Effective restoration of chiral symmetry in excited mesons

A fast restoration of chiral symmetry in excited mesons is demonstrated. A minimal"realistic"chirally symmetric confining model is used, where the only interaction between quarks is the linear instantaneous Lorentz-vector confining potential. Chiral symmetry breaking is generated via the nonperturbative resummation of valence quarks self-energy loops and the meson bound states are obtained from the Bethe-Salpeter equation. The excited mesons fall into approximate chiral multiplets and lie on the approximately linear radial and angular Regge trajectories, though a significant deviation from the linearity of the angular trajectory is observed.

There are certain phenomenological evidences that in highly excited hadrons, both in baryons [1,2,3] and mesons [4,5] chiral and U(1) A symmetries are approximately restored, for a short overview see [6]. This "effective" restoration of chiral and U(1) A symmetries should not be confused with the chiral symmetry restoration at high temperatures and/or densities. What actually happens is that the excited hadrons gradually decouple from the quark condensates. Fundamentally it happens because in the high-lying hadrons the semiclassical regime is manifest and semiclassically quantum fluctuations of the quark fields are suppressed relative to the classical contributions which preserve both chiral and U(1) A symmetries [6,7]. The microscopical reason is that in high-lying hadrons a typical momentum of valence quarks is large and hence they decouple from the quark condensate and consequently their Lorentz-scalar dynamical mass asymptotically vanishes [1,3,7,10,14]. Restoration of chiral symmetry requires a decoupling of states from the Goldstone bosons [3,8,9,10] which is indeed observed phenomenologically since the coupling constant for h * → h + π decreases fast higher in the spectrum.
At the moment there are two main paths to understand this phenomenon. In the first one one tries to connect the high-lying states to the short-range part of the two-point correlation function where the Operator Product Expansion is valid [2,11]. However, the OPE is an asymptotic expansion. Then, while the correct spectrum of QCD must be consistent with the OPE, there is an infinite amount of incorrect spectra that can also be in agreement with the OPE. Hence the results within the present approach crucially depend on additional assumptions [12,13].
In the second approach the authors try to understand this phenomenon within the microscopical models [7,8,10,14]. There are also interesting attempts to formulate the problem on the lattice [16,17], though extraction of the high-lying states on the lattice is a task of future.
In the absence of the controllable analytic solutions to QCD an insight into phenomenon can be achieved only through models. Clearly the model must be field-theoretical (in order to be able to exhibit the spontaneous breaking of chiral symmetry), chirally symmetric and contain confinement. In principle any possible gluonic interaction can contribute to chiral symmetry breaking and it is not known which specific interaction is the most important one in this respect. However, at the first stage it is reasonable to restrict oneselves to the simplest possible model that contains all three key elements. Such a model is known, it is a generalized Nambu and Jona-Lasinio model (GNJL) with the instantaneous Lorentzvector confining kernel [18,19,20]. This model is similar in spirit to the large N c 't Hooft model (QCD in 1+1 dimensions) [21]. In both models the only interaction between quarks is the instantaneous infinitely raising Lorentz-vector linear potential. Then chiral symmetry breaking is described by the standard summation of the valence quarks self-interaction loops Conceptually the underlying physics is very clear in the 't Hooft model in the sense that once the proper gauge is chosen, the linear Lorentz-vector confining potential appears automatically as the Coulomb interaction in 1+1 dimensions. In 3+1 dimensions, once the Coulomb gauge is used for the gluonic field [22], an almost linearly raising confinement potential has been obtained [23,24,25].
An obvious advantage of the GNJL model is that it can be applied in 3+1 dimensions to systems of arbitrary spin. In 1+1 dimensions there is no spin, the rotational motion of quarks is impossible, and the states are characterized by the only quantum number, which is the radial quantum number. Then it is known that the spectrum represents an alternating sequence of positive and negative parity states and chiral multiplets never emerge. This happens because in 1+1 dimensions the valence quarks can perform only an oscillatory motion. In 3+1 dimension, on the contrary, the quarks can rotate and hence can always be ultrarelativistic and chiral multiplets should emerge naturally [3].
