A Multiquark Approach to Excited Hadrons and Regge Trajectories

Hindawi Advances in High Energy Physics Volume 2019, Article ID 1701939, 7 pages https://doi.org/10.1155/2019/1701939

Research Article A Multiquark Approach to Excited Hadrons and Regge Trajectories

S. S. Afonin

Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia

Correspondence should be addressed to S. S. Afonin; [email protected]

Received 11 December 2018; Accepted 28 January 2019; Published 12 February 2019

Academic Editor: Juan Jos´e Sanz-Cillero

Copyright © 2019 S. S. Afonin. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te publication of this article was funded by SCOAP3.

We propose a novel approach to construction of hadron spectroscopy. Te case of light nonstrange mesons is considered. By assumption, all such mesons above 1 GeV appear due to creation of constituent quark-antiquark pairs inside � or �(�)mesons. Tese spin-singlet or triplet pairs dictate the quantum numbers of formed resonance. Te resulting classifcation of light mesons turns out to be in a better agreement with the experimental observations than the standard quark model classifcation. It is argued that the total energy of quark components should be proportional to the hadron mass squared rather than the linear mass. As a byproduct a certain relation expressing the constituent quark mass via the gluon and quark condensate is put forward. We show that our approach leads to an efective mass counting scheme for meson spectrum and results in the linear Regge and radial Regge trajectories by construction. An experimental observation of these trajectories might thus serve as evidence not for string but for multiquark structure of highly excited hadrons.

1. Introduction For instance, it does not explain the scalar mesons below 1 GeV for which a tetraquark structure is ofen assumed [4]. Te regularities in the masses of atomic nuclei tell us Aquestionariseswhydonotweseemanylightmultiquark that nuclei certainly consists of some building blocks, the hadrons above 1 GeV? Among other questions one can nucleons,andthemassofnucleiisjustproportionaltothe mention the experimental absence of many predicted states number of nucleons. Te masses squared of light mesons also �� −− (e.g., with � =2 ) and, on the other hand, observation reveal many regularities [1, 2] but they are usually attributed of many not predicted states (e.g., too rich scalar sector and to the properties of strong interactions acting between valent exotic �1). quarks. In the present paper, we will try to develop an alternative point of view on the masses of light hadrons: Tere are some deep theoretical questions as well. Te within the presented scheme, they could be also viewed as light hadrons represent highly relativistic quantum systems � originating from some efective “building blocks”. in which is not a conserved quantity. Te quark angu- Historically the main basic framework for description of lar momentum � is nevertheless a standard ingredient in hadronsisthequarkmodel.Tismodelhadagreatsuccess constructing dynamical models for light hadrons. Another in making order in numerous hadronic zoo. In particular, question concerns the observable quantities like hadron all mesons within this picture are composed of a quark masses: they must be renorminvariant in the feld-theoretical and antiquark having the relative angular momentum � sense. Stated simply, they must represent some constants which dictates the spatial and charge parities for the quark- independent of energy scale. Te quark masses, for instance, �+1 �+� antiquark systems, �=(−1) and �=(−1) ,where are not such constants since they have anomalous dimension � is the total quark spin. By now we know that the ensuing described by QCD. A relation of calculated observables to classifcation of mesons has many phenomenological faws the renormalization invariance is obscure in many known (see, e.g., the corresponding reviews in the Particle Data [3]). phenomenological models of hadrons. 2 Advances in High Energy Physics

