Axial Resonances in the Open and Hidden Charm Sectors
aa r X i v : . [ h e p - ph ] M a r Axial Resonances in the Open and Hidden CharmSectors
D. Gamermann , E. Oset Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain
May 31, 2018
Abstract A SU (4) flavor symmetrical Lagrangian is constructed for the interaction of thepseudo-scalar mesons with the vector mesons. SU (4) symmetry is broken to SU (3)by suppression of terms in the Lagrangian where the interaction should be drivenby charmed mesons. Chiral symmetry can be restored by setting this new SU (4)symmetry breaking parameters to zero. Unitarization in coupled channels leads to thedynamical generation of resonances. Many known axial resonances can be identifiedincluding the new controversial X (3872) and the structure found recently by Bellearound 3875 MeV in the hidden charm sector. Also new resonances are predicted,some of them with exotic quantum numbers. Recent years have been very exciting for the hadron spectroscopy because of thediscovery of many new and controversial states that do not fit well the interpretationof baryons as qqq states or mesons as q ¯ q states. In particular two charmed resonancesdiscovered by BaBar [1] and confirmed by other experiments [2], [3], [4], the D s (2317)and D s (2460) have animated the debate about non q ¯ q mesons. Also non-strangepartners of these resonances have been observed [5], [6].The predictions for the masses of these states with quark model potentials al-ready existed [7] and turned out to be off by more than 100 MeV. The fact thatthe D s (2317) lies just below the DK threshold and the D s (2460) just below the D ∗ K threshold made many theoreticians speculate that these states could be mesonmolecules [8], [9], [10], [11], [12], [13]. Others support a tetraquark assignment [14],[15], [16], or usual q ¯ q states with more sophisticated quark model potentials or withinQCD sum rules calculations [17], [18], [19], [20] and there is also the possibility ofadmixture between these configurations [21], [22], [23].In the hidden-charm sector also new controversial resonances have been found. Inparticular the X (3872), observed in four different experiments [24], [25], [26], [27], has ttracted much attention. The narrow width of this state makes its interpretationas a usual charmonium c ¯ c state very difficult. For this resonance too, many exotictheoretical interpretations have been investigated such as tetraquarks, hybrids andmolecules [28], [29], [30], [31], [32]. For a good review on heavy mesons one can referto [33].In this work unitarization in coupled channels is used to explore the pseudo-scalarmeson interaction with vector mesons. In the works of Kolomeitsev [9] and Guo [11] asimilar approach has been done using a Lagrangian based on heavy quark chiral sym-metry that allowed the investigation only of the open-charm sector and constrainedthe interaction for only light pseudo-scalars with heavy vector mesons. For our phe-nomenological model we construct a Lagrangian based on SU (4) flavor symmetryand this symmetry is broken to SU (3) by suppressing exchanges of heavy mesonsin the implicit Weinberg-Tomozawa term. Chiral symmetry can be restored fromour model by setting the SU (4) symmetry breaking parameters to zero. This newLagrangian, based on the ideas of a previous paper [34], includes also the possibilityto investigate the hidden-charm sector and the interaction of heavy pseudo-scalarswith light vector mesons, which enriches the spectrum of axial resonances generated.The paper is organized as follows: in the next section the construction of theLagrangian is explained in detail and also the mathematical framework for solvingthe scattering equations in a unitarized approach is presented. Section 3 is devotedto the presentation and discussion of the results and section 4 contemplates overviewand conclusions. We will start by constructing two fields, one for the SU (4) 15-plet of pseudo-scalarsand another one for the 15-plet of vector mesons:Φ = X i =1 ϕ i √ λ i == π √ + η √ + η c √ π + K + ¯ D π − − π √ + η √ + η c √ K D − K − ¯ K − η √ + η c √ D − s D D + D + s − η c √ (1) V µ = X i =1 v µi √ λ i = ρ µ √ + ω µ √ + J/ψ µ √ ρ + µ K ∗ + µ ¯ D ∗ µ ρ ∗− µ − ρ µ √ + ω µ √ + J/ψ µ √ K ∗ µ D ∗− µ K ∗− µ ¯ K ∗ µ − ω µ √ + J/ψ µ √ D ∗− sµ D ∗ µ D ∗ + µ D ∗ + sµ − J/ψ µ √ . (2)Now for each one of these fields a vector current is build: J µ = ( ∂ µ Φ)Φ − Φ ∂ µ Φ (3) J µ = ( ∂ µ V ν ) V ν − V ν ∂ µ V ν . (4)The Lagrangian is then constructed by connecting the two currents: L = − f T r ( J µ J µ ) . (5)Note that the ω appearing in eq. (2) is not the physical ω but ω . The additionof a singlet state, which is diagonal and proportional to the identity matrix in therepresentation of eq. (2) does not give any contribution to J µ in eq. (4) and hencedoes not modify the Lagrangian in eq. (5). However, there is a caveat since the singletand octet mix strongly to give the ω and φ states which have different masses. Inthe results section we shall come back to this problem and will discuss the effect ofthis mixing.Next step is to break SU (4) symmetry, this will be done by suppressing exchangesof heavy mesons. To identify these terms, one should first decompose each one of thefields into its SU (3) components:Φ = φ + √ φ ˆ1 φ φ ¯3 − √ φ ! (6) V µ = V µ + √ V µ ˆ1 V µ V ¯3 µ − √ V µ ! . (7)The ˆ1 is the 3x3 identity matrix and the fields φ i and V iµ contain the meson fieldsfor each i -plet of SU (3) into which the 15-plet of SU (4) decomposes: φ = π √ + η √ π + K + π − − π √ + η √ K K − ¯ K − η √ = ¯ D D − D − s φ ¯3 = (cid:16) D D + D + s (cid:17) φ = η c V µ = ρ µ √ + ω µ √ ρ + µ K ∗ + µ ρ − µ − ρ µ √ + ω µ √ K ∗ µ K ∗− µ ¯ K ∗ µ − ω µ √ V µ = ¯ D ∗ µ D ∗− µ D ∗− sµ V ¯3 µ = (cid:16) D ∗ µ D ∗ + µ D ∗ + sµ (cid:17) V µ = J/ψ µ In terms of the SU (3) fields the Lagrangian reads: L = − f (cid:16) T r (cid:18) J µ J µ + J µ J µ + J µ J µ + J µ J µ + J µ J µ ¯38 +2 √ J µ J µ ¯31 + J µ J µ ¯38 ) + 43 J µ J µ ¯31 (cid:19) + J ¯33 µ J µ ¯33 + J ¯38 µ J µ +2 √ J ¯38 µ J µ + J ¯31 µ J µ ) + 43 J ¯31 µ J µ (cid:17) . (8)In eq.