The origin of light 0 + scalar resonances
aa r X i v : . [ h e p - ph ] F e b IPPP/10/56, DCPT/10/112, SHEP 10-30, USTC-ICTS-10-13
Origin of light + scalar resonances Zhi-Yong Zhou ∗ Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China andInstitute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, United Kingdom
Zhiguang Xiao † School of Physics and Astronomy, University of Southampton Highfield, Southampton, SO17 1BJ, United Kingdom andInterdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: November 27, 2018)We demonstrate how most of the light J P = 0 + spectrum below 2 . SU (3) breaking effects into account. These resonances appear as poles in thecomplex s plane in a unified picture as q ¯ q states strongly dressed by hadron loops. Through the large N c analysis, these resonances are found to naturally separate into two kinds: σ, κ, f (980) , a (980)are dynamically generated and run away from the real axis as N c increases, while the others movetowards the q ¯ q seeds. In this picture, the line shape of a (980) is produced by a broad pole belowthe K ¯ K threshold, and exhibits characteristics similar to the σ and κ . PACS numbers: 12.39.Ki, 11.55.Bq, 13.75.Lb, 14.40.Be
I. INTRODUCTION
The enigmatic spectrum of light J P = 0 + scalar res-onances are of great interest for its importance in un-derstanding chiral symmetry breaking and confinementin QCD. Despite many theoretical efforts, the currentunderstanding of the microscopic structures of these res-onances is in a well-known unclear situation as summa-rized in Particle Data Group (PDG) [1]: q ¯ q models [2],the unitarized meson model [3], a tetraquark model withand without q ¯ q mixing [4, 5], the J¨ulich meson exchangemodel [6], the unitarized σ model [7], glueball [8] or usingthe inverse amplitude method (IAM) [9], NJL model [10]and lattice simulations [11], and so on. Most of thesestudies focus on the lowest putative nonet or explain thelighter and heavier resonances in different ways. In thepresent paper, we show that, all the light scalar spec-trum below 2 . q ¯ q seedsstrongly dressed by hadron loops. The picture bringsmore insights on the origin of the resonances, which aregenerated as the poles of the S matrix and have no one-to-one correspondence with the nonet in the Lagrangian.At the weak coupling limit as N c increases, σ , κ , a (980),and f (980) move away from the real axis on the com-plex energy plane, whereas all the other heavier J P = 0 + resonances move to the bare seeds. This reveals the dif-ferences between the lighter mesons and heavier states.We use the unitarized quark model (UQM) [2] pro-posed by T¨ornqvist, which played a pioneering role inthe resurrection of the σ meson. The merit of this modelis that it naturally respects the unitarity of the S-matrix ∗ [email protected] † [email protected] but also incorporates some dynamics at the same time.Besides, the Adler zeros [12], as the constraints from chi-ral symmetry, can also be easily implemented. Neverthe-less, the κ resonance was not found in his explicit analysisof experimental data, and those resonances with highermasses than 1 . κ resonance can re-ally be found in this picture. Moreover, most resonancesin I = 0, I = 1 /
2, and I = 1 channels can find theircorresponding poles on the complex plane.The paper is organized as follows: In Sec.II, we brieflyintroduce the basic scheme of UQM and the three non-trivial improvements we make to this model. Our numer-ical results are elaborately discussed in Sec.III. SectionIV is devoted to a further study on the characteristics ofthese resonances based on the large N c technique. Sec-tion V summarizes our main results. II. THE THEORETICAL SCHEMEA. Unitarized quark model