Te quark model represents a concept rather than a Masses of other mesons must be also renorminvariant. On model. A real model appears only when a defnite interaction the other hand, the light nonstrange mesons can be viewed, between quarks is postulated. Any such model should be in one way or another, as some quantum excitations of pion. able to explain general features of observed hadron spectrum. We will assume that the masses of these mesons can be In the light hadrons, perhaps the most spectacular general represented as feature is the observation of approximately linear Regge and 2 2 radial trajectories of the kind: �ℎ =Λ(�ℎ +2��)=Λ�ℎ +��. (3) 2 � � =��+��+�, �,�=0,1,2,.... (1) �,� Below we will motivate a proportionality of the energy � Here � denotes the spin, � is the radial quantum number parameter ℎ to the renorminvariant gluon condensate, � � ∼�⟨�2 ⟩/⟨��⟩ (enumerating the daughter Regge trajectories), and �, � , ℎ � �] . But frst we would like to develop and � are the slope and intercept parameters. Tere is no a phenomenological mass counting scheme based on the general agreement on the position of diferent states on the relation (3) and demonstrate how it describes the meson trajectories (1) (see, e.g., [1, 2, 5–20]) but the existence of these spectroscopy. trajectories seems to be certain for many phenomenologists Our second assumption is that the parameter �ℎ can [12–27]. Tere are also some indications on a similar behavior be interpreted as an efective energy of some constituents in the heavy quark sector [28–32]. Te nonrelativistic and diferent from the current quarks. Te problem is to pro- partially relativistic potential models cannot explain the pose a model for these constituents which should represent recurrences (1) in a natural way. Usually the observation of some excitations inside the pion. We postulate three basic linear trajectories (1) is interpreted as an evidence for string excitations. Te frst one appears when one of quarks absorbs picture of mesons. In spite of many attempts (see, e.g., [33– agluonofcertainenergy�ℎ =��.Tespinofexcited 40]), however, a satisfactory quantized hadron string has not quark changes its direction to the opposite one, converting been constructed. Among typical faws of this approach one the original spin-singlet ��-pair (�-meson) to the spin-triplet can mention the absence of spontaneous chiral symmetry one (�-meson). Te second kind of excitation emerges due breaking, rapidly growing (with mass) size of meson exci- to formation of spin-singlet ��-pair with efective mass �ℎ = tations that is not supported experimentally, unclear role of �0 (the lower index stays for total spin of �-wave ��-pair). higher Fock components in the hadron wavefunction. We will call this pair �0. Te third basic excitation is the Te purpose of the present work is to propose a novel formation of spin-triplet ��-pair with efective mass �ℎ =�1. realization of quark model concept, a realization that leads Tis pair will be called �1. Te formation of constituent pairs to a natural (and alternative to hadron strings) explanation �0 and �1 may be a QCD analogue of formation of para- and of Regge recurrences (1) and that is potentially free of typical orthopositronia from photons. We assume that any excitation shortcomings inherent to string, potential, and some other inside pion leading to an observable resonance can be approaches. represented as a combination of these basic excitations so that the total efective energy �ℎ in (3) is just a sum (appropriate � � � 2. Masses and Classification of Light Mesons number of times) of �, 0,and 1. Simultaneously this will dictate the quantum numbers of constructed resonance (�0 �� −+ −− In hadron physics, the pion is known to be the most impor- and �1 possess � =0 and 1 , respectively). In a sense, tant and best studied meson both experimentally and theoret- the introduction of �0 and �1 is our model for excitations of ically. Concerning the theoretical aspect, the pion is the only higher Fock components in the pion wavefunction. hadron (along with � and �)forwhichweknowamodel- It is convenient to divide the nonstrange meson reso- independent mass formula: the Gell-Mann–Oakes–Renner nances above 1 GeV into excited pions and excited �(�) (GOR) relation [41], mesons. Te excited � mesons containing � pairs �0 and � �� pairs �1 will be labeled as �� ,where� means the total ⟨��⟩ 0 1 �2 =− (� +� ) =Λ⋅2�, spin. According to our mass counting scheme, masses of these � 2 � � � (2) �� excitations are given by the relation