(8) the currents are defined as: J ijµ = ( ∂ µ φ i ) φ j − φ i ∂ µ φ j (9) J ijµ = ( ∂ µ V iν ) V νj − V iν ∂ µ V νj . (10)The interaction in the Lagrangian of eq. (8) is usually visualized, in the vector-meson dominance picture, as a t-channel exchange of a vector meson in betweenthe meson pairs. Since this Lagrangian is SU (4) flavor symmetric, it assumes equalmasses for the virtual vector meson exchanged in between the meson pairs. In orderto break SU (4) symmetry we will suppress the terms in this Lagrangian where thevirtual vector meson exchanged is a heavy one. If the transferred momentum inbetween the meson pairs is neglected, the propagator of the exchanged meson isproportional to the inverse of its squared mass ( m V ). Considering the terms where alight vector mesons is exchanged of the order of magnitude of the unit, the suppressedterms should be thought to be of the order (cid:16) m L m H (cid:17) where m L and m H are scales ofthe order of magnitude of the light and heavy vector mesons masses respectively.Terms in the Lagrangian where the two currents of eqs. (9) and (10) have ex-plicit charm quantum number should be suppressed since these currents can only e connected through the exchange of a charmed, and hence heavy, vector meson.There are still two terms in the Lagrangian where a hidden-charm meson can beexchanged in between the meson pairs. These are the terms involving only mesonsbelonging to the SU (3) triplet and anti-triplet. In this case the interaction is drivensimultaneously by light and heavy vector mesons and in order to correctly suppressthese terms one should isolate each one of these contributions and suppress only theone coming from the heavy vector meson ( J/ψ ). These contributions have alreadybeen calculated in a previous paper [34] and they can be either and for lightand heavy vector mesons respectively, if two equal currents are connected, or − and for light and heavy vector mesons respectively, if two different currents areconnected. So, following the steps in [34], the corrected Lagrangian, accounting forthe masses of the heavy vector mesons, reads as: L = − f (cid:16) T r (cid:18) J µ J µ + J µ J µ + J µ J µ + J µ J µ + γJ µ J µ ¯38 +2 γ √ J µ J µ ¯31 + J µ J µ ¯38 ) + 4 γ J µ J µ ¯31 (cid:19) + ψJ ¯33 µ J µ ¯33 + γJ ¯38 µ J µ + 2 γ √ J ¯38 µ J µ + J ¯31 µ J µ ) + 4 γ J ¯31 µ J µ (cid:17) , (11)with γ = (cid:16) m L m H (cid:17) and ψ = − + (cid:16) m L m ′ H (cid:17) .The parameters m L , m H and m ′ H should be chosen of the order of magnitude of alight vector-meson mass, charmed vector-meson mass and the J/ψ mass, respectively.The masses of the light vector mesons vary in between 770 MeV, for the ρ mesonand 892 MeV for the K ∗ mass, the charmed vector-mesons have masses 2008 MeVand 2112 MeV ( D ∗ and D ∗ s , respectively) and the J/ψ mass is approximately 3097MeV. We will chose m L =800 MeV, m H =2050 MeV and m ′ H =3 GeV. Changing theseparameters over the whole allowed physical range (770 MeV ≤ m L ≤
892 MeV and2008 MeV ≤ m H ≤ f asthe coefficient in the Lagrangian of eq. (5).From the Lagrangian in eq. (11) one gets the transition amplitudes between aninitial and a final state: M Cij ( s, t, u ) = − ξ Cij f ( s − u ) ǫ.ǫ ′ . (12)The super-index C refers to the charge basis, and the labels i and j to the initialand final channels while s , t and u are the usual Mandelstam variables. In appendixA we give tables for the coefficients ξ I in an isospin basis. hese amplitudes will first be projected in s-wave: V Iij ( s ) = 12 Z − d ( cosθ ) M Iij (cid:18) s, t ( s, cosθ ) , u ( s, cosθ ) (cid:19) (13)This potential will then be used as kernel in a Bethe-Salpeter equation, in anon-shell formalism [37], [38], [39], [40]. In this way the unitary T-matrix assumes theform [36]: T = − (ˆ1 + V ˆ G ) − V −→ ǫ . −→ ǫ ′ (14)In this equation ˆ G is a diagonal matrix with each element given by:ˆ G l = G l p M ! (15) G l = 116 π (cid:18) α i + Log m l µ + M l − m l + s s Log M l m l + p √ s (cid:16) Log s − M l + m l + 2 p √ s − s + M l − m l + 2 p √ s + Log s + M l − m l + 2 p √ s − s − M l + m l + 2 p √ s (cid:17)(cid:19) (16)in the above equations p is the three-momentum in the center of mass frame of thetwo mesons in channel l , while M l and m l are the masses of the vector and pseudo-scalar mesons respectively, α is the subtraction constant, which should be fitted asa free parameter and µ is a cut-off scale which we will set to 1.5 GeV. Note that α and µ are not independent, this justifies setting one to a fixed value and adjustingjust the other one to data.The three-momentum p is calculated with: p = p ( s − ( m l + M l ) )( s − ( m l − M l ) )2 √ s (17)Unitarity is ensured by the imaginary part of the loop function of equation (16): Im ( G l ) = − p π √ s (18)When looking for poles in the complex plane one should be careful because ofthe cuts of the loop function beyond each threshold. Bound states appear as polesover the real axis and below threshold in the first Riemann sheet. Resonances showthemselves as poles above