# Chiral and angular momentum content of mesons

###### Abstract

First, we overview the present status of the effective chiral restoration in excited hadrons and an alternative explanation of the symmetry observed in the highly excited hadrons. Then we discuss a method how to define and measure in a gauge invariant manner the chiral and angular momentum content of mesons at different resolution scales, including the infrared scale, where mass is generated. We illustrate this method by presenting results on chiral and angular momentum content of and mesons obtained in dynamical lattice simulations. The chiral symmetry is strongly broken in the and neither the nor the can be considered as its chiral partners. Its angular momentum content in the infrared is approximately the partial wave, in agreement with the quark model language. However, in its first excitation, , the chiral symmetry breaking is much weaker and in the infrared this state belongs predominantly to the (1/2,1/2) chiral representation. This state is dominated in the infrared by the partial wave and cannot be considered as the first radial excitation of the -meson, in contrast to the quark model.

## I Parity doubling and higher symmetry seen in highly excited hadrons

The spectra of highly excited hadrons, both baryons G1 and mesons G2 , reveal almost systematical parity doubling. This parity doubling can be interpreted as an indication of effective chiral and restorations, for reviews see G3 . The effective chiral restoration means that dynamics of chiral symmetry breaking in the vacuum is almost irrelevant to the mass generation of these highly excited hadrons and their mass comes mostly from the chiral invariant dynamics. This is just in contrast to the lowest lying hadrons such as , or , where the chiral symmetry breaking in the vacuum is of primary importance for their mass origin. The latter can be seen from the SVZ sum rules Shifman ; Ioffe and many different microscopical models.

However, there could be other reasons for parity doubling JPS ; SV ; K ; A and one needs alternative evidences. If the effective chiral restoration is correct, then the highly excited hadrons should have small diagonal axial coupling constants. It is not possible, unfortunately, to measure these quantities experimentally. The effective chiral restoration also predicts that the states with almost restored chiral symmetry should have small decay coupling constants into the ground state and the pion. The decay coupling constants can be obtained from the known decay widths. It turns out that all excited nucleons that have an approximate chiral partner have a very small decay coupling constant (as compared to the pion-nucleon coupling constant). In contrast, the state, in which case a chiral partner cannot be identified from the spectrum, has a decay coupling that is even larger than the pion-nucleon coupling. One observes a 100% correlation of the spectroscopic patterns with the decays as predicted by effective chiral restoration G4 .

The observed high lying spectra have higher degeneracy. The states group not only into possible chiral multiplets, but also states with different spins are approximately degenerate. Chiral symmetry cannot connect states with different spins. This means that higher symmetry is observed, that includes chiral and as subgroups. It is a key question to understand this high symmetry and its dynamical origin. The answer to this question would clarify the origin of confinement and its interconnection with dynamical chiral symmetry breaking, the mass and the angular momentum generation in QCD. It is possible to explain this degeneracy of states with different spins if one assumes a principal quantum number on top of chiral and restorations GN1 .

If chiral restoration is correct, then there must be chiral partners to mesons with the highest spin states at the bands around 1.7 GeV, 2 GeV and 2.3 GeV, that are presently missing, see Fig. 2 of ref. G3 . Consequently, a key question is whether these states do not exist or they could not be seen due to some kinematical reasons. It turns out that the latter is correct and a centrifugal repulsion in the incoming wave suppresses all missing states as compared to to all observed ones GS . There is a weak signal for missing states once a careful analysis is done. Obviously, the missing states should be also searched in other types of experiments. The same centrifugal suppression in the pion-nucleon scattering is present for all missing chiral partners in the nucleon and delta spectra K .

The alternative explanation of the large degeneracy seen in both nucleon and meson spectra would be existence of the relation , where L is the orbital angular momentum in the state. The total angular momentum is constructed from the quark spins and the orbital angular momentum according to the standard nonrelativistic rules. The parity of the state is connected with by the standard nonrelativistic relation SV ; K ; A . In such case the parity doubling is accidental and is not related with chiral symmetry in the states. This scenario requires that there must not be parity partners to the highest spin states in every band. Such relation implies that there are three independent conserved angular momenta, . If the high lying states behaved non-relativistically and assuming absence of the spin-orbit force, it would be indeed possible to obtain a principal quantum number , like in the nonrelativistic Hydrogen atom.

Such a scenario is inconsistent with QCD and can be ruled out on very general grounds. (i) In QCD, that is a highly relativistic quantum field theory, there is only one conserved angular momentum, . There are no representations of the Poincaré group that would contain the orbital angular momentum as a good quantum number. (ii) QCD is a renormalizable quantum field theory. The hadron mass is a renormalization group invariant and does not depend on the renormalization scale. At the same time is not a renormalization group invariant. Then the relation cannot exist within QCD.

From the theoretical side, there exists a transparent model that manifestly exhibits effective chiral restoration in hadrons with large WG ; NB . While this model is a simplification of QCD, it gives the insight into phenomenon. The model is confining, chirally symmetric and provides dynamical breaking of chiral symmetry in the vacuum Y ; ADLER . The chiral symmetry breaking is important only at small momenta of quarks. But at large the centrifugal repulsion cuts off the low-momenta components in hadrons and consequently the hadron wave function and its mass are insensitive to the chiral symmetry breaking in the vacuum. The chiral symmetry breaking in the vacuum represents only a tiny perturbation effect: Practically the whole hadron mass comes from the chiral invariant dynamics.