2 � =� ≐� Λ≐−⟨��⟩/� 2 2 where we set � � � and � .Wewilluse � =Λ(� +�� +�� )+� . � � � � 0 1 � (4) the standard values for the quark condensate and masses of �0�1 current quarks at the scale of the pion mass from the QCD sum rules [42, 43] and Chiral Perturbation Teory (ChPT) By assumption, the whole system is in the �-wave state (one 3 �>0 [44], ⟨��⟩ = −(250 MeV) , �� +�� =11MeV. Using can include into this scheme the higher waves; we � �� = 92.4 MeV (the value of weak pion decay constant), will consider the simplest possibility); hence, at fxed and � relation (2) yields �� = 140 MeV and Λ = 1830 MeV. , relation (4) defnes mass of a set of degenerate states with Te famous relation (2) was derived in various approaches various spins up to �=�+1. � and � parities obey the standard �+�+1 �+1 assuming the spontaneous chiral symmetry breaking (CSB) multiplicative law: (�, �) = ((−1) ,(−1) ).Tetotal in strong interactions. isospin of system of �+�constituent �� pairsiszeroasthis Te renormalization invariance of pion mass follows from system, by assumption, arises from an isosinglet combination the renorminvariance of operator �� in QCD Lagrangian. of gluons. Advances in High Energy Physics 3

�� = Te excited pions follow the same principle. Te states �0�1 and �0�1 have quantum numbers � −+ � � � have mass (0, 1, 2) �0 �1 , i.e., 3 possible spins and pion parities. Tese states describe resonances �, �1,and�2 above 1 GeV. In contrast � �2 =Λ(�� +�� )+�2 , to the standard quark model, the 1-meson is not exotic � � � 0 1 � (5) �0�1 in our scheme! Te relation (5) predicts mass about �= 2 √Λ(�� +�0 +�1)+�� ≈ 1550 − 1610 MeV. Te possible �+�+1 � maximal spin �=�,andparities(�, �) = ((−1) ,(−1)). candidates are �1(1600) and �2(1670) [3]. � � �� −− Since a simultaneous excitation of two close pairs 0 0 Te states �2 and �2 have � = (1, 3) as the total does not change neither spin nor parities, it is natural to refer 1 1 spin of the pair �1�1 can be 0 or 2. Natural candidates are to a meson containing even number 2� of �0 and no one �1 as �(1700) and �3(1690) (and �(1650) with �3(1670) for �)[3]. � the -th radial excitation. Te radial excitations appear thus 2 Our fts yield �=√Λ(�� +2�1)+�� ≈ 1630 − 1650 with a “period” 2�0Λ. Consider a formation of two close vector pairs �1�1.As MeV. As in the standard quark model, the exotic states with �� −− they are formed from originally massless gluons we must quantum numbers � =0 are absent and �(1700) is not combine their spins as if they were massless particles. Tis the second radial excitation (in the potential quark models, follows from conservation of total spin of original gluons. �(1700) (and �(1650))isaD-wave state, while the radial We recall that the spin of massless gluons has only two excitations of �(770) should be S-wave ones) of �(770).On �� −− projections: two possible helicities. Te total helicity of two the other hand, we do not obtain states with � =2 gluons can be 0 or 2. Tis rule (in other words, the addition which are predicted by the standard quark model but were law