Effect of Transverse Gluons on Chiral Restoration in Excited Mesons

Effect of Transverse Gluons on Chiral Restoration in Excited Mesons

M. Pak markus.pak@uni-graz.at    L. Ya. Glozman leonid.glozman@uni-graz.at Institut für Physik, FB Theoretische Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria
12th July 2019

The effect of transverse gluons on the chiral symmetry patterns of excited mesons is studied in a Coulomb gauge QCD model. The linear rising static quark-antiquark potential and the transverse gluon propagator known from lattice studies are input into the model. The non-perturbative quark propagator, which enters the meson bound state equations, is derived from the Dyson–Schwinger equations and a complete set of mesons for general spin quantum number is presented. From analyzing the bound state equations for large spins it is demonstrated, that chiral and axial symmetry are restored. In this limit a complete degeneracy of all multiplets with given spin is observed. The effect of the transverse gluon interaction is shown to vanish rapidly as the spin quantum number is increased. For vanishing dynamical quark mass the expected meson degeneracies are recovered.

QCD, Chiral symmetry breaking, Bethe–Salpeter equations, Excited Mesons
11.30.Rd, 12.38.Aw, 14.40.-n

I Introduction

Understanding the two cornerstones of non-perturbative QCD - confinement and chiral symmetry breaking - is one of the most important challenges of modern particle physics. The hadron spectrum offers a unique possibility to get insight into the underlying mechanisms of both these phenomena and their interplay. For instance, we certainly know that for the low energy quark sector of QCD the chiral symmetry and its spontaneous breaking are crucially important. Quark condensation in the vacuum breaks the symmetry group of massless QCD to the vector subgroup , preventing the ground state parity partners to have equal masses and giving rise to pseudoscalar Goldstone bosons. The mass of the particle is shifted up due to the anomalous breaking of the symmetry.

However, higher states in the light hadron spectrum show a different picture: mesons and baryons tend to form chiral multiplets, signaling the “effective” restoration of the symmetry Glozman:1999tk (); *Cohen:2001gb; *Glozman:2002cp; *Glozman:2003bt, see Ref. Glozman:2007ek () for an overview. However, for a final proof missing states have to be experimentally confirmed. The physical picture is the following: for high-lying states the valence quarks should become unaffected by the condensate, since one enters the semi-classical regime, where the quantum fluctuations are suppressed, see Ref. Glozman:2004gk (); *Glozman:2005tq. This phenomenological picture has been tested, however, with only a static quark-antiquark Lorentz-vector confining potential entering the system, which has either been chosen to be of harmonic oscillator-type Kalashnikova:2005tr () or linearly rising Wagenbrunn:2006cs (). Here we close a gap and show that highly excited high-spin mesons form approximately degenerate chiral multiplets also for a transverse gluon type of interaction.

The Nambu–Jona-Lasinio type mechanism of spontaneous breaking of chiral symmetry has been successfully analyzed in effective potential models, Refs. Finger:1981gm (); LeYaouanc:1983iy (); Adler:1984ri (), however, with the low energy chiral properties of the theory predicted too low. It has been observed that an additional transverse gluon exchange increases the values of the quark condensate and dynamical quark mass substantially towards their phenomenological values, Refs. Alkofer:1988tc (); Szczepaniak:2002ir (); LlanesEstrada:2004wr (); Pak:2011wu (); Fontoura:2012mz (). Most recently, in Ref. Pak:2011wu () this has been shown by applying the variational approach to the quark sector of Coulomb gauge QCD. Since transverse gluons have a significant effect on chiral symmetry breaking we here ask the question if they also affect high-spin mesons, where chiral symmetry is expected to be effectively restored.

In Refs. Wagenbrunn:2006cs (); Wagenbrunn:2007ie () it has been shown that in a model with chiral quarks and a color-Coulomb linear potential mesons fall into parity multiplets for large spins. Here we generalize the model and include transverse gluons. We demonstrate that the additional interaction respects chiral symmetry as well.

