Search

The discovery of four-quark states attracted a lot of attention from the theoretical as well as the experimental side. To study their properties from QCD we use a functional framework which combines (truncated) Dyson-Schwinger and Bethe-Salpeter equations in Landau gauge. This approach allows us to extract qualitative results for mass spectra, decay widths and wavefunctions of candidates for bound as well as resonant four-quark states. Furthermore, we can investigate the possible internal structure of such states. We report on recent developments and results using this functional framework and give an overview about the current status as well as future developments., Comment: 5 pages, 2 figures; contribution to the proceedings of the FAIR next generation scientists workshop (FAIRNESS), 7th edition, 23-27 May 2022, Paralia, Greece

The possible occurrence of crystalline or inhomogeneous phases in the QCD phase diagram at large chemical potential has been under investigation for over thirty years. Such phases are present in models of QCD such as the Gross-Neveu model in 1+1 dimensions, Nambu-Jona-Lasinio (NJL) and quark meson models. Yet, no unambiguous confirmation exists from actual QCD. In this work, we propose a new approach for a stability analysis that is based on the two-particle irreducible effective action and compatible with full QCD calculations within the framework of functional methods. As a first test, we reproduce a known NJL model result within this framework. We then discuss the additional difficulties which arise in QCD due to the non-locality of the quark self-energy and suggest a method to overcome them. As a proof of principle and as an illustration of the analysis, we consider the Wigner-Weyl solution of the quark Dyson-Schwinger equation (DSE) within a simple truncation of QCD in the chiral limit and analyse its stability against homogeneous chiral-symmetry breaking fluctuations. For temperatures above and below the tricritical point we find that the boundary of the instability region coincides well with the second-order phase boundary or the left spinodal, respectively, obtained from the direct solutions of the DSEs. Finally, we outline how this method can be generalized to study inhomogeneous fluctuations., Comment: refs added; data files organised

We investigate the quality of the extrapolation procedure employed in Ref. [1] to extract the crossover line at real chemical potential from lattice data at imaginary potential. To this end we employ a functional approach that does not suffer from the sign problem. We utilize a well-studied combination of lattice Yang--Mills theory with a truncated set of Dyson--Schwinger equations in Landau gauge for $2 + 1$ quark flavors. This system predicts a critical endpoint at moderate temperatures and rather large (real) chemical potential with a curvature comparable to recent lattice extrapolations. We determine the light quark condensate and chiral susceptibility at imaginary chemical potentials and perform an analytic continuation along the lines described in [1]. We find that the analytically continued crossover line agrees very well (within one percent) with the explicitly calculated one for chemical potentials up to about 80 % of the one of the critical end point. The method breaks down in the region where the chiral susceptibility as a function of the condensate cannot any longer be well described by a polynomial., Comment: 7 pages, 5 figures

A quark+diquark Faddeev equation treatment of the baryon bound state problem in Poincaré-invariant quantum field theory is used to deliver parameter-free predictions for all six $Δ$-baryon elastic weak form factors. Amongst the results, it is worth highlighting that there are two distinct classes of such $Δ$-baryon form factors, $(g_1, g_3, G_{πΔΔ})$, $(h_1, h_3, H_{πΔΔ})$, the functions within each of which are separately linked via partial conservation of axial current (PCAC) and Goldberger-Treiman (GT) relations. Respectively within each class, the listed form factors possess qualitatively the same structural features as the nucleon axial, induced pseudoscalar, and pion-nucleon coupling form factors. For instance, the $Δ$-baryon $g_1$ axial form factor can reliably be approximated by a dipole function, characterised by an axial charge $g_A^{Δ^+}=0.71(9)$ and mass-scale $m_A^Δ=0.95(2)m_Δ$. Moreover, the two distinct $Δ$-baryon PCAC form factor relations are satisfied to a high degree of accuracy on a large range of $Q^2$; the associated GT relations present good approximations only on $Q^2/m_Δ^2 \simeq 0$; and pion pole dominance approximations are reliable within both classes. There are two $πΔΔ$ couplings: $g_{πΔΔ} = 10.46(1.88)$; $h_{πΔΔ}= 35.73(3.75)$; and the associated form factors are soft. Such couplings commonly arise in phenomenology, which may therefore benefit from our analyses. A flavour decomposition of the axial charges reveals that quarks carry $71$\% of the $Δ$-baryon spin. The analogous result for the proton is $\approx 65$\%., 19 pages, 9 figures, 2 tables

In this contribution, we summarize a truncation-independent method to compute the equation of state within nonperturbative functional approaches. After demonstrating its viability, the method is applied to solutions obtained from a set of truncated Dyson-Schwinger equations for the quark and gluon propagators of (2+1)-flavor QCD to obtain thermodynamic quantities across the phase diagram of strong-interaction matter., Comment: 6 pages, 2 figures; contribution to the proceedings of the FAIR next generation scientists workshop (FAIRNESS), 7th edition, 23-27 May 2022, Paralia (Pieria), Greece