The unitarized quark model begins by assuming thatthere are q ¯ q bare bound states generated in QCD andthey are coupled with the pseudoscalar mesons. Themain idea is to take into account the hadron loop dress-ing effect in the propagators of the bare q ¯ q states [2, 13].The bare propagator of a q ¯ q bound state is P = 1 m − s , (1)where m is the bare mass. For example, m is 2 ˆ m for u ¯ u or d ¯ d , ˆ m + ∆ m for u ¯ s , and 2 ˆ m + 2∆ m for s ¯ s state,respectively. The vacuum polarization function, Π( s ),which represents all the possible two pseudoscalar mesonloops, will contribute to the full propagator as P = 1 m − s + Π( s ) . (2)As an analytic function with a right-hand cut, its realand imaginary parts are related by a dispersive integralReΠ( s ) = 1 π P Z ∞ s th d z ImΠ( z ) / ( z − s ) , (3)where ImΠ( s ) = − X i G i ( s ) = − X i g i k i ( s ) √ s F i ( s ) θ ( s − s th,i ) , (4)where the general coupling function G i ( s ) includes thecoupling constants g i ’s, the phase space factor k i ( s ) / √ s ,and a Gaussian form factor F i ( s ) = exp [ − k i ( s ) / k ]. k i ( s ) is the i th channel c.m. momentum with k i ( s ) = q λ ( s, m A i , m B i ) / s and the θ ( s − s th,i ) is a unit stepfunction.If there exists more than one bare state in the i → j channel, the partial-wave amplitude can be representedin a more general matrix form: T ij = X α,β G iα P αβ G ∗ jβ , { P − } αβ ( s ) = ( m ,α − s ) δ αβ + Π αβ ( s ) , ImΠ αβ ( s ) = − X i G iα ( s ) G ∗ βi ( s )= − X i g αi g βi k i ( s ) √ s F i ( s ) θ ( s − s th,i ) , (5)where ReΠ αβ is determined by a similar dispersion inte-gral of ImΠ αβ as Eq.(3). The off-diagonal terms of Π αβ produce the mixing between different bare states coupledwith the same intermediate states.The Adler zeros are incorporated into the UQM modelin a direct and easily operated phenomenological way [2]: G αi ( s ) G βi ( s ) → γ αi γ βi ( s − z A,i ) F i ( s ) k i ( s ) √ s θ ( s − s th,i ) , (6)where the z A,i ’s denote the Adler zeros, and γ βi are di-mensionless coupling constants . B. The Adler zeros in the Chiral PerturbationTheory
Normally, the T matrix also contains left-hand cuts.Because the Adler zero is usually located nearer to thephysical threshold than the left-hand cut, it is natural to expect that it plays a more important role than theleft-hand cut in determining the scattering amplitudesalong the right hand cut. Since such zeros reflect theconstraints of the chiral symmetry, in the I = 0 , / I = 1 chan-nel, we also fix the Adler zero according to ChPT butthere are some subtleties which will be addressed later.Being within the convergence radius of the chiral expan-sion, these Adler zeros should be reliably determined byChPT and, hence, is a reasonable starting point for aphenomenological study.After partial-wave expansion, one obtains the Adlerzero of I = 0 ππ S wave at about m π / s ≃ . m π by includingthe contribution up to two-loop SU (2) ChPT[15]. Sim-ilarly, in the Kπ scattering, the SU (3) ChPT to O ( p )gives S-wave amplitude of I = 1 / T l =0 I =1 / ( s ) = − (cid:0) m K − m π (cid:1) − (cid:0) m K + m π (cid:1) s + 5 s πf K f π s , (7)which has two zeros located at ( m K + m π ± √ m K − m K m π + 4 m π ) in the unphysical region.One is on the negative real axis inside the circular cut,and the other is on the positive axis between the circularcut and the Kπ threshold. With the O ( p ) contribution,the left zero on the negative axis moves to the complex s plane at about s = 0 . ± . i GeV and the right oneon the positive axis will also be slightly shifted (at about s = 0 . ). As for the πη scattering, at leading or-der O ( p ), ChPT recovers the current-algebra result[16]: T (2) πη ( s, t, u ) = m π f π , (8)which contains no zero on the real s axis after the partial-wave projection. Including the higher order contributionis not helpful to obtain a real-valued Adler zero. A pair ofzeros, at about 0 . ± i .