threshold and in the second Riemann sheet of the channelswhich are open.Over the real axis the discontinuity of the loop function is known to be twotimes its imaginary part [41] so, knowing the value of the imaginary part of the loopfunction over the axis, eq. (18), one can do a proper analytic continuation of it forthe whole complex plane: IIl = G Il + i p π √ s , Im ( p ) > G II and G I refer to the loop function in the second and first Riemann sheets, respec-tively.Until now our formalism worked only with stable particles, but in some cases onehas a ρ or a K ∗ meson in the coupled channels, and these particles have relativelylarge widths. The consideration of the mass distributions of these particles can berelevant whenever thresholds are open thanks to this mass distribution.In order to take this into account we follow the procedure of [42] and convolutethe loop function with the spectral function of the particle, hence, using a new loopfunction:˜ G ( √ s, m, M R ) = 1 N Z ( M R +2Γ R ) ( M R − R ) d ˜ M (cid:18) − π (cid:19) Im M − M R + iM R Γ R . ˆ G ( √ s, m, ˜ M ) (20) N = Z ( M R +2Γ R ) ( M R − R ) d ˜ M (cid:18) − π (cid:19) Im M − M R + iM R Γ R (21)In the next section we will comment further on this issue and present results, forthe heavy resonances, by taking into account the finite width of the ρ and K ∗ vectormesons in the few cases where the generated resonances have important coupling tochannels involving these mesons and a mass close to the threshold of these channels. The 15-plet of SU (4) breaks down into 4 multiplets of SU (3):15 −→ ⊕ ⊕ ¯3 ⊕ . (22)Knowing this, one can study the SU (3) structure of the interaction betweenpseudo-scalar and vector mesons. Table 1 shows the SU (3) decomposition of theinteraction. The irreducible representations (irreps) marked with a ∗ refer to thevector meson multiplet.Since now one can differentiate between the vector and the pseudo-scalar repre-sentations in the irrep products, the coupled channel space gets enlarged with respectto the case of the scalar resonances and therefore, a much richer spectrum is gen-erated for the axial resonances. In contrast with previous works of Kolomeitsev [9]and Guo [11] where light pseudo-scalar mesons are scattered off heavy vector mesons,in our work we have an enlarged coupled channel basis accounting also for channelswhere light vector mesons are scattered off heavy pseudo-scalars.Using SU (3) isoscalar factors [34], [44], [45] the ξ Iij can be transformed to a SU (3)basis, by means of which one knows in which multiplets there is attraction and, ⊗ ¯3 ∗ → ⊕ ¯61 ¯3 ⊗ ∗ → ¯15 ⊕ ¯3 ⊕ ⊗ ¯3 ∗ → ¯15 ⊕ ¯3 ⊕ ⊗ ∗ → ¯31 ⊗ ¯3 ∗ → ¯30 ¯3 ⊗ ∗ → ⊕ ⊗ ¯3 ∗ → ⊕ ⊗ ∗ → ⊗ ∗ → ⊗ ∗ → ⊗ ∗ → ⊕ s ⊕ a ⊕ ⊕ ¯10 ⊕ SU (3) decomposition of the interaction between pseudo-scalar and vector mesonsin SU (4). The sectors not shown in the table correspond to the C = − , − therefore, the possibility to generate resonances. In the C=2 sector the interactionis not attractive in any multiplet, in the C=1 sector there is attraction in the anti-triplets and in the sextets and in the C=0 sector the two octets and the singletcoming from 8 ⊗ ∗ and the two heavy singlets from the ¯3 ⊗ ∗ and the 3 ⊗ ¯3 ∗ areattractive.Table 2 shows the channel content in each sector. We will work with states ofdefined charge conjugation or G-parity, where it applies.Our model assumes that isospin symmetry is exact so, all particles belonging toa same isospin multiplet have the same mass. For the pions we use m π =138 MeV,for kaons m K =495 MeV and for the eta m η =548 MeV. In the heavy sector we use,for the pseudo-scalars m D =1865 MeV, m D s =1968 MeV and m η c =2979 MeV. Themasses of the light vector mesons are: m ρ =771 MeV, m K ∗ =892 MeV and m ω =782MeV. While for the heavy vector mesons we use: m D ∗ =2008 MeV, m D ∗ s =2112 MeVand m J/ψ =3097 MeV.It is also possible to restore SU (3) symmetry by setting the masses of all particlesin a same SU (3) multiplet to a common value. For this purpose we introduce theparameter x , x = 0 is the case when SU (3) symmetry is restored and x = 1 the casewe see in Nature with SU (3) broken. The meson masses as a function of x are givenby: m ( x ) = ¯ m + x ( m phys. − ¯ m ) (23)where ¯ m is the meson mass in the symmetric limit. We will use for the pseudo-scalars: G ( J P C ) Channels1 1 1(1 + ) πD ∗ s , D s ρKD ∗ , DK ∗ + ) DK ∗ , KD ∗ , ηD ∗ s D s ω , η c D ∗ s , D s J/ψ (1 + ) πD ∗ , Dρ , KD ∗ s , D s K ∗ ηD ∗ , Dω , η c D ∗ , DJ/ψ -1 0(1 + ) DK ∗ , KD ∗ (1 + ) πK ∗ , Kρ , ηK ∗ , Kω ¯ DD ∗ s , D s ¯ D ∗ , KJ/ψ , η c K ∗ + (1 + − ) √ ( ¯ KK ∗ + c.c. ), πω , ηρ √ ( ¯ DD ∗ + c.c. ), η c ρ , πJ/ψ − (1 ++ ) πρ , √ ( ¯ KK ∗ − c.c. ), √ ( ¯ DD ∗ − c.c. )0 + (1 ++ ) √ ( ¯ KK ∗ + c.c. ), √ ( ¯ DD ∗ + c.c. ), √ ( ¯ D s D ∗ s − c.c. )0 − (1 + − ) πρ , ηω , √ ( ¯ DD ∗ − c.c. ), η c ωηJ/ψ , √ ( ¯ D s D ∗ s + c.c. ), √ ( ¯ KK ∗ − c.c. ), η c J/ψ