for spins of spin-1 massless particles is the same as for not observed. massive spin -1/2 particles; just the fnal result must be � multiplied by 2) extends to arbitrary number of close 1 3. Regge Trajectories pairs. It is easy to see now that the excitations of the kind �� 2� , � = 0, 1, 2, . . ., will give rise to a degenerate family of �1 Relations (4) and (5) lead to linear Regge, equidistant daugh- �� −− resonances with � = (1,3,...,2�+1) . Te states on the ter Regge, and radial trajectories. Below we give examples for � some of them. main -meson Regge trajectory follow thus with a “period” 0 Te states � 2� , � = 0, 1, 2, . . ., form linear trajectory for 2�1Λ. Tese excitations generate also daughter trajectories. �0 �� −− For instance, the additional � mesons (� =1 )arise the radial excitations of pion, whichappearwiththesame“period”2�1 Λ. Tey are diferent 2 2 �� (�) =2Λ�0�+��. (6) from the “radial” � mesonsthatappearedwith anotherperiod 1� 2�0Λ. Te radial excitations of �(770), the resonances 2� ,lieon �0 In order to demonstrate how our scheme works in the frst radial �-trajectory, practice let us consider some examples. � � � = √Λ� +�2 2 � 2 Te mass of -meson is � � �.Tisrelation �� (�)� =2Λ�0 (�+ ) +��. (7) 2�0 fxes �� ≈ 310 MeV from the averaged mass of charged � in � the hadronic processes, �� = 766.5 ± 1.1 MeV [3]. Te second radial -trajectory is composed of the states 1 1 �� +− 1� � � � =1 �2�2� , �0 and �0 have quantum numbers and, 1 0 2 � according to (4), mass �=√Λ(�� +�0)+��. Fitting to the 2 �1 � 2 �� (�)�� =2Λ�0 (� + + )+��. (8) masses of experimental states ℎ1(1170) and �1(1230) [3], we �0 2�0 � ≈ 430 − 510 obtain the estimate 0 MeV. 1� 0� It is evident that the states �2(�−1)�2� formally give rise to Te state � has the quantum numbers of scalar 1 0 0 � � �� =0++ �= the -th radial -trajectory. Te resonances having structure particle, . According to (5) its mass is 1� 2� form the frst radial �1-trajectory, 2 �1�0 √Λ�0 +�� ≈ 900 − 980 MeV.Tisestimateisclosetothe � (980) � �� mass of 0 [3]. �2 (�) =2Λ� (� + 1 + )+�2 . � � �� ++ �1 � 0 � (9) � � � = (0, 1, 2) 2�0 2�0 �1 and �1 have .Teyshouldbe � (1260) � (1285) � (1270) � (1320) the series of states 1 , 1 , 2 , 2 , Te expressions for further axial radial trajectories can be � (1370) and likely 0 [3]. Substituting the experimental masses easily written. It should be remarked that the standard quark � (1285) � (1270) of well measured resonances 1 and 2 to our model predicts only two radial �-trajectories (�-and�-wave 2 relation for their mass, �=√Λ(�� +�1)+��,weobtain ones) and one �1-trajectory (a �-wave one). Te scheme � ≈ 570 − 580 under discussion is much richer. the estimate 1 MeV. �� 1 1 Te spin �-trajectory is composed of the states �−1 with � � 2 �1 �2 and �2 have mass �=√Λ(�� +2�0)+�� ≈ 0 0 � = 1, 3, 5, . . .. Te corresponding masses are 1470 − 1590 MeV. Tey are the frst radial excitations of � � and � mesons and describe the resonance regions �(1450) 2 � 2 � =Λ�1 (� − 1 + )+� . (10) �(1420) �� and [3]. �1 4 Advances in High Energy Physics