178 GeV , can be found whenthe O ( p ) terms are taken into account using the low-energy constants from Ref.[17]. Since there is no accuratedata of the πη scattering, we could choose a zero pointon the real axis to simulate them as mentioned later. C. The scalar-pseudoscalar-pseudoscalar coupling
The coupling of pseudoscalar-pseudoscalar to the 0 + states could be described by effective interaction terms inthe Lagrangian: L SP P = αT r [ SP P ] + βT r [ S ] T r [ P P ] + γT r [ S ] T r [ P ] T r [ P ]. The first term has an SU (3)-symmetric quarkonium coupling as used in many phe-nomenological models. We also include the Okubo-Zweig-Iizuka violation terms in the last two terms in ageneral way. Moreover, the decay of the quarkonium into TABLE I. The effective scalar quarkonium coupling to pseu-doscalar mesons up to a global constant.I Coupling coefficient0( n ¯ n ) ππ −√ α − √ βK ¯ K − ρα − βηη cos φ α + 2 β + 2(1 + cos φ − √ φ ) γηη ′ sin2 φ √ α + (4cos2 φ + √ φ ) γη ′ η ′ sin φ α + 2 β + 2(1 + sin φ + √ φ ) γ s ¯ s ) ππ −√ βK ¯ K −√ α − √ βηη √ ρ sin φ α + β + (1 + cos φ − √ φ ) γ ) ηη ′ − ρ sin2 φα + 2 √ φγ + sin2 φγη ′ η ′ √ ρ cos φ α + β + (1 + sin φ + √ φ ) γ )1 ηπ √ φαK ¯ K − ραη ′ π √ φα Kπ − p / αKη ( cos φ √ − ρ sin φ ) αKη ′ ( sin φ √ + ρ cos φ ) α a pair of mesons Q ¯ Q → M ( Q ¯ q i ) M ( q i ¯ Q ) involves the cre-ation of a q i ¯ q i pair from the vacuum. The ratio of thecreation rates of s ¯ s and u ¯ u or d ¯ d from the vacuum is usu-ally defined as ρ = h | V | s ¯ s i / h | V | u ¯ u i , representing thebreaking of SU (3) symmetry [18]. SU (3) breaking ef-fects have proved to be important, so we allow for thesein our version of the UQM. To be explicit about our de-scription of the coupling of quarkonium to mesons, weexpress the scalar and pseudoscalar 3 × q ¯ q configurations as S = √ a + √ f n a + κ + a − − √ a + √ f n κ κ − ¯ κ f s , (9) P = √ π + η √ π + K + π − −√ π + η √ K K − ¯ K − q η + 1 √ η , (10)where f n = n ¯ n ≡ ( u ¯ u + d ¯ d ) / √ f s = s ¯ s . Thephysical states, η and η ′ , are conventionally defined as η = cos φ | n ¯ n i − sin φ | s ¯ s i , η ′ = sin φ | n ¯ n i + cos φ | s ¯ s i , with φ = tan − √ θ P , where θ P = − . ◦ being the pseu-doscalar octet-singlet mixing angle [1]. Thus, by stan-dard derivation, a general form of effective coupling con-stants between the scalar quarkonium and pseudoscalarpair is obtained, as shown in Table I. s th, − + s th, ++ s th, + + ++ − + + − + − ++ − −− + + − + −− − + − − −−−− + FIG. 1. The right-hand cuts and their signatures in a three-channel case.