Table 2: Channel content in each sector9 s1 (2460)D (2430) Figure 1: Pole trajectories for one of the anti-triplets in the C=1 sector while breaking SU (3) symmetry in steps of ∆ x =0.1. The two degenerate poles at x =0 become twodifferent resonances at x =1, in the two extremes of the curve (real world). ¯ m =430 MeV, ¯ m =1900 MeV and ¯ m =1900 MeV, and for the vector mesons:¯ m ∗ =800 MeV, ¯ m ∗ =2050 MeV and ¯ m ∗ =2050 MeVIn our model we still have to fit the subtraction constants in the loop function. Asdone in our previous work [34] we use two different values, α L for channels involvingjust light mesons and α H for channels involving at least one heavy meson. This isjustified because the heavy and light sector nearly decouple from each other. Theresult of our fit is α L = − . α H = − . f appearing in theLagrangian: for light mesons f = f π = 93 MeV, while for heavy mesons f = f D = 165MeV [43].When SU (3) symmetry is restored it is possible to identify 3 poles and one cuspin the open charm sector. Two poles come from the two anti-triplets where theinteraction was attractive, their positions are at 2432.63 MeV and (2535.07-i0.08)MeV. The other pole is broad and comes from one of the sextets at (2532.57-i199.36)MeV, the other sextet appears as a narrow cusp around 2700 MeV, it becomes a polewhen the heavy light threshold at 2700 MeV moves because of the SU (3) symmetrybreaking.Figure 1 shows the pole trajectories for the anti-triplet starting at 2432.63 MeVwhile changing x from 0 to 1 in steps of 0.1.In the hidden charm sector two octets and three singlets are expected, one light nd two heavy. The two octets are nearly degenerated at 1161.06 MeV and 1161.37MeV. In the work of Roca [36] the two octets are degenerated, but in our model theinteraction with the heavy sector removes this degeneracy by a very small amount,indicating a weak coupling between the light and heavy sectors. The light singletappears as a pole at 1055.77 MeV and the two heavy ones at 3867.59 MeV and(3864.62-i0.00) MeV this second one is not exactly a bound state as the others, buta narrow state with a width smaller than 1 KeV.Table 3 shows the pole positions for the case x = 1 within our model and thepossible identification of each one.We will now discuss separately the particle identification in each sector. In contrast with the scalar resonances where the sextet state became very broad [34],the axial sextets are narrower, hence easier to detect experimentally. One shouldnote also that these states are truly exotics since quark models cannot generate q ¯ q pairs with such quantum numbers. We found two poles in this sector at positions(2529.30-i238.56) MeV and (2756.52-i32.95) MeV.The couplings, g i , of the poles to each channel i have been calculated from theresidues of each pole. Close to the pole position one can write: T ij ∼ = g i g j s − s pole (24)Table 4 shows the results of g i for the poles in this sector. With the couplings itis possible to do a rough estimate of the partial decay widths for the resonances andthus identify the channels with largest contribution to the width in order to motivateexperimental searches in this direction. At tree level one has:Γ A → P V = | g i | πM A p (25)where p is the center of mass three-momentum of the two particles in the final state.For the pole at (2756.52-i32.95) MeV the estimate reads: Γ Dsρ Γ πD ∗ s ∼ . Γ Dsρ Γ KD ∗ ∼ . Γ πD ∗ s Γ KD ∗ ∼ . πD ∗ s and KD ∗ , or the heavier one to D s ρ and DK ∗ make these states qualify as roughly quasi-bound states of these channelsrespectively. Note that they separate two basic configurations: heavy vector-lightpseudo-scalar and heavy pseudo-scalar-light vector.When taking into account the finite ρ and K ∗ widths the resonance at (2756.52-i32.95) becomes a little bit higher in mass, crossing the KD ∗ thresholds and disap-pearing as a pole.
11 Irrep S I G ( J P C ) RE( √ s ) (MeV) IM( √ s ) (MeV) Resonance IDMass (MeV)1 ¯3 1 0(1 + ) 2455.91 0 D s (2460)2432.63 0 (1 + ) 2311.24 -115.68 D (2430)6 1 1(1 + ) 2529.30 -238.56 (?)2532.57 0 (1 + ) Cusp (2607) Broad (?)-i199.36 -1 0(1 + ) Cusp (2503) Broad (?)1 0(1 + ) 2573.62 -0.07 D s (2536)¯3 [-0.07]2535.07 0 (1 + ) 2526.47 -0.08 D (2420)-i0.08 [-13]6 1 1(1 + ) 2756.52 -32.95 (?)[cusp]Cusp (2700) 0 (1 + ) 2750.22 -99.91 (?)[-101]Narrow -1 0(1 + ) 2756.08 -2.15 (?)[-92]0 1 0 0 − (1 + − ) 925.12 -24.61 h (1170)1055.778 1 (1 + ) 1101.72 -56.27 K (1270)1161.06 0 1 + (1 + − ) 1230.15 -47.02 b (1235)0 − (1 + − ) 1213.00 -5.67 h (1380)1 0 0 + (1 ++ ) 3837.57 -0.00 X (3872)3867.598 1 (1 + ) 1213.20 -0.89 K (1270)1161.37 0 1 − (1 ++ ) 1012.95 -89.77 a (1260)0 + (1 ++ ) 1292.96 0 f (1285)1 0 0 − (1 + − ) 3840.69 -1.60 (?)3864.62-i0.00Table 3: Pole positions for the model. The column Irrep shows the results in the SU (3)limit. The results in brackets for the Im √ s are obtained taking into account the finitewidth of the ρ and K ∗ mesons. 12hannel (2529.30-i238.56) MeV (2756.52-i32.95) MeV | g i | (GeV) | g i | (GeV) πD ∗ s D s ρ DK ∗ KD ∗ | g i | (GeV) | g i | (GeV) DK ∗ KD ∗ ηD ∗ s D s ω η c D ∗ s D s J/ψ
The two poles found in this sector have the proper quantum numbers to be identifiedwith the two D s resonances. The first pole appears as an exact bound state at2455 MeV and we identified it with the D s (2460) state. Experimentally the mainhadronic decay channel for this resonance is D ∗ s π which is an isospin violating decayand therefore not taken into account by our model. Other decays for this resonanceare three body decays or electromagnetic ones, which are also not included in ourframework.The other pole appears at (2573.62-i0.07) MeV and couples mainly to the DK ∗ and D s ω channels. The only open channel for it to decay is the KD ∗ channel but,because of the dynamics of the interaction, this resonance barely couples to it. Thisexplains the small width of this resonance, 140 KeV, despite the 70 MeV phase spaceavailable for it to decay. We identify this pole with the D s (2536) which is alsoobserved in the decay channel KD ∗ with a small width (Γ < . | g i | for each channel for the twopoles in this sector.Once more we see that the lighter state couples strongly to KD ∗ and ηD ∗ s while thesecond one couples strongly to DK ∗ and D s ω . Hence