a$ have the same type (Gaussian type) momentum de…
At very high densities, QCD is in the color-flavor-locked phase, which is a color-superconducting phase. The diquark condensates break chiral symmetry in the same way as it is broken in vacuum QCD and gives rise to an octet of pseudo-Goldstone bosons and a superfluid mode. The lightest of these are the charged and neutral kaons. For energies below the superconducting gap, the kaons are described by an $O(2)\times O(2)$-symmetric effective scalar field theory with chemical potentials. We use this effective theory to study Bose-condensation of kaons and their properties as functions of the tempe…
When applying the maximum entropy method (MEM) to the analysis of hadron correlation functions in QCD a central issue is to understand to what extent this method can distinguish bound states, resonances and continuum contributions to spectral functions. We discuss these issues by analyzing meson and diquark correlation functions at zero temperature as well as free quark anti-quark correlators. The latter test the applicability of MEM to high temperature QCD.
The reactions gamma p -> K+ Lambda and gamma p -> K* Lambda are analyzed within perturbative QCD, allowing for diquarks as quasi-elementary constituents of baryons. The diquark-model parameters and the quark-diquark distribution amplitudes of proton and Lambda are taken from previous investigations of electromagnetic baryon form factors and Compton-scattering off protons. Unpolarized differential cross sections and polarization observables are computed for different choices of the K and K* distribution amplitudes. The asymptotic form of the K distribution amplitude (proportional to x1 x2) is f…
We analyse fall-apart decays of poliquark (tetra, penta and molecule type) hadrons within the constituent quark picture. For processes in which a poliquark hadron goes to final states containing a light pseudoscalar meson the constraints given by chiral symmetry are implemented. As an application of the approach developed, fall-apart decays of $a(980)$ and X(3872) are studied, assuming these mesons are poliquark hadrons. Two extreme options - confined diquark-diquark states and molecular states - are considered. For $a^0(980)$, the observed width can be obtained assuming that this meson is a d…
In this article, we take the scalar diquarks as point particles and describe them as basic quantum fields, then introduce the SU(3) color gauge interaction and new vacuum condensates to study the nonet scalar mesons as tetraquark states with the QCD sum rules. Comparing with the conventional quark currents, the diquark currents have the outstanding advantage to satisfy the two criteria of the QCD sum rules more easily.
We investigate the distance and temperatures scale for which the string in baryonic quark configuration approaches the limiting behavior of mesonic strings in pure Yang-Mills SU(3) lattice gauge theory. We calculate and compare the numerical values of the Polyakov loop correlators and the width profile of both diquark-quark $(QQ)Q$ and mesonic $Q\bar{Q}$ strings. We find the diquark-quark configuration to exhibit an identical behavior to the mesonic string for the potential and energy-density width profile for temperature near the end of QCD plateau. In the vicinity of the deconfinement point;…
We propose a model aimed at explaining jet quenching, large azimuthal asymmetry and baryon/meson ratio at large transverse momenta pt=2-10 GeV observed at RHIC. Its main point is that a QCD string can be cut by matter quarks and/or diquarks before its natural breaking. We model the early quark/diquark production via QCD sphalerons.
The phase diagram of three-flavor quark matter under compact star constraints is investigated within a Nambu--Jona-Lasinio model. Local color and electric charge neutrality is imposed for beta-equilibrated superconducting quark matter. The constituent quark masses and the diquark condensates are determined selfconsistently in the plane of temperature and quark chemical potential. Both strong and intermediate diquark coupling strengths are considered. We show that in both cases, gapless superconducting phases do not occur at temperatures relevant for compact star evolution, i.e., below T ~ 50 M…
We compute the rates of real and virtual photon (dilepton) emission from dense QCD matter in the color-flavor locked (CFL) phase, focusing on results at moderate densities (3-5 times the nuclear saturation density) and temperatures $T\simeq80$ MeV. We pursue two approaches to evaluate the electromagnetic (e.m.) response of the CFL ground state: (i) a direct evaluation of the photon self energy using quark particle/-hole degrees of freedom, and (ii) a Hidden Local Symmetry (HLS) framework based on generalized mesonic excitations where the $ρ$ meson is introduced as a gauge boson of a local SU(3…