D. The analyticity of S matrix By definition, a resonance is specified as a pole of the S matrix analytically continued to the complex s plane.Extracting the poles of the partial-wave amplitude of i → j process described in the UQM is actually to findthe zeros of the determinant of the inverse propagator.Sometimes, it can be obtained in some other equivalentway. For example, in a two-channel occasion the analyt-ically continued S matrices on different Riemann sheetscould be written down using those on the first sheet[19]: S II = S iS S iS S detSS ! , S III = S detS − S detS − S detS S detS ! ,S IV = detSS − iS S − iS S S ! , (11)which implies that a pole on the second Riemann sheetis just located at a zero point of S on the first sheet,a third-sheet pole at a zero point of detS , and a fourth-sheet pole at a zero of S , respectively.In the literature it is common to define a Breit-Wignermass of a resonance as the solution of m BW = m B + Re Π( m BW ). This is a good approximation for narrowresonances and is commonly used in experimental anal-ysis. However, if the propagator is strongly dressed byhadron loops where Im Π( s ) is large, the mass and widthare no longer suitably determined by the Breit-Wignerform, but should only be defined by the pole position ofthe S matrix, i.e., the solution of m B − s p + Π( s p ) = 0with s p = ( M − i Γ / . Because of the analyticity of the S matrix, the determinant of inverse propagator vanishesonly on unphysical Riemann sheets.The general character of the poles on different Rie-mann sheets has been discussed widely in the literature,(see, for example, [20, 21]). Every physical cut will dou-ble the Riemann sheets in the analytical continuation, sothere are 2 n Riemann sheets in a process with n coupledchannels, as shown in Fig.1. The physical sheet is definedas the sheet where all the c.m. momenta are positive onthe physical cuts, denoted as (+ + + · · · +) signature. Inthe same fashion, the ( n + 1) th sheet ( n ≤ N ), attachedto the physical sheet between s th,n and s th,n +1 along thephysical cut, is denoted by ( − · · · − + + · · · ) with the n TABLE II. The results of the fit parameters with a Gaussianform factor. m , m + m s , and m + 2 m s are the bare massesof n ¯ n , n ¯ s , and s ¯ s respectively. α : β : γ . ± . : ( − . ± . ) : 0 . ± . ρ . ± . k (GeV) 0 . ± . m (GeV) 1 . ± . m s (GeV) 0 . ± . consecutive “ − ” signs before the other “+” signs. A res-onance is represented by a pair of conjugate poles on theRiemann sheet, as required by the real analyticity. Themicro-causality tells us the first Riemann sheet is free ofcomplex-valued poles, and the resonances are representedby those poles on unphysical sheets. The resonance be-havior is only significantly influenced by those nearbypoles, and that is why only those closest poles to the ex-periment region could be extracted from the experimentdata in a phenomenological study. Those poles on theother n > N sheets, which are reached indirectly, makeless contribution and are thus harder to determine. III. NUMERICAL ANALYSIS
Now, we apply the partial-wave formulation [2] withour new ingredients to study I = 1 / Kπ , I = 0 ππ ,and I = 1 πη S-wave scattering. The main purpose ofthis paper is not to make an exhaustive fit, so the onlydata used in the combined fit are: (1) the I = 1 / Kπ scattering amplitude [22, 23], (2) the phase shiftof I = 0 S-wave ππ scattering [24], and (3) the phaseshift of ππ → K ¯ K [25] below 1 . χ /d.o.f. ≃ .
8. The central values andthe statistical errors of the seven parameters are listed inTable II. The good agreement between our theoreticalresults and experimental data can be seen in Fig. 2, eventhough some of these data have not been used in thefit. The parameter values in Table II are all in realisticranges. The bare masses of q ¯ q states are slightly larger,but not in conflict with NJL modelings [10]. The SU (3)breaking effect parameter is also consistent with the valuein the literature [18]. A general comparison of the massesand widths of the resonances from our results and thevalues from the PDG table is presented in Fig. 3 whichwill be discussed in detail.The I = 1 / Kπ S-wave scattering provides an idealillustration and the best testing ground because of itslarge threshold spans and clean experimental informa-tion. With the parameters of Table II, there is only onesolution to s = ( m + m s ) + Re Π( s ) for real s at about s / = 1 .