the decoupling into two familiesof heavy vector-light pseudo-scalar and light vector-heavy pseudo-scalar shows up inthis sector too.This sector has one resonance with a strong coupling to the D s ω channel ( ω = ω here). Hence, this is one case where the φ − ω mixing can be relevant. Since we saw | g i | (GeV) | g i | (GeV) DK ∗ KD ∗ ηD ∗ s D s ω D s φ η c D ∗ s D s J/ψ D s φ channel that the singlet ( ω ) does not give any contribution to the Lagrangian, the explicitintroduction of the ω and φ states can be done by substituting: ω = 1 √ ω − r φ (26)This splitting into two fields introduces a new column and a new row in the secondtable of appendix A.1., by simply multiplying the fourth column and row by √ andintroducing a D s φ column and row with weights − q those of the original D s ω .When we look now for poles we obtain the results in Table 6.Comparing Tables 5 and 6, we can see that in the case where the coupling of theresonance to the D s ω channel is weak (first resonance) the effects of the mixing arevery small in the energy and couplings of the resonance. In the case of the secondresonance, where the coupling to the D s ω channel was large, the effects of the mixingare more visible. There is a shift of the mass of about 25 MeV, which is well withinour theoretical uncertainties. This effect can be considered an upper bound for allother cases, since we have chosen the resonance with strongest coupling to ω .It is also interesting to see that the sums of the squares of the couplings to ω and φ are close to the square of that to ω , indicating a redistribution of the strength ofthe coupling to ω between ω and φ .The widths of the light vector mesons have no significant effects over the reso-nances generated in this sector, because the mass of the resonances are far away fromthe threshold of the DK ∗ channel. Here the companions of the two anti-triplets and the two sextets should be found.Note that when we refer to the SU (3) multiplet we are talking about the case whenone has SU (3) symmetry. This correspond to x = 0 in the pole trajectories. At x = 1,since SU (3) symmetry is broken, the physical states mix the SU (3) multiplets. Yet,the study of the trajectories allows us to trace back any pole to its origin in the SU (3) | g i | (GeV) | g i | (GeV) | g i | (GeV) πD ∗ Dρ KD ∗ s D s ¯ K ∗ ηD ∗ Dω η c D ∗ DJ/ψ sector symmetric case, and we have used this information for the classification of states inTable 3.The anti-triplet companion of the pole for the D s (2460) is the pole located at(2311.24-i115.68) MeV that we identify with the D (2430) because of its naturallylarge width, since it is strongly coupled to the πD ∗ channel into which it is free todecay. On the other hand the pole at (2526.47-i0.08) MeV, companion of the oneidentified with the D s (2536), has its coupling to the πD ∗ channel strongly suppressedand therefore has a very narrow width. Because of this unnatural narrow width weare tempted to identify it with the D (2420) although the mass of our dynamicallygenerated resonance is around 100 MeV off the experimental value for this state.Moreover, when considering the finite widths of the ρ and K ∗ mesons, this pole getsa larger width, its imaginary part goes to −
13 MeV, implying a width of about 26MeV, in fair agreement with experiment.As for the sextets, one of the poles becomes a broad cusp at the ¯ KD ∗ s thresholdas one gradually breaks SU (3) symmetry through the parameter x , and the otherpole emerges from a cusp into a pole at (2750.22-i99.91) MeV. The channel to whichit is most strongly coupled is closed, the D s ¯ K ∗ , but it also has sensitive couplings toall channels into which it is allowed to decay. The consideration of the finite widthof the vector mesons has very small effect over this resonance ( ∼ The two remaining exotic members of the sextet should be found in this sector. Oneof them becomes a broad cusp at the ¯ KD ∗ threshold when x = 1 while the other oneis a narrow resonance with pole position (2756.08-i2.15) MeV. The couplings of thispole are given in Table 8. | g i | (GeV) D ¯ K ∗ KD ∗ | g i | (GeV) | g i | (GeV) πK ∗ Kρ ηK ∗ Kω DD ∗ s D s ¯ D ∗ KJ/ψ η c K ∗ sector When taking into account the 50 MeV width of the K ∗ meson, this resonance getsa much bigger width, of the order of 180 MeV. In this case, and in all other sectorswhere the effect of the finite width of the vector mesons were taken into account, theonly significant effect one could observe was over the width of the resonance. Themass of the resonances were affected in less than 0.5 % and the important couplingsin less than 5%. Two poles are found here coming from the two octets in the scattering of the lowlying pseudo-scalar with the light vector mesons. In principle one could be temptedto assign these two poles to the two axial kaons from PDG [43], but the mass of oneof these, the K (1400) is about 200-300 MeV off the pole positions we found and itswidth is much smaller than that. With this in mind we followed the interpretationof Roca [36] that the K (1270) should have a two pole structure.The couplings of the two poles to the different channels are in Table 9.This sector is explained in more detail in [36]. The novelty here is that, in spiteof including now the heavy channels, the results are basically unaltered compared tothose of [36] where only the light sector was used. This indicates a very weak mixingof the heavy and light sectors.Concerning the two K states it is also opportune to mention that in [46] someexperimental information was reanalyzed giving strong