Restoration of chiral symmetry in excited heavy-light mesons has been studied with the quadratic confining potential [14] and it was also mentioned in a model with the instantaneous potential of a more complicated form [15]. Here we report our results for excited light-light mesons with the linear potential. To our best knowledge this is the first explicit demonstration of the restoration of the chiral symmetry within the solvable "realistic" fieldtheoretical model. The results for the lowest mesons within the similar model have been previously reported in ref. [26].
The GNJL model is described by the Hamiltonian [19] interaction parametrised by an instantaneous confining kernel K ab µν ( x − y) of a generic form. In this paper, we use the linear confining potential, and absorb the color Casimir factor into string tension, λ a λ a 4 V 0 (r) = σr. The Schwinger-Dyson equation for the self-energy operator Σ where so that the dressed Dirac operator becomes where, due to the instantaneous nature of the interaction the time-component of the Dirac operator is not dressed. The Lorentz-scalar dynamical mass A p as well as the Lorentz-vector spatial part B p contain both the classical and quantum contributions, the latter coming from loops [7]: where tan ϕ p = Ap Bp . Solution of the Schwinger-Dyson equation (3) with the linear potential is well-known, see e.g. [27], and the mass-gap equation has a nontrivial solution which breaks chiral symmetry, by generating a nontrivial dynamical mass function A p . This dynamical mass is a very fast decreasing function at larger momenta. Then the quark condensate is given as The homogeneous Bethe-Salpeter equation for the quark-antiquark bound state with mass M in the rest frame, i.e. with the four momentum P µ = (M, P = 0), is where χ( p, M) is the mesonic Salpeter amplitude in the rest frame. Eq. (9) is written in the ladder approximation for the vertex which is consistent with the rainbow approximation for the quark mass operator and which is well justified in the large-N C limit.
The Salpeter amplitude can be decomposed into two components for mesons with J P C = (2n) −+ , (2n + 1) +− , (2n + 1) ++ , (2n + 2) −− , or 0 ++ and into four components for mesons with J P C = (2n + 1) −− or (2n + 2) ++ , respectively. Here J is the spin, P the parity and C the charge conjugation parity of the meson and n ∈ N 0 . In that way the Bethe-Salpeter equation becomes a system of coupled integral equations for the components which we solve by expanding them into a finite number of properly chosen basis functions. This leads to a matrix eigenvalue problem which can be solved by standard linear algebra methods. We vary the meson mass until one of the eigenstates is equal to one.
In the gap as well as in the Bethe-Salpeter equations the infrared divergences are removed by introducing a finite "mass" into a confining potential, which is a standard trick. Then in the infrared limit ("mass" goes to 0) the quark propagator consists of a finite and diverged parts, while the mesons masses are finite. Recently it was demonstrated that also the masses of quark-quark subsystems in the color-antitriplet state go to infinity in this limit and hence are removed from the physical spectrum [28]. The results presented in the following were obtained by calculating the infrared limit numerically, i.e. the quoted meson masses were extrapolated to the infrared limit from a few points with a very small but finite mass of the infrared regulator. It turned out that in this region of the small mass of the infrared regulator the squares of the meson masses depend almost linearly on the mass of the infrared regulator making the extrapolation reliable. The presented results are accurate within the quoted digits at least for states with small J and for states with higher J but small n. At larger J for larger n numerical errors accumulate in the second digit after comma.
By definition an effective chiral symmetry restoration means that (i) the states fall into   In Table 1 we present our results for the spectrum for the two-flavor (u and d) mesons  A key feature of this dynamical mass is that it is strongly momentum-dependent and vanishes very fast once the momentum is increased. When one increases excitation energy of a hadron, one also increases a typical momentum of valence quarks. Consequently, the chiral symmetry violating dynamical mass of quarks becomes small. Hence the mixing of the independent chiral Bethe-Salpeter amplitudes becomes small. A given state in the table is then assigned to the chiral representation according to the chiral Bethe-Salpeter amplitude that dominates in the given state.
In Fig. 2  the value obtained here [29,30]. This would increase the value for the condensate but on the other hand lead to an unrealistically small pion decay constant [31,32].
In the limit n → ∞ and/or J → ∞ one observes a complete degeneracy of all multiplets, representation that combines all possible chiral representations for the systems of two massless quarks [5]. This means that in this limit the loop effects disappear completely and the system becomes classical [6,7].