33 GeV, which is the Breit-Wigner-like massmentioned previously. However, there are three poles of d ( deg ) s (GeV) Cern-Munich
Wetzel et. al. Martin et. al. Kaminski et. al. h s (GeV) P ha s e [ T ( pp - > KK ) ] ( deg ) s (GeV) Cohen et al. Etkin et al. E v en t s / ( . G e V / c ) s (GeV) Etkin et. al. Cohen et. al. Constants*|T( pp ->KK)| V a l ue s (GeV) Re part of I=1/2 T( p K-> p K) Im part of I=1/2 T( p K-> p K) E v en t s / . G e V s (GeV) WA76 Constant*|T( ph -> ph )| FIG. 2. The left three figures show the fit quality. The rightthree ones are predictions. Compared data are from [25, 26]. W i d t h Mass k K (1430) K (1950) W i d t h Mass s f (980) f (1370) f (1500) f (1710) f (2020) W i d t h Mass a (980) a (1450) a (2020) FIG. 3. The filled symbols represent the resonances’ massesand widths from the PDG table, and the empty ones representthe pole masses and widths we obtained. The reason of thediscrepancy between the value sets has been discussed in thetext. the S matrix found near the physical region at √ s II = 0 . ± . − i . ± . , √ s III = 1 . ± . − i . ± . , √ s IV = 1 . ± . − i . ± . , (12)where the superscript denotes the number of the sheetand the units are in GeV. Simply comparing these poleswith the tables of Particle Data Group [1], a good agree-ment in quality is instantly found (see Fig. 3). The lowestsecond-sheet pole is just the κ resonance and consistentwith the values determined by those model-independentmethods [27]. The third-sheet pole corresponds to the K ∗ (1430) and a second-sheet “shadow” pole (due to theweak coupling constant of the Kη channel in our resultand also found in [21]) is also found at almost the samelocation. Although the fit is only carried out with thedata below 1 . SU (3) couplings and unitarity constraints, which cor-responds to the higher K ∗ (1950) resonance. The widthof K ∗ (1950) is larger than its PDG value, but is quali-tatively acceptable compared with the average value cal-culated from both solutions A and B of the original dataanalysis [22, 28].The poles in I = 0 ππ S wave are inevitably morecomplicated than those in the Kπ S wave because of themixing of n ¯ n and s ¯ s states. We find √ s II = 0 . ± . − i . ± . , √ s II = 0 . ± . − i . ± . , √ s IV = 1 . ± . − i . ± . , √ s V = 1 . ± . − i . ± . , √ s V I = 1 . ± . − i . ± . , √ s V I = 2 . ± . − i . ± . . (13)All these poles could be assigned to those light resonancesof I G ( J P C ) = 0 + (0 ++ ) listed in the PDG table, exceptfor the f (1710). The position of the σ pole is in agree-ment with the results of model-independent analysis [29],while the f (980) is a narrow pole below the K ¯ K thresh-old. The f (1370) is a fourth-sheet pole and its position iswithin the uncertainty of the PDG value. The pole massis consistent with that preferred by the Belle Collabora-tion from γγ → π π , at 1.47GeV [30]. Here, the res-onance shape of f (1500) is generated by the fifth-sheetpole and ηη ′ threshold together, as found in Ref. [31].This may be the reason why the width of the pole ismuch wider than the PDG value. The 4 π and ρρ thresh-olds turn out to be increasingly important beyond about1 . SV V and
V P P interactions to be taken intoaccount. This would introduce many new parameters andthis case is beyond the scope of this paper. Not incor-porating the
SV V interaction might explain why thereis no f (1710) pole in this picture. The other possibil-ity is that the main ingredient of f (1710) could be thelowest scalar glueball, as preferred by recent quenchedLattice calculations [32]. This would add a narrow res-onance structure with its own hadron cloud [33]. Thetwo sixth-sheet poles are assigned to the f (2020) andthe f (2330), respectively.As for the I = 1 πη scattering, owing to the poornessof data, this channel is not included in the fit and sothe plot in the lower right corner of Fig. 1 is wholly aprediction. As we have mentioned previously, no real-value Adler zero is found up to O ( p ) ChPT amplitudein this partial wave. We simply set z A = 0 .