## Ii The chiral content of mesons from first principles

To resolve the issue one needs direct information about the chiral structure of states, which can be obtained from ab initio lattice simulations. Here we define and reconstruct in dynamical lattice simulations a chiral as well as an angular momentum decomposition of the leading quark-antiquark component of mesons GLL1 ; GLL2 .

The variational method var represents a tool to study the hadron wave function. One chooses a set of interpolators with the same quantum numbers as the state of interest and computes the cross-correlation matrix

If this set is complete and orthogonal with respect to some transformation group, then it allows to define a content of a hadron in terms of representations of this group.

In G2 ; G3 ; CJ a classification of all non-exotic quark-antiquark states (interpolators) in the light meson sector with respect to the and was done. If no explicit excitation of the gluonic field with the non-vacuum quantum numbers is present, this basis is a complete one for a quark-antiquark system and we can define and investigate chiral symmetry breaking in a state. The eigenvectors of the cross-correlation matrix describe the quark-antiquark component of the state in terms of different chiral representations.

For example, when we study the meson and its excitations, two different chiral representations exist that are consistent with the quantum numbers of the -mesons. Assume that chiral symmetry is not broken. Then there are two independent states. The first one is ; it can be created from the vacuum by the standard vector current, Its chiral partner is the meson. The other state is , which can be created by the pseudotensor operator, The chiral partner is the meson.

Chiral symmetry breaking in a state implies that the state should in reality be a mixture of both representations. If the state is a superposition of both representations with approximately equal weights, then the chiral symmetry is maximally violated in the state. If, on the contrary, one of the representations strongly dominates over the other representation, one could speak about effective chiral restoration in this state.

Diagonalizing the cross-correlation matrix one can extract energies of subsequent states from the leading exponential decay of each eigenvalue

The corresponding eigenvectors give us information about the structure of each state. Namely, the coefficients define the overlap of the physical state with the interpolator ,

While the absolute value of the coupling constant cannot be defined in lattice simulations (because a normalization of the quark fields on the lattice is arbitrary), their ratio for two different operators and for a given state is well defined GLL1 . Consequently, the ratio of the vector to pseudotensor couplings, , tells us about the chiral symmetry breaking in the states .

## Iii The angular momentum content of mesons from first principles

The chiral representations can be transferred into the basis, using the unitary transformation GN1 ; GN2

(1) |

where is given by

(2) |

Thus, using the interpolators and for diagonalization of the cross-correlation matrix, we are able also to reconstruct a partial wave content of the leading Fock component of the -mesons. Note, that it is a manifestly gauge-invariant definition of the angular momentum content of mesons.

## Iv Scale dependence of the chiral and angular momentum decompositions

The ratio as well as a partial wave content of a hadron are not the renormalization group invariant quantities. Hence they manifestly depend on a resolution scale at which we probe a hadron. If we probe the hadron structure with the local interpolators, then we study the hadron decomposition at the scale fixed by the lattice spacing . For a reasonably small this scale is close to the ultraviolet scale. However, we are interested in the hadron content at the infrared scales, where mass is generated. For this purpose we cannot use a large , because matching with the continuum QCD will be lost. Given a fixed, reasonably small lattice spacing a small resolution scale can be achieved by the gauge-invariant smearing of the point-like interpolators. We smear every quark field in spatial directions with the Gaussian profile over the size in physical units such that , see Fig. 1. Then even in the continuum limit we probe the hadron content at the resolution scale fixed by . Such definition of the resolution is similar to the experimental one, where an external probe is sensitive only to quark fields (it is blind to gluonic fields) at a resolution that is determined by the momentum transfer in spatial directions.

## V The chiral and angular momentum content of and mesons

To explore the chiral structure of mesons and possible effective chiral restoration it is important to have a Dirac operator with good chiral properties. We use specifically the Chirally Improved Dirac operator GAT . The set of dynamical configurations is used for two mass-degenerate light sea quarks, see for details ref. GLL2 .

Our cross-correlation matrix is calculated with the following four interpolators

where is one of the spatial Dirac matrices, is the -matrix in (Euclidean) time direction. The subscripts and (for narrow and wide) denote the two smearing widths, fm and fm, respectively. Both the ground state mass and the mass of the first excited state of the -meson are shown on the l.h.s. of Fig. 2.

On the r.h.s. of Fig. 2 we show the -dependence of the ratio both for the ground state -meson and its first excited state. For the ground state at the smallest resolution scale of fm this ratio is approximately , i. e., we see a strong mixture of the and representations in the -meson. Consequently, there is no chiral partner to . Such ratio implies that the vector meson in the infrared is approximately a state with a tiny admixture of a wave.

However, the situation changes dramatically for the first excited state, . In this case a strong dependence of the ratio on the resolution scale is observed. Although we do not have the precise value of the ratio for at large fm, it is indicative that this value is very small. One observes a significant contribution from the representation and a contribution of the other representation is suppressed. This indicates a smooth onset of effective chiral restoration. The approximate chiral partner is . This small ratio also implies a leading contribution of the wave. This result is inconsistent with to be a radial excitation of the ground state -meson, i. e., a state, as predicted by the quark model.

Acknowledgements

The author is thankful to Christian Lang and Markus Limmer for a fruitful collaboration on the lattice aspects of this talk. Support of the Austrian Science Fund through the grant P21970-N16 is acknowledged.

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