We present results for the volume dependence of baryon number fluctuations in the vicinity of the (conjectured) critical endpoint of QCD. They are extracted from the nonperturbative quark propagator that is obtained as a solution to a set of truncated Dyson-Schwinger equations of (2+1)-flavor QCD in Landau gauge, which takes the backcoupling of quarks onto the Yang-Mills sector explicitly into account. This well-studied system predicts a critical endpoint at moderate temperatures and rather large chemical potential. We investigate this system at small and intermediate finite, three-dimensional, cubic volumes and study the resulting impact on baryon number fluctuations and ratios thereof up to fourth order around the critical endpoint. We find that the fluctuations are visibly affected by the finite volume, particularly for antiperiodic boundary conditions, whereas their ratios are practically invariant., Comment: 7 pages, 5 figures

In a recent comment [1] Blankleider and Kvinikhidze claim that our work [2] is based on a set of inconsistent Bethe-Salpeter equations for the coupling of four-quark to two-quark states. Here we demonstrate that their argument is insubstantial., 3 pages, 1 figure

We calculate masses and decays of the (lightest) hybrid nonet with exotic quantum numbers $J^{PC}=1^{-+}$ and the nonet of their chiral partners with $J^{PC}=1^{+-}$ in the framework of the extended Linear Sigma Model (eLSM). As an input, we identify $\pi_{1}^{hyb}=\pi_{1}(1600)$ as a low-lying hybrid. We investigated interaction terms which fulfil chiral symmetry. For what concerns $\pi_{1}^{hyb},$ the most important decays are $\pi_{1}^{hyb}\rightarrow b_{1}\pi,$ $\pi_{1}(1600)\rightarrow\rho\pi\eta,$ $\pi_{1}^{hyb}% \rightarrow\rho\pi,$ and $\pi_{1}^{hyb}\rightarrow KK^{\ast}(892).$ The decays $\pi_{1}^{hyb}\rightarrow\eta\pi$ and $\pi_{1}^{hyb}\rightarrow\eta^{\prime }\pi$ are expected to be small but non-zero: they follow from a chirally symmetric interaction term that breaks explicitly the axial anomaly. For all the other members of the two hybrid nonets (for which no experimental candidates exist yet) we report decay ratios that may guide ongoing and future experiments., Comment: 26 pages, 6 tables. To appear in EPJPlus

We study the dynamical generation of resonances in isospin singlet channels with mixing between two- and four-quark states. To this end we generalise a Bethe-Salpeter approach to four-quark states employed previously \cite{Heupel:2012ua} to accommodate for mixing diagrams. The $q\bar{q}q\bar{q}$ and $q\bar{q}$ components of the Bethe-Salpeter wave function (with light quarks $q\in\{u,d\}$) are determined consistently in a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. As a prominent example we deal with the isospin-singlet $0^{++}$ meson with light quark content. We find that the $����$ contribution of the four-quark component is mainly responsible for the low (real part of the) mass of the resulting state. We also study the analytic structure in the complex momentum plane and find a branch cut at the two-pion threshold and a singularity in the second Riemann sheet indicating a considerable decay width. Our findings are in excellent qualitative agreement with the general picture for the $��/f_0(500)$ that emerged in the past two decades from dispersive approaches \cite{Pelaez:2015qba}., 10 pages, 7 figures, v2: minor changes, version accepted by PRD

Glueballs are bound states in the spectrum of quantum chromodynamics which consist only of gluons. They belong to the group of exotic hadrons which are widely studied experimentally and theoretically. We summarize how to calculate glueballs in a functional framework and discuss results for pure Yang-Mills theory. Our setup is totally self-contained with the scale being the only external input. We enumerate a range of tests that provide evidence of the stability of the results. This illustrates the potential of functional equations as a continuum first-principles method complementary to lattice calculations., 8 pages, 5 figures, 1 table, contribution to the proceedings of "XVth Quark Confinement and the Hadron Spectrum", Aug. 1-6, 2022

We study the phenomenology of the ground-state (pseudo-)scalar charmonia $��_c$ and $��_{c0}$ in the framework of a $U(4)_r \times U(4)_l$ symmetric linear sigma model with (pseudo-)scalar and (axial-) vector mesons. Based on previous results for the spectrum of charmonia and the spectrum and (OZI-dominant) strong decays of open charmed mesons, we extend the study of this model to OZI-suppressed charmonia decays. This includes decays into 'ordinary' mesons but also particularly interesting channels with scalar-isoscalar resonances $f_0(1370),\, f_0(1500),\, f_0(1710)$ that may include sizeable contributions from a scalar glueball. We study the variation of the corresponding decay widths assuming different mixings between glueball and quark-antiquark states. We also compute the decay width of the pseudoscalar $��_c$ into a pseudoscalar glueball. In general, our results for decay widths are in reasonable agreement with experimental data where available. Order of magnitude predictions for as yet unmeasured states and channels are potentially interesting for BESIII, Belle II, LHCb as well as the future PANDA experiment at the FAIR facility., 20 pages, 3 figures, 6 tabes