078 GeV in TABLE III. The results of the fit parameters with anotherform factor. α : β : γ . ± . : ( − . ± . ) : 0 . ± . ρ . ± . m (GeV) 1 . ± . m s (GeV) 0 . ± . the calculation, which is close to the complex zeros. It isnot difficult to exhibit the a (980) line shape below the K ¯ K threshold, as shown in Fig. 2. The pole positions arenot sensitive to small deviations from the value of Adlerzero we choose: √ s II = 0 . ± . − i . ± . , √ s III = 1 . ± . − i . ± . , √ s IV = 1 . ± . − i . ± . . (14)The second-sheet pole below the K ¯ K threshold is broad,but still produces the a (980) line shape combined withthe threshold effect, as proposed by Flatt´e [34]. Thiseffect was also found by studying the πη amplitude inthe J¨ulich model [35] and implied in the unitarized σ model [7]. The third-sheet pole could represent the a (1450), although it plays a minor role in our picture.There is also a higher a (1830) predicted, which has notbeen widely observed in experiments, but might be re-lated to the a (2020) seen by the Crystal Barrel Collab-oration [36].The use of a Gaussian form factor is the most dras-tic assumption we have made. This has been widelyused in experimental analyses and in many other mod-els. However, an exact representation of the form factor,satisfying unitarity and analyticity and easily applied inphenomenological studies, has not been found. To testthe stability of our results, we also use a different formfactor, [( M + s th,i ) / ( M + s )] ( M is the mass of theresonance) proposed in [37]. The latter form factor isnot better than the Gaussian form factor since it suffersfrom a spacelike pole, but it might provide a reasonablequalitative cross-check, especially for the N c analysis weaddressed later. Incorporating the latter form factor re-moves one parameter, k , so the fit quality is worse and χ /d.o.f. ≃ .
21. The central values of the other sixparameters listed in Table III are different from thosein Table II. Nevertheless, the poles below 1 . √ s II = 0 . ± . − i . ± . , √ s III = 1 . ± . − i . ± . , (15)for the I = 1 / √ s II = 0 . ± . − i . ± . , √ s II = 0 . ± . − i . ± . , √ s IV = 1 . ± . − i . ± . , (16)for the I = 0 channel and √ s II = 0 . ± . − i . ± . , (17)for the I = 1 channel. The corresponding poles beyond1 . . I = 1 data, they movefarther away from the previous values. A posteriori , thismeans the second kind of form factor may not be a goodchoice.
IV. LARGE N c ANALYSIS OF THE POLETRAJECTORIES
The success of describing such a broad range of spec-trum and their decays in a unified and consistent waysuggests that it is a reasonable model to study these res-onances and could be used to gain further insights intotheir nature. The large N c behavior of the pole trajecto-ries serves to shed light on the origin of these resonances.The lowest order of α is 1 / √ N c , β , and γ by a factor of1 /N c and 1 /N c . The bare mass, the location of Adlerzero, and the form factor are of order 1, while the mixingangle φ is of order 1 /N c [38]. Whichever of our two formfactors we use, the poles exhibit similar trajectories as N c increases. Those for the I = 1 / , σ , κ , and a (980)poles move farther away from the real axis. In contrast,the K ∗ (1430) and K ∗ (1950) become narrower and movetowards the n ¯ s bound state. Analogously, the a (1450)and a (1830) move to the ( u ¯ u − d ¯ d ) / √ f poles other than σ and f (980) move towards eitherthe n ¯ n or s ¯ s bare seeds. At N c = 3, if the couplingto the ππ channel is switched off, the f (980) will movedown below the K ¯ K threshold and form a bound state.This behavior implies that the f (980) is more like a K ¯ K molecule state. However, when N c increases, it exhibitsa peculiar trajectory: it moves to the real axis rapidlyand then crosses the cut onto the (+ − + + +) sheet, andthen moves away from the real axis as the σ pole does,as seen in Fig.4. If the coupling to the lowest thresholdsare increased, respectively, by hand when N c = 3, the σ , κ , and a (980) will move to the real axis and become vir-tual bound states different from the seeds either. Whilethe coupling becomes strong enough, the virtual boundstates will move onto the first sheet and become boundstates. It is worth mentioning that, in using the IAMto unitarize ChPT [39], Pel´aez has observed