For the even spins, � = 2, 4, 6, . . ., trajectory (10) describes 1.14/(2 ⋅ 1.83) ≈ 0.31 GeV that coincides with our previous �� (��) mesons. Te Regge trajectory (10) describes thus estimate for ��. states with alternating parities and this agrees with the Let us clarify further how the excited resonances with phenomenology. identical quantum numbers can have diferent origin in the Te principal �-meson Regge trajectory (10) is accom- proposed scheme. Tey may represent the radial states, states panied by the daughter trajectories following with the step on daughter Regge trajectories and various “mixed” ones. 1� � 2Λ�1.Tespin-1�-mesons of the kind ��−1 are the lowest For instance, the second -meson excitation with the same 1 parities appears in three forms: the second radial excitation states on the daughters. For example, the lowest state in (8) is 1 1 1 � � � �4 , the vector state on the second daughter trajectory �4 , the lowest state on the frst daughter. Te spectrum of ��−1 0 1 1 1� and the mixed one 2 2 . Tey are degenerate only in the excitations reads �0�1 � =� � limit 0 1. It is likely difcult to detect such a splitting 2 � 2 �1 =2Λ� (� + )+� , experimentally because of overlapping widths. In reality, one � �−1 1 � (11) � 2� 1 1 would observe rather a “broad resonance region”. We note also that placing of the observed “radial” states � = 0,1,2,... where enumerates the daughters. Similar on a certain trajectory should be made with care: an incorrect relations can be written for the axial and other mesons. interpretation of states leads to a false (or more precisely, An obvious consequence of the emerging spectrum is the introduced by hands) nonlinearity of the trajectory. Take degeneracy of spin and daughter radial excitations of the type � 2 again the meson as a typical example. Te frst three radial � (�, �) ∼ � + �, which is typical for the Veneziano dual 1� 1� 1� excitations of � are the states �2 , �4 ,and �6 .Tey amplitudes [45] and the Nambu–Goto open strings. Tis kind 0 0 0 are accompanied by the following states with the quantum of degeneracy was observed in the experimental spectrum of 1� 1� 1� numbers of �: �2 , �2�2 ,and �4 .Since(3/2)�0 >�1 > light nonstrange mesons [5–11]. 1 1 0 1 1� �� 0 �1 �0 in our fts, the sequence of frst 7 �-mesons is �, 2 , Te ftted values of , ,and are rather close. Tis �0 allows considering reasonable limits where some of them are 1� 1� 1� 1� 1� �2 , �4 , �2�2 , �4 ,and �6 .Teylikelycorrespond equal. Te notions of radial and daughter radial trajectories 1 0 1 0 1 0 to the vector resonances �(770), �(1450), �(1700), �(1900), coincide in the limit �0 =�1.Inthelimit�� =�1,theradial �(2000), �(2150),and�(2270) [3]. vector and axial trajectories are related by