The organization of the paper is as follows: in Section II we review the model Hamiltonian generalized to transverse gluon interaction and specify the interaction kernels from lattice Coulomb gauge studies. In Section III the quark propagator dressing functions are obtained from the Dyson–Schwinger equations in a rainbow truncation and the infrared divergencies of these quantities are discussed. The symmetry properties of non-interacting quarks are listed in Section IV. A complete spectrum of meson Bethe–Salpeter equations for arbitrary orbital quantum number are presented in Section V. The limit of vanishing dynamical quark mass is discussed in Section VI. In Section VII the equations are studied for large spins . It is shown by numerically analyzing the angular integrals of the coupled system that the effect of transverse gluons rapidly vanishes for increasing spin. Effective chiral symmetry restoration is found to persist for such an interaction. In Section VIII we summarize our main findings. Irreducible tensor relations and the Bethe–Salpeter equations are collected in Appendices A and B.

Ii Model Hamiltonian

The model Hamiltonian generalized to transverse gluons is defined as (see Refs. Wagenbrunn:2007ie (); Alkofer:1988tc ())




is the free Hamiltonian of the quark field . Here are the usual Dirac matrices satisfying and , is the current quark mass. The interaction Hamiltonian is split up into the instantaneous linear confining Coulomb term


with being the Hermitian generators of the gauge group in the fundamental representation and the transverse gluon interaction of the form


Here a comment is in order: the interaction Hamiltonian (3) arises naturally in Coulomb gauge QCD from the kinetic energy of the longitudinal modes after resolving Gauss’ law. Using a perturbative gluon propagator and integrating out the gluonic degrees of freedom leads to an additional quark interaction due to transverse gluons, Eq. (4). A recent Coulomb gauge study supports such an interaction kernel. Using the so-called variational approach to QCD in Coulomb gauge, a quark vacuum wave functional is suggested, which goes beyond the typical BCS approximation and includes a coupling between quarks and transverse gluons, see Ref. Pak:2011wu (). The additional transverse gluon interaction increases the chiral condensate and the constituent quark mass of about . These findings are in agreement with the study of Ref. Fontoura:2012mz (), where a potential of the form (4) is used to describe transverse gluons. For a gluon propagator which vanishes for small momenta, both these approaches lead to a constituent quark mass of MeV. In Fig. 1 on the left-hand side the dynamical quark mass is plotted and compared to the cases without the additional transverse gluon interaction and to most recent lattice Coulomb gauge data, see Ref. Burgio:2012ph (). For the purpose of our work it is much more convenient to use a model Hamiltonian as considered in Eq. (4).

Figure 1: Left Panel: Dynamical quark mass for and MeV comparing the results from the variational approach to QCD in Coulomb gauge, Ref. Pak:2011wu (), to the lattice data obtained in Ref. Burgio:2012ph (). The dashed curve is obtained with the additional coupling of quarks to transverse gluons while the dash-dotted curve is obtained using the color-Coulomb potential only, for details see Ref. Pak:2011wu (). Right Panel: Instantaneous transverse gluon propagator . Data points show the lattice results of Ref. Burgio:2008jr (). The dashed curve is the result of the variational analysis using a Gaussian vacuum, Ref. Feuchter:2004mk () and the full curve shows the extension to non-Gaussian wave functionals. The plot is taken from Ref. Campagnari:2010wc ().

We now specify the interaction kernels and , appearing in Eqs. (3), (4). Both are diagonal in color space. From the variational approach to Coulomb gauge, Refs. Feuchter:2004mk (); Epple:2006hv (), one finds a potential which at large distances rises linearly


The same behavior is found on the lattice, Ref. Voigt:2008rr (); *Nakagawa:2011ar; *Greensite:2003xf, with a Coulomb string tension of


where MeV is the Wilsonian string tension. It is well known that the linearly rising confinement potential in Eq. (3) leads to infrared divergences in momentum space, which, however, cancel from the observable quantities as shown in Ref. Adler:1984ri (). For the actual computation we introduce an (infrared) cut-off parameter, which, in the end of the calculation, is send to zero. For the definition of the infrared regular confinement potential we adopt the convention of Refs. Alkofer:1988tc (); Wagenbrunn:2007ie (), given as (in momentum space)


with the Coulomb string tension and the infrared cut-off parameter. Due to asymptotic freedom the color-Coulomb potential also has an ordinary short-range part, which influences the UV-part of the dynamical quark mass. Since we want to investigate the role of transverse gluons for chiral symmetry restoration in highly excited mesons, the Coulomb potential is not required and can be neglected.