support to the existence of | g i | (GeV) | g i | (GeV) πρ - 4.49 K ¯ K ∗ ± c.c. πω ηρ D ¯ D ∗ ± c.c. η c ρ πJ/ψ these two states. In this sector too there are two poles coming from the two octets but, since this isthe non-strange sector, this two states have defined G-parity and therefore cannotmix.The pole with positive G-parity we associate with the b (1235) resonance. Thesmall discrepancy between the experimental width and the value found from ourtheoretical model is explained since, experimentally, some decay channels of thisresonance are three or four body decays while our model contemplates just two bodyhadronic decays.The negative G-parity pole should be identified with the a (1260) but here themodel gives a worse description of the resonance, the mass of the pole is smaller thanexpected although the huge width of the resonance makes this a minor problem.Also the width found within the model is very large, of the order of magnitude ofthe experimental one which is estimated with large errors. Again one should notethat an important fraction of the width of this resonance could be due to many bodydecays not included in the present model.The couplings of the resonances to the channels are given in Table 10 and they arevery similar to those found in [36]. There the ω was substituted in terms of ω and φ and the sum of | g | for πω and πφ is similar to the | g | for πω of our calculation. Five poles are found in this sector. Tree have negative charge conjugation parityand two of them positive C-parity. In the light sector the positive C-parity pole isassociated with the f (1285), it appears in our model as a truly bound state, as itshould, since none of its observed decay channels is a pseudo-scalar vector mesonone, the possible decay channels within the model. The results obtained here and inthe other two sectors for the light axial resonances without ω − φ mixing are very imilar to those obtained in [36] where the mixing was explicitly taken into accountthus, corroborating the moderate effects of the mixing found for the charmed sector.The heavy singlet with positive C-parity obtained at 3837 MeV is a good candi-date to be associated with the controversial state X (3872). In this case this stateis interpreted as being mainly a mixed molecule of D ¯ D ∗ + c.c. and D s ¯ D ∗ s − c.c. , itsonly possible decay channel within the model being the K ¯ K ∗ + c.c. which is highlysuppressed. In Table 11 the couplings of the two poles are presented. We can seethere the strong decoupling of the heavy and light sectors.The low lying negative C-parity resonances can be associated with the two h resonances. The singlet at (925.12-i24.61) MeV we identify with the h (1170) and,since we get it with a lower mass, our width is much smaller than the experimentalone because our state has less phase space for decay. With the octet pole at (1213.00-i5.67) MeV the same thing happens, and we associate it with the h (1380) despitethe smaller mass and width compared with experimental values.In the heavy sector we find another state at 3840 MeV and negative C-parity.The decays of the X (3872), reported in [47], into γJ/ψ and ωJ/ψ indicate that theC-parity of this state is positive. The decay into π + π − J/ψ reported in [24] assumingthat the π + π − comes from a ρ state would give the same C-parity but would implyisospin breaking. The large branching fraction B ( X → π + π − π J/ψ ) B ( X → π + π − J/ψ ) = 1 . ± . ± . X particle.There is a more appealing explanation for eq. (27) if one had two X (3872)states with different G-parity and correspondingly C-parity. Should the π + π − in thedenominator of eq. (27) correspond to an I=0 state one would not have to invokeisospin violation, but instead the existence of a negative G-parity (and hence C-parity) state. This would imply that there is strength of these events in the σ regionof the ππ invariant mass, and this seems to be the case as reported in [27], althoughthe statistics is low. This latter scenario would fit with our predictions of two statesnearly degenerate with opposite C-parity. A similar argumentation has been made in[48] to justify the existence of two degenerate X (3872) states with different C-parity.The difference of 35 MeV in the binding energy between our model and experiment( ∼
1% difference) is perfectly acceptable for a theoretical model that looks at thewhole spectrum of axial vector mesons in a broad range of masses with only twofree parameters ( α H , α L ). Yet, we can do some fine tuning to get a mass like inexperiment by changing the subtraction constants in the loops. In this case we cantake α H = − .
30 (from − .
55 before) and then we find the mass of the positiveC-parity pole at 3872.67 MeV and simultaneously the state with negative C-parityjust disappears as a pole, by crossing the ¯ DD ∗ threshold, and leads to a markedcusp structure in the ¯ DD ∗ − c.c. amplitude. Indeed, changing α H to slightly morenegative values we regain the pole just below this threshold.Recently a new possible state at 3875 MeV was reported at Belle [50] decaying into¯ D D ∗ . The 3 MeV difference of this state with the X (3872) is precisely the difference c oun t s / b i n E(MeV)a) Belle data|T| for G=+ -2 0 2 4 6 8 10 12 14 16 18-10 0 10 20 30 c oun t s / b i n E(MeV)b) Belle data|T| p for G=-|T| p for G=+ Figure 2: a) | T | for the positive G-parity state in the D ¯ D ∗ channel compared with theBelle Data (in this plot α H = − .
23) b) | T | p for both G-parity states in the D ¯ D ∗ channelcompared with the Belle Data (in this plot α H = − .