�2 (�) =�2 (�) +�2 −�2 . �1 � � � (12) 4. A Theoretical Motivation �2 ∼ Tis relation holds both for radial and for daughter radial It is interesting to get a theoretical hint both on the rule ℎ � trajectories. In the chiral limit, �� =0,relation(12)forthe �,0,1 underlying the presented mass counting scheme and � ground states, �=0, reduces to the old Weinberg relation, on the value of energy parameters �,0,1.Telatterpointis 2 2 intriguing by itself since the scale about 0.3 GeV is ubiquitous �� =2�� [46]. In the most symmetric limit, �� =�0 =�1, 1 in the hadron physics; for instance, it ofen emerges in mass the vector and axial radial spectrum in the chiral limit reduce relations between hadrons containing heavy quarks [48]. to a very simple form, Te mass of a hadron state |ℎ⟩ can be related to the trace of 1 energy momentum tensor Θ�] in QCD (the sign convention �2 (�) =2�2 (� + ), � � 2 is mostly positive), (13) � � 2 2 2 � �� (�) =2� (�+1) . 2�ℎ =−⟨ℎ�Θ�� ℎ⟩ , (14) �1 � � � � Relations (13) frst appeared in the variants of Veneziano where Θ� is given by the scale anomaly, amplitude which incorporated the Adler self-consistency � condition [45]. Tis condition (the amplitude of �� scattering Θ� = �2 +∑� ��. � 2� �] � (15) is zero at zero momentum) incorporates the CSB removing � �=�,�,... degeneracy between the � and �1 spectra. Within the QCD � sum rules, relations (13) may be interpreted as a large- � Here � denotes the QCD beta-function and �� is the coupling generalization of the Weinberg relation [47]. constant. Relation (14) represents trace taken in the following We see thus that in certain limits the Regge phenomenol- Ward identity known in deep inelastic phenomenology [49]: ogy of our approach reproduces various known relations. 2��] =−⟨ℎ(��)|Θ�]|ℎ(�])⟩. As follows from derivation In addition, it is not excluded that the assumption �� = of the identity (14), the squared mass (not the linear one!) �0 =�1 is compatible with the available data if a global ft in the l.h.s. appears due to the relativistic invariance. Te is performed. According to the phenomenological analysis in r.h.s. of (14) is a renorminvariant quantity. One can build two review [2] the averaged slopes of spin and radial trajectories � substantially diferent renormalization invariant operators in light nonstrange mesons, � and � in (1), are equal; the in QCD; both of these operators are present in (15). Te � 2 reported value is �≈� ≈1.14GeV .Tisremarkable l.h.s. of (14) defnes the gravitational mass of a hadron. In observation leads to a large degeneracy in the spectrum [5– reality, the vast majority of hadron resonances probably have 11, 21–27]. From the given value we have �0 ≈�1 ≈�/(2Λ)≈ no well-defned gravitational (and inertial) mass as they do Advances in High Energy Physics 5 not propagate in space: their typical lifetime of the order of spontaneous CSB in QCD and related physics, e.g., into the −23 −24 10 –10 sdoesnotallowleavingthereactionareaofthe Nambu–Jona-Lasinio model [51] or the linear sigma-model, �2 =�2 +6�2 order of 1 fm which is comparable to their size. Tey show a model-independent relation emerges: �1 � � . up only as some structures in the physical observables at On the other hand, the idea of CSB was also exploited in certain energy intervals. Te resonance mass is commonly �2 =2�2 the derivation of famous Weinberg relation, �1 � associated with the real part of an �-matrix pole on the √ [46]. Combining both relations, one has �� =��/ 6. second (unphysical) sheet. How this defnition is related to Substitution of the value �� ≈ 766 MeV used in our fts leads the gravitational mass is by far not obvious in a theory with to �� ≃0.31GeV), �� ≈��/3,where�� is the proton mass. confnement. � � It is interesting to observe the numerical coincidence In the sector of light and quarks, there are only two � ≃� hadrons with well-defned gravitational mass: the pion and � �. Tis suggests that the “spin fip” converting pion to � is essentially equivalent to creation of constituent quark. nucleon. Te pion mass is given by the GOR relation (2). Tis � relation should somehow follow from the Ward identity (14). Tus the meson in our approach may be interpreted as a An explicit derivation of (2) from (14) would likely give an system of one constituent quark and one current antiquark (or vice versa) which interact with the QCD vacuum. Tis analytical proof for spontaneous CSB in QCD. Our present � aim is to show, at least on a rather intuitive level, how the mass picture is diferent from the usual quark model where gap may emerge in nonperturbative gluon vacuum and get consists of two constituent quarks interacting with each other some quantitative insight. Consider the case of nucleon, |ℎ⟩ = (in view of these speculations one can ask what is the exact |�⟩, for which (14) must yield the proton mass. Let us insert relation for the nucleon mass within the given approach? We � � a complete set of vacuum states |0⟩⟨0| from both sides of Θ can propose the following guess. In the case of meson, we � motivated “a rule of transition” from the nonrelativistic quark in (14); the nucleon mass is then given by 2 model to our scheme: �ℎ = ∑ �� + interactions �→ � ℎ = � � 2 � � Λ⋅(1/2)∑�� +��. Applying this “rule” to the ground state of 2 1 2 � � 2 � 2 � =− ⟨0|�⟩ ⟨0 � � +∑� �� 0⟩ . nucleon (3 constituent quarks) we arrive at the relation � = � 2 �2� �] � � (16) � � � �=�,� � 2 � � (3/2)Λ�� +�� that gives �� ≃0.93GeV. Te factor 3/2 � =0 could be also qualitatively interpreted as follows: the creation Tis expression suggests that, in the chiral limit, � ,the of 3 constituent quarks and 3 constituent antiquarks (i.e., the ⟨0|�2 |0⟩ nucleon mass is determined by the vacuum average �] . creation of �� pair in vacuum) is equivalent to creation of 3 From the one-loop QCD beta-function we have �/2�� = constituent quark-antiquark pairs, so the relation of energies 2 −(�0/8)(��/�),where�0 = 11−(2/3)��, �� =�� /4�.Relation of constituents in nucleon and vector meson is 3 to 2.). (16)canbethenrewrittenas Te given interpretation can be further substantiated by 2 the observation that the same “spin fip” should convert the 1 � (��/�) ⟨��]⟩ Δ Λ� �2 = ⟨0|�⟩2 ⟨��⟩ (−∑� + 0 ). nucleon to baryon. As this fip “costs” � for the hadron � 2 � 8 (17) 2 2 � ⟨��⟩ mass squared, we should expect the relation �Δ =��+Λ��. It indeed yields �Δ ≃1.2GeV in a good agreement with Relation (17) demonstrates that the masses of current quarks experimental data for Δ(1232) [3]. in nucleon acquire a contribution from nonperturbative Coming back to motivation of our approach, the pre- gluon vacuum. Since nucleon at rest is interpreted in the sented mass counting scheme is based on the assumption that quark model as a bound system of three constituent quarks, the origin of hadron masses is similar to the case of nucleon wecanestimatefrom(17)anefectiveenergyperquark,i.e., mass in (17); the hadron mass squared (not the linear one as in the value of constituent quark mass; in the chiral limit, it is many other approaches!) is proportional to efective energy of just one-third of gluon contribution in (17): hadron constituents. Tis rule is conjectured from the Ward identity (14). � (� /�) ⟨�2 ⟩ 0 � �] Te second assumption refers to the postulated form of �� ≃− . (18) 24 ⟨��⟩ meson constituents; the valent �� pair plus constituent �� pairs which efectively parametrize contributions to hadron Substituting into (18) the standard value of gluon condensate 2 4 mass from strong gluon feld. Te given choice is a model from QCD sum rules, (��/�)⟨��]⟩ = 0.012(3) GeV [42, 43], which works in the meson spectroscopy above and near 3 and ⟨��⟩ = −(0.25 GeV) , �� =2, we obtain the estimate 1 GeV. Besides the spectroscopy it could explain why the �� ≃ 310 MeV. It is a typical estimate for the value of con- highly excited nonstrange mesons prefer to decay to more stituent quark mass (strictly speaking, the value of constituent than two pions or, say, to have sometimes �-meson in the quark mass is very model dependent; as far as we know, it decay products instead of ��-pair [3]; the corresponding ranges from 220 to 450 MeV in various models. Te value possibilities are likely enciphered in the expression for the 2 ��(� ) ≃ 310 MeV at small momentum �,however,proves mass of a resonance as is seen in the examples of Section 2. to be seen in unquenched lattice simulations in the chiral It should be mentioned that hadron structure strongly limit (see, e.g., [50]). We can indicate a simple qualitative way depends on a reference frame. For instance, the proton leading to this estimate. When one incorporates the vector structure experimentally looks very diferent if it probed for and axial mesons into the low-energy models describing the a proton at rest or for ultrarelativistic proton. It might be that 6 Advances in High Energy Physics thehadronmodeldevelopedinthispaperismoreappropriate [5] E. Klempt and A. Zaitsev, “Glueballs, hybrids, multiquarks: as a picture of light hadrons near the light cone while the experimental facts versus QCD inspired concepts,” Physics traditional quark models refer to the rest frame. In this case Reports,vol.454,pp.1–202,2007. there is no contradiction between diferent approaches. [6] M. Shifman and A. Vainshtein, “Highly excited mesons, linear Te constituent �� pairs are not operative degrees of Regge trajectories, and the pattern of the chiral symmetry freedom noticeably below 1 GeV.It is interesting to note, how- realization,” Physical Review D,vol.77,ArticleID034002,2008. ever, that the spectroscopy in this sector can be constructed [7] S. S. Afonin, “Light meson spectrum and classical symmetries following essentially the same relation (3) if the pseudoscalar of QCD,” Te European Physical Journal A,vol.29,pp.327–335, 2006. pair �0 is replaced by the pseudo-goldstone bosons �, �, � [8] S. S. Afonin, “Experimental indication on chiral symmetry [52]. restoration in meson spectrum,” Physics Letters B,vol.639,no. 3-4,pp.258–262,2006. 5. Summary [9] S. S. 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