As static transverse gluon propagator in Eq. (4) we use the result from lattice studies, Ref. Burgio:2008jr () in Coulomb gauge QCD, which can nicely be fitted by Gribov’s formula


where MeV is a mass scale referred to as Gribov mass and the transverse projector is defined as


On the right-hand side of Fig. 1 the transverse gluon propagator is presented and compared to results in the variational approach to QCD, Refs. Feuchter:2004mk (); Campagnari:2010wc ().

Iii Quark Propagator and Gap Equation

Due to the static interaction kernels (3), (4), the dressed inverse quark propagator can be decomposed as


with denoting scalar and vector dressing functions. Within this large- model only the rainbow diagrams contribute to the gap equation (see Refs. LeYaouanc:1983iy (); Adler:1984ri () for a detailed derivation)


with and . We stress that the quark-gluon vertex is taken to be bare and it is assumed that no infrared divergence occurs in the transverse gluon sector. Spontaneous chiral symmetry breaking reflects itself in a non-vanishing scalar dressing function and in an infrared finite dynamical quark mass function , which is defined as (see Fig. 1)


The plateau value at zero momentum is interpreted as constituent quark mass. For highly excited states, where the typical quark momentum is large, the valence quarks become less affected by the quark condensate, hence in this regime the behavior of the quark dressing functions (11), (12) for large momenta is essential. At tree-level in the chiral limit they approach


and the dynamical quark mass goes to zero. For later purpose it is important to project out the infrared divergences, which has explicitly been done in Ref. Wagenbrunn:2007ie () for the case of a purely linear color-Coulomb potential and is not changed by the additional transverse gluon interaction. We therefore quote the result of Ref. Wagenbrunn:2007ie (), given as


with being infrared-finite.

Iv Symmetry Properties of Non-Interacting Quarks

Before we proceed with deriving the meson bound state equations, we discuss the expected degeneracies in case of non-broken chiral symmetry. The two-quark amplitudes then follow from Poincaré invariance and symmetry. The complete list of -meson multiplets is given as (see Refs. Glozman:2002cp (); Glozman:2003bt (); Glozman:2007ek () for a detailed explanation) ()


The axial symmetry mixes the states from the and chiral multiplets with the same isospin but opposite parity. Moreover, from theoretical arguments, Ref. Glozman:2002kq (), and from the analysis with a color-Coulomb potential in the limit , Ref. Wagenbrunn:2007ie (), one expects an even higher degree of degeneracy


i.e., all states with given spin should become degenerate.

Next we turn to the interacting case, where the quark self-energy breaks chiral and axial symmetry. Chiral symmetry restoration then means that the splittings within the multiplets vanish in the limit . It will be interesting to observe if the transverse gluon interaction still allows for such a degeneracy for large orbital quantum numbers. To investigate the effect of transverse gluons we derive bound state equations for the model Hamiltonian (1).

V Bethe–Salpeter Equations for Mesons

The homogeneous ()-meson Bethe–Salpeter equation (BSE) with total and relative four momenta and individual momenta of the (anti-) quarks is conveniently set up in momentum space


with the Bethe–Salpeter kernel, the quark propagator and the meson vertex function. Dirac and color indices are suppressed. As a first step one usually chooses the rest frame, where the (anti-) quark momenta have equal magnitude, so that . Due to the instantaneous interactions of our model the Bethe–Salpeter kernel and the vertex function depend only on three momentum and the BSE (in the ladder approximation, which is exact within this large- model) simplifies as (see Ref. Alkofer:1988tc ())


This equation is solved in the standard fashion: the -integral is taken explicitly, the vertex function of a given meson type expanded in terms of all possible Poincaré-invariant amplitudes (see Appendix B) and, by applying appropriate traces of Dirac matrices, the independent tensor components (see Eqs. (B)-(B)) are projected out. Since we are interested in meson bound states for general orbital excitations , irreducible tensor products occur in the construction of the independent vertex components, which can be simplified by using formulas listed in Appendix A.