30 for the G=- state) of masses between the positive and negative C-parity states that we obtain. It is thustempting to associate to the new X (3875) state our negative C-parity state. Suchscenario is not excluded by the data as we show in the analysis below.In the decay of a B particle to Kπ + π − J/ψ we have, defining E = M inv ( π + π − J/ψ ) − M ¯ D D ∗ : dBr + dE ∝ | T ( ¯ D D ∗ + c.c. → ¯ D D ∗ + c.c. ) | Γ( X → π + π − J/ψ ) (28)which is approximately proportional to | T | in a few MeV range of E around M ¯ D D ∗ since Γ( X → π + π − J/ψ ) barely changes in this range of energies given the relativelylarge phase space for this decay. On the other hand for the B particle decaying into K ¯ D D ∗ we have (now E = M inv ( ¯ D D ∗ ) − M ¯ D D ∗ ): dBr − dE ∝ | T ( ¯ D D ∗ − c.c. → ¯ D D ∗ − c.c. ) | Γ( X ′ → ¯ D D ∗ ) (29)which is approximately proportional to | T | p , where p is the three-momentum of the¯ D particle in the center of mass frame of the ¯ DD ∗ system, since now Γ( X ′ → ¯ D D ∗ )is proportional to this three-momentum.In fig 2 we plot dBr + dE and dBr − dE as a function of E and we compare our results withthe experimental data from Belle [50] [51]. We can see that in the case of the positiveC-parity state the π + π − J/ψ distribution is very sharp, while the distribution of theinvariant mass of ¯ D D ∗ leads to a strong enhancement of the distribution aroundthe ¯ D D ∗ threshold, in both cases in fair agreement with experiment.We should note that an empirical analysis of the data in a recent paper [52], takingonly one resonance, produces a similar behavior assuming that the resonance couplesvery strongly to D ¯ D ∗ , as it is also our case (see couplings in Tables 11-12). In viewof the results of [52] it is also instructive to see what our model would give assumingthat the resonance with positive C-parity is responsible for the two distributions in | g i | (GeV) | g i | (GeV) √ ( D ¯ D ∗ + c.c. ) 0.15 13.61 √ ( D s ¯ D ∗ s − c.c. ) 0.54 10.58 √ ( K ¯ K ∗ + c.c. ) 7.15 0.03Table 11: Residues for the C=0,S=0,I P =0 + sectorChannel (925.12-i24.61) MeV (1213.00-i5.67) MeV (3840.69-i1.60) MeV | g i | (GeV) | g i | (GeV) | g i | (GeV) πρ ηω √ ( D ¯ D ∗ − c.c. ) 3.93 1.03 13.44 η c ω ηJ/ψ √ ( D s ¯ D ∗ s + c.c. ) 2.31 1.26 9.96 √ ( K ¯ K ∗ − c.c. ) 0.73 6.14 0.01 η c J/ψ P =0 − sector fig. 2. We show the result obtained for | T | p with the positive C-parity resonance infigure 2 (b) with dotted line. As we can see, the shape and strength obtained in thiscase is not very different from that of the negative C-parity resonance. Hence, thisalternative scenario, which would correspond to the one in [52], is not ruled out bythese combined data.As mention above the ratio of eq. (27) and the possible strength seen for the B ( X → π + π − J/ψ ) in the region of invariant ππ masses around m σ , is so far thestrongest indications favoring the two C-parity states. We studied the dynamical generation of axial resonances by looking for the poles inthe scattering T-matrix of pseudo-scalars with vector mesons. For the interactionLagrangian we first constructed a SU (4) flavor symmetrical Lagrangian for the in-teraction of the 15-plet of pseudo-scalar mesons with the 15-plet of vector mesons.The symmetry was broken down to SU (3) by suppressing exchanges of heavy vectormesons in the implicit Weinberg-Tomozawa term, following a prescription developedin a previous paper [34]. From the Lagrangian, tree level amplitudes were evaluated,projected in s-wave and collected in a matrix for the various channels. This matrixwas used as the potential, transformed to an isospin basis, in the Bethe-Salpeter quation, which provides the unitarized amplitudes between the channels.The poles generated within the model can be associated with the various axialresonances listed by the Particle Data Group [43], and also many new resonances arepredicted: three broad ones in the mass range between 2.5 and 2.6 GeV, and threenarrower ones around 2.75 GeV, these resonances belong to two SU (3) sextets, theyhave, therefore, exotic quantum numbers and they have not yet been experimentallyobserved. We summarize these states with their quantum numbers in Table 13. C S I G ( J P C ) Re( √ s ) (MeV) Im( √ s ) (MeV) Channel1 1 1(1 + ) 2529.30 -238.56 πD ∗ s , KD ∗ (1 + ) Cusp (2607) Broad -1 -1 0(1 + ) Cusp (2503) Broad -1 1 1(1 + ) 2756.52 -32.95 [cusp] D s ρ , DK ∗ (1 + ) 2750.22 -99.91 [-101] Dω + ) 2756.08 -2.15 [-92] D ¯ K ∗ − (1 + − ) 3840.69 -1.60 η c ω , ηJ/ψ Table 13: List of predicted and not yet observed resonances with quantum numbers andthe open channels to which they couple most strongly. The results in brackets for the Im √ s are obtained taking into account the finite width of the ρ and K ∗ mesons. Remaining discrepancies between our model and experiment can be attributedto possible many body decays of these objects or possible higher order terms thatcould be included in the Lagrangian. The size of the discrepancies, typical of anysuccessful hadronic model describing hadronic spectra, can be used to estimate theuncertainties in the predictions for the new states that we obtain.Some states obtained here have been also studied before in a similar framework,but with more restricted coupled channels space. The similarity of the results rein-forces these findings. Many other states are reported here for the first time within thisunitary coupled channel framework. In the light scalar sector the poles and couplingsfound within our approach coincide very well with the ones found by Roca [36]. Thishappens because, despite the enlarged coupled channel space including heavy mesons,this new sector couples very weakly with the light one. In the open-charm sector thepoles found for the lightest anti-triplet coincide with the results found by Kolomeit-sev [9] and Guo [11]. As already happened for the scalar resonances, the poles foundwithin our model for the sextet state in this sector have broader widths because ofthe use, in our model, of a different meson decay constant for heavy mesons. Besides,our model allows also for the inclusion of channels with heavy pseudo-scalar mesonsinteracting with light vector ones, as a result of which our model generates a richerspectrum, with poles for an extra anti-triplet and an extra sextet. In the charmedsector, C=1, we find six resonances not yet observed. Three of them are either toobroad, or they degenerate into cusps as SU (3) is gradually broken. However, threeof them remain sufficiently narrow, such that they could in principle be detected.Moreover our Lagrangian incorporates the hidden-charm sector and an attractive nteraction in the 3 ⊗ ¯3 ∗ and ¯3 ⊗ ∗ charmed mesons is responsible for the generationof two resonances. One of them can be associated with the new X (3872) state. Theother one appears as a strong cusp in our case, or a resonance with slightly increasedattraction, and could be associated to the peak structure found recently around the¯ D D ∗ threshold at Belle, although this structure can also be explained in terms ofonly one state. On the other hand, the ratio of partial decay width into two or threepions and J/ψ (eq. (27)) provides strong support for the existence of two states withdifferent C-parity.The agreement of the theory with data for the known resonances, together withthe success of the theory for the charmed scalar mesons [34], gives us confidence onthese new predicted states, such as to strongly suggest their experimental search.