We present the mini-proceedings of the workshops Hadronic contributions to the muon anomalous magnetic moment: strategies for improvements of the accuracy of the theoretical prediction and $(g-2)_��$: Quo vadis?, both held in Mainz from April 1$^{\rm rst}$ to 5$^{\rm th}$ and from April 7$^{\rm th}$ to 10$^{\rm th}$, 2014, respectively., 84 pages, 23+18 contributions. Editors: Tom Blum, Pere Masjuan, and Marc Vanderhaeghen

We summarize recent results for the spectrum and properties of light pseudoscalar and scalar hadrons in a functional approach to QCD using Dyson-Schwinger and Bethe-Salpeter equations. We focus in particular on the role of pions as virtual constituents in light mesons ('pion cloud effects') as well as their role as constituents of tetraquark bound states. An extension of our framework into the charm quark sector reveals an all-charm tetraquark in the mass region around 5.5 GeV., Comment: 10 pages, 7 figures, contribution to the 51st International Winter Meeting on Nuclear Physics, 21-25 January 2013, Bormio (Italy)

We investigate the phase structure of QCD at finite temperature and chemical potential by solving a coupled set of truncated Dyson-Schwinger equations for the quark and gluon propagator. In contrast to previous calculations we take into account the full back-reaction of the quarks onto the Yang-Mills sector and we include the effects of strange quarks. We discuss the resulting thermal mass of the unquenched gluon propagator and extract order parameters for the chiral and deconfinement transition from the quarks. Our result for the temperature dependence of the quark condensate at zero chemical potential agrees well with corresponding lattice calculations. We determine the phase diagram at finite chemical potential and find a potential critical endpoint at (mu_q,T) = (190,100) MeV., Comment: 11 pages, 9 figures, 1 table

We summarise results for the running coupling from the four-gluon vertex in Landau gauge, SU($N_c$) Yang-Mills theory as given by a combination of dressing functions of the vertex and the gluon propagator. These functions have been determined numerically from the corresponding set of Dyson-Schwinger equations. In the infrared we obtain a nontrivial infrared fixed point which is three orders of magnitude smaller than the corresponding one in the coupling of the ghost-gluon vertex., Comment: 6 pages, 1 figure. Talk given at ExcitedQCD 2009, Feb. 8-14, Zakopane, Poland

We investigate masses of light mesons from a coupled system of Dyson--Schwinger (DSE) and Bethe--Salpeter equations (BSE), taking into account dominant non-Abelian, sub-leading Abelian, and dominant pion cloud contributions to the dressed quark-gluon vertex. The axial-vector Ward-Takahashi identity preserving Bethe-Salpeter kernel is constructed and the spectrum of light mesons calculated. Our model goes significantly beyond the rainbow-ladder. We find that sub-leading Abelian corrections are further dynamically suppressed, and that our results supersede early qualitative predictions from simple truncation schemes., Comment: 4 pages, 3 figures, 1 table. Talk given at the 5-th Int. Conf. on Quarks and Nuclear Physics, Beijing, September 21-26,2009

In this talk we present results on the chiral and the deconfinement transition of quenched QCD from Dyson-Schwinger equations. We determine the ordinary quark condensate signaling the chiral transition and the dual quark condensate signaling the deconfinement transition from the Landau gauge quark propagator evaluated at generalized boundary conditions. We find only slightly different transition temperatures at finite quark masses, whereas the transition temperatures coincide in the chiral limit., Comment: 10 pages, 5 figures, Talk given at the Workshop CPOD 2009, June 08 - 12, BNL, USA

We summarize results on finite-volume effects in the propagators of Landau gauge Yang-Mills theory using Dyson-Schwinger equations on a 4-dimensional torus. We demonstrate explicitly how the solutions for the gluon and the ghost propagator tend towards their respective infinite volume forms in the corresponding limit. We discuss the relation of our solutions with results from lattice Monte-Carlo simulations., Comment: 7 pages, 2 figures, Presented by CF at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, Germany

We study dynamical mass generation in pure Yang-Mills theory and report on a recently developed ansatz that exactly solves the tower of Dyson-Schwinger equations in Landau gauge at low Euclidean momentum, featuring enhanced gluon-gluon vertices, a finite ghost-gluon vertex in agreement with an old argument of Taylor, and an IR suppressed gluon propagator. This ansatz reinforces arguments in favor of the concept of a gluon mass gap at low momentum (although the minimum of the gluon's dispersion relation is not at zero momentum). As an application, we have computed the spectrum of oddballs, three-gluon glueballs with negative parity and C-parity. The three body problem is variationally solved employing the color density-density interaction of Coulomb gauge QCD with a static Cornell potential. Like their even glueball counterparts, oddballs fall on Regge trajectories with similar slope to the pomeron. However their intercept at t=0 is smaller than the omega Regge trajectory and therefore the odderon may only be visible in experimental searches (for example at BNL) with higher -t than conducted to date at DESY., Comment: 10 pages, 1 figure, contribution to HSQCD 05, St. Petersburg, Russia, September 2005