As in Ref. Wagenbrunn:2007ie () three different types of mesons with total angular momentum, parity and -parity are constructed

A fourth possible category with is absent in a system with only instantaneous interactions. In Appendix B the coupled integral equations for all meson categories are presented.

In Ref. Wagenbrunn:2007ie () it has been shown that in the limit all states for a given are completely degenerate. Here we repeat the calculation for an additional transverse gluon interaction and investigate if the same degree of degeneracy is still present. As a first step towards this study we have to analyze the BSEs regarding the infrared limit . Mesons of category one are described by two coupled integral equations, see Eqs. (67a), (67) ()


where we have defined the following quantities


which appear for the transverse gluon interaction. With help of Eqs. (15), (16), (17) we rewrite Eqs. (24a), (24) as


In addition we have used that


and that the integral


vanishes in the limit . We note, that the transverse gluon propagator , Eq. (8), is finite for .

Four linearly independent equations describe mesons in category two (74)-(74) given in the infrared limit as ()


with the quantities , Eqs. (25), (26),




entering the transverse gluon part. We note that for the state several vertex function components in Eq. (B) are absent and therefore only two independent functions and remain.

Finally, let us turn to the Bethe–Salpeter equations for mesons of category three (78a), (78), which for have the form


where we have used the definitions (31), (32), (33). For the state there would only be one independent vertex tensor component, see Eq. (B). However, inserting it into the BSE gives zero. Hence, such a state is absent in our instantaneous model, Ref. Wagenbrunn:2007ie ().

All equations listed in this section now serve as a laboratory to study the influence of transverse gluons on mesons for large orbital excitations. By analyzing the occurring angular integrals we are going to observe that the transverse gluon interaction is suppressed for highly excited meson states.

Vi Chiral Symmetry Structure of the Bethe–Salpeter Equations
for the Non-Interacting Case

In this section we discuss the special case of non-interacting quarks and show that the BSEs fulfill the expected meson degeneracies, Eqs. (18)-(20). For vanishing dynamical quark mass the integral equations (27a), (27) for mesons of category one simplify as (using Eq. (14) and )


with and given in Eqs. (25), (26). Inserting the tree-level dressing functions into the last two equations for mesons of category two, (30c), (30d) yields


and one ends up with identical equations, which yield the expected degeneracies and . The remaining two equations (30a), (30b) for mesons of category two


then coincide with the equations for mesons of category three:


The meson degeneracies and follow from these equations. The state only applies for the last two equations of category two, Eqs. (36a), (36), as shown in the last section. We have thus shown, that for vanishing dynamical mass the BSEs fall into the expected chiral multiplets and and are recovered. Next we analyze this question for the interacting case and discuss the role played by transverse gluons.

Vii Effective Chiral Symmetry Restoration for Large Orbital Angular Momentum

The transverse gluon interaction influences the meson states in two ways, namely via the BSEs and via the quark dressing functions. In this section we analyze the BSEs in order to answer if transverse gluons change the asymptotic degeneracy for high-spin bound states.

By comparing mesons of category one and three, Eqs. (27a), (27) and (34a), (34), in the limit of large spins for a purely linear interquark potential , the degeneracy of all chiral multiplets, Eq. (21), has been demonstrated in Refs. Wagenbrunn:2006cs (); Wagenbrunn:2007ie (). Here we discuss the effect of the additional transverse gluon interaction, Eq. (4), for the degeneracy within these two categories in the limit . For mesons in category three, Eqs. (34a), (34), we make the replacement . For a clear reading we once again list the equations for category one