We would like to thank prof. K. Terasaki for some useful discussions.. This work ispartly supported by DGICYT contract number BFM2003-00856 and the Generali-tat Valenciana. This research is part of the EU Integrated Infrastructure InitiativeHadron Physics Project under contract number RII3-CT-2004-506078. The ξ Coefficients
A.1 Open-Charm Sector (C=1)
S=1, I G ( J P C )=1(1 + )Channels πD ∗ s D s ρ KD ∗ DK ∗ πD ∗ s γD s ρ γ -1 KD ∗ -1 - γ DK ∗ − γ -1 0 0S=1, I G ( J P C )=0(1 + )Channels DK ∗ KD ∗ ηD ∗ s D s ω η c D ∗ s DJ/ψDK ∗ -2 0 - γ √ - √ q γ KD ∗ √ γ √ q γηD ∗ s − γ √ √ γ − √ γ D s ω - √ γ √ γ √ γ η c D ∗ s − q γ − √ γ γ DJ/ψ q γ - √ γ γ G ( J P C )= (1 + )Channels πD ∗ Dρ ¯ KD ∗ s D s ¯ K ∗ ηD Dω η c D ∗ DJ/ψπD ∗ -2 γ q γ √ γDρ γ -2 0 - q γ √ γ KD ∗ s q q - q γ γ √ D s ¯ K ∗ − q q γ - q
32 2 γ √ ηD γ - q - q γ γ √ γ Dω - γ q γ − q γ √ γ η c D ∗ √ γ γ √ √ γ γ DJ/ψ - √ γ γ √ √ γ γ G ( J P C )=0(1 + )Channels D ¯ K ∗ ¯ KD ∗ D ¯ K ∗ -1 - γ ¯ KD ∗ - γ - 1 .2 Hidden-Charm Sector (C=0) S=1, I G ( J P C )= (1 + )Channels πK ∗ Kρ ηK ∗ Kω ¯ DD ∗ s D s ¯ D ∗ KJ/ψ η c K ∗ πK ∗ -2 - q γ Kρ -2 - q γ ηK ∗ − − γ √ q γ Kω q γ − γ √ DD ∗ s − q γ − γ √ q γ − ψ − γ √ − γ √ D s ¯ D ∗ q γ q γ − γ √ − ψ − γ √ − γ √ KJ/ψ − γ √ − γ √ η c K ∗ − γ √ − γ √ G ( J P C )=1 + (1 + − )Channels πω ηρ √ ( ¯ KK ∗ + c.c. ) √ ( ¯ DD ∗ + c.c. ) η c ρ πJ/ψπω √ γ √ ηρ √ γ √ √ ( ¯ KK ∗ + c.c. ) √ √ − γ √ ( ¯ DD ∗ + c.c. ) γ √ γ √ − γ − ψ q γ q γη c ρ q γ πJ/ψ q γ G ( J P C )=1 − (1 ++ )Channels √ ( ¯ KK ∗ − c.c. ) πρ √ ( ¯ DD ∗ − c.c. ) √ ( ¯ KK ∗ − c.c. ) -1 −√ γπρ −√ √ γ √ ( ¯ DD ∗ − c.c. ) γ √ γ − ψ S=0, I G ( J P C )=0 + (1 ++ )Channels √ ( ¯ KK ∗ + c.c. ) √ ( ¯ DD ∗ + c.c. ) √ ( ¯ D s D ∗ s − c.c. ) √ ( ¯ KK ∗ + c.c. ) -3 - γ √ γ √ ( ¯ DD ∗ + c.c. ) − γ − ψ − −√ √ ( ¯ D s D ∗ s − c.c. ) √ γ −√ − ψ − =0, I G ( J P C )=0 − (1 + − )Channels πρ ηω √ ( ¯ DD ∗ η c ω ηJ/ψ √ ( ¯ D s D ∗ s √ ( ¯ KK ∗ η c J/ψ − c.c. ) + c.c. ) − c.c. ) πρ -4 0 √ γ −√ ηω − γ − √ γ √ ( ¯ DD ∗ − c.c. ) √ γ − γ − ψ − − √ γ − √ γ −√ γ − γ η c ω − √ γ γ ηJ/ψ − √ γ γ √ ( ¯ D s D ∗ s + c.c. ) 0 − √ γ −√ γ γ − ψ − √ γ − √ γ √ ( ¯ KK ∗ − c.c. ) −√ γ √ γ -3 0 η c J/ψ − γ − √ γ References [1] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. , 242001 (2003)[arXiv:hep-ex/0304021].[2] D. Besson et al. [CLEO Collaboration], Phys. Rev. D , 032002 (2003)[arXiv:hep-ex/0305100].[3] P. Krokovny et al. [Belle Collaboration], Phys. Rev. Lett. , 262002 (2003)[arXiv:hep-ex/0308019].[4] K. Abe et al. , Phys. Rev. Lett. , 012002 (2004) [arXiv:hep-ex/0307052].[5] K. Abe et al. [Belle Collaboration], Phys. Rev. D , 112002 (2004)[arXiv:hep-ex/0307021].[6] J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B , 11 (2004)[arXiv:hep-ex/0312060].[7] S. Godfrey and N. Isgur, Phys. Rev. D , 189 (1985).[8] T. Barnes, F. E. Close and H. J. Lipkin, Phys. Rev. D , 054006 (2003)[arXiv:hep-ph/0305025].[9] E. E. Kolomeitsev and M. F. M. Lutz, Phys. Lett. B (2004) 39[arXiv:hep-ph/0307133].[10] J. Hofmann and M. F. M. Lutz, Nucl. Phys. A , 142 (2004)[arXiv:hep-ph/0308263].[11] F. K. Guo, P. N. Shen and H. C. Chiang, Phys. Lett. B , 133 (2007)[arXiv:hep-ph/0610008].[12] F. K. Guo, P. N. Shen, H. C. Chiang and R. G. Ping, Phys. Lett. B (2006)278 [arXiv:hep-ph/0603072].[13] Y. J. Zhang, H. C. Chiang, P. N. Shen and B. S. Zou, Phys. Rev. D , 014013(2006) [arXiv:hep-ph/0604271].
14] Y. Q. Chen and X. Q. Li, Phys. Rev. Lett. , 232001 (2004)[arXiv:hep-ph/0407062].[15] M. Nielsen, R. D. Matheus, F. S. Navarra, M. E. Bracco and A. Lozea, Nucl.Phys. Proc. Suppl. , 193 (2006) [arXiv:hep-ph/0509131].[16] P. Bicudo, Phys. Rev. D , 036008 (2006) [arXiv:hep-ph/0512041].