similar polebehaviors of σ and κ in some parameter region. Usingthe Pad´e technique to unitarize ChPT amplitudes, thesimilar σ pole trajectory is also found [40]. The pole be-haviors of f (980) and a (980), as we pointed out herefor the first time, may explain the strange behavior ofthe line shape in large N c shown in [39].So, the general N c behavior separates the poles intotwo types: σ , κ , a (980), and f (980) are the first type (or k K (1430) K (1950) Re[s ] (N C =3) (N C =3)(N C =3) I m [ s / ] (N C =3)(N C =3) I m [ s / ] Re[s ] s f (980) on (-++++) sheet f (980) on (+-+++) sheet FIG. 4. Left: I=1/2 poles’ trajectories; Right: the pole tra-jectories of σ and f (980). the unconventional type) of resonances, like bound statesof mesons, which are dynamically generated by the pseu-doscalar interactions. This may be the reason why theycould be described by the tetraquark model. All the otherresonances except the glueball candidate, as the secondtype (or the conventional type) of resonances, are directlygenerated from q ¯ q seeds by renormalization effect, whichindicates that they all belong to the same bare q ¯ q nonet.As the interactions are turned on and different channelsare open, the bare seeds are copied to different Riemannsheets and get renormalized by the hadron clouds in var-ious ways [21]. Some of them run too far away from thephysical region to be detectable. In this picture, there isno need to distinguish parts of them to be a nonet. V. SUMMARY
In conclusion, this paper demonstrates that the wholelow-energy scalar spectrum below 2.0 GeV, except fora possible glueball f (1710), could be described in oneconsistent picture, with the bare “ q ¯ q seeds” dressed bythe hadron loops. All the resonances are dynamicallygenerated by the same mechanism, and there is no di-rect correspondence between the poles and the originalnonet in the Lagrangian. In a large N c analysis of thispicture, the pole trajectories exhibit a general behaviorwhich agrees with other models. In particular, the σ , κ , f (980), a (980) resonances, though running away fromthe real axis when N c is larger, are also generated in thismodel, which means this large N c behavior does not con-flict with the q ¯ q dressed by the hadron loop picture. Theyare produced by large hadron loop effects and this mayalso imply their large four-quark components. Thus, inthis paper, we present that the usual speculation in par-ticle physicist community, that the lighter scalars behavelike the tetraquark states and the heavier scalars do asthe q ¯ q states, could be actually realized in such a coherentpicture of improved UQM model.We also show how the line shape of a (980) is possiblygenerated by a deep pole, like the σ or κ , encountering the K ¯ K threshold. This whole treatment could be extendedto other spectra, e.g., the charmonium states [41], andprovide theoretical suggestions for further experimentalinvestigation. ACKNOWLEDGMENTS
We are grateful to Mike Pennington and Han-qing Zheng for instructive discussions and thank Yan-rui Liu for helpful discussion. Z. X. thanks the Science and Technology Facilities Council in the United King-dom for financial support. This work is partly supportedby China Scholarship Council and China National Nat-ural Science Foundation under Contracts No.10705009,No.10647113, and No.10875001. [1] C. Amsler et al. (Particle Data Group),Phys. Lett.
B667 , 1 (2008).[2] N. A. Tornqvist, Z. Phys.
C68 , 647 (1995); N. A. Torn-qvist and M. Roos, Phys. Rev. Lett. , 1575 (1996);P. Geiger and N. Isgur, Phys. Rev. D47 , 5050 (1993);M. Boglione and M. R. Pennington,Phys. Rev.
D65 , 114010 (2002).[3] E. van Beveren et al. , Z. Phys.
C30 , 615 (1986).[4] R. L. Jaffe, Phys. Rev.
D15 , 267 (1977); L. Maiani et al. , Phys. Rev. Lett. , 212002 (2004); G. ’t Hooft et al. , Phys. Lett. B662 , 424 (2008).[5] A. H. Fariborz, R. Jora, andJ. Schechter, Phys. Rev.
D77 , 094004 (2008),arXiv:0801.2552 [hep-ph].[6] D. Lohse et al. , Nucl. Phys.
A516 , 513 (1990).[7] D. Black et al. , Phys. Rev.
D64 , 014031 (2001).[8] P. Minkowski and W. Ochs,Eur. Phys. J. C9 , 283 (1999).[9] J. A. Oller, E. Oset, and J. R. Pelaez,Phys. Rev. D59 , 074001 (1999).[10] M. X. Su, L. Y. Xiao, and H. Q. Zheng,Nucl. Phys.
A792 , 288 (2007).[11] M. G. Alford and R. L. Jaffe,Nucl. Phys.
B578 , 367 (2000), arXiv:hep-lat/0001023.[12] S. L. Adler, Phys. Rev. , B1638 (1965).[13] V. Weisskopf and E. P. Wigner, Z. Phys. , 54 (1930).[14] J. Gasser and H. Leutwyler, Ann. Phys. , 142 (1984).[15] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, and M. E.Sainio, Phys. Lett. B374 , 210 (1996).[16] V. Bernard, N. Kaiser, and U.-G. Meissner,Phys. Rev.
D44 , 3698 (1991).[17] J. Gasser and H. Leutwyler,Nucl. Phys.
B250 , 465 (1985).[18] C. Amsler and F. E. Close, Phys. Rev.
D53 , 295 (1996).[19] Z. Xiao and H.-Q. Zheng,Commun. Theor. Phys. , 685 (2007).[20] R. G. Newton, J. Math. Phys. , 188 (1961).[21] R. J. Eden and J. R. Taylor,Phys. Rev. , B1575 (1964).[22] D. Aston et al. , Nucl. Phys. B296 , 493 (1988).[23] J. M. Link et al. (FOCUS Collaboration),Phys. Lett.
B653 , 1 (2007).[24] W. Ochs, Ph.D. Thesis, Munich Univ. (1974); G. Grayer et al. , Nucl. Phys.
B75 , 189 (1974).[25] D. H. Cohen et al. , Phys. Rev.
D22 , 2595 (1980);A. Etkin et al. , Phys. Rev.
D25 , 1786 (1982).[26] W. Wetzel et al. , Nucl. Phys.
B115 , 208 (1976);A. D. Martin and E. N. Ozmutlu,Nucl. Phys.
B158 , 520 (1979); R. Kaminski, L. Les-niak, and K. Rybicki, Z. Phys.
C74 , 79 (1997);T. A. Armstrong et al. (WA76 Collaboration),Z. Phys.
C52 , 389 (1991).[27] H. Q. Zheng et al. , Nucl. Phys.
A733 , 235 (2004); Z. Y.Zhou and H. Q. Zheng, Nucl. Phys.
A775 , 212 (2006);S. Descotes-Genon and B. Moussallam,Eur. Phys. J.
C48 , 553 (2006).[28] A. V. Anisovich and A. V. Sarantsev,Phys. Lett.
B413 , 137 (1997); M. Jamin, J. A. Oller,and A. Pich, Nucl. Phys.
B587 , 331 (2000).[29] Z. Y. Zhou et al. , JHEP , 043 (2005); I. Caprini,G. Colangelo, and H. Leutwyler, Phys. Rev. Lett. ,132001 (2006).[30] S. Uehara et al. (Belle Collaboration),Phys. Rev. D78 , 052004 (2008).[31] M. Albaladejo and J. A. Oller,Phys. Rev. Lett. , 252002 (2008).[32] W.-J. Lee and D. Weingarten,Phys. Rev.
D61 , 014015 (1999); C. Liu,Chin. Phys. Lett. , 187 (2001); Y. Chen et al. ,Phys. Rev. D73 , 014516 (2006).[33] M. Boglione and M. R. Pennington,Phys. Rev. Lett. , 1998 (1997).[34] S. M. Flatte, Phys. Lett. B63 , 224 (1976).[35] G. Janssen et al. , Phys. Rev.
D52 , 2690 (1995).[36] A. V. Anisovich et al. (Crystal Barrel Collaboration),Phys. Lett.
B452 , 173 (1999).[37] F. Ravndal, Phys. Rev. D4 , 1466 (1971).[38] G. ’t Hooft, Nucl. Phys. B72 , 461 (1974); E. Witten,Nucl. Phys.
B160 , 57 (1979).[39] J. R. Pelaez, Phys. Rev. Lett. , 102001 (2004).[40] Z. X. Sun, L. Y. Xiao, Z. Xiao, andH. Q. Zheng, Mod. Phys. Lett. A22 , 711 (2007),arXiv:hep-ph/0503195.[41] M. R. Pennington and D. J. Wilson,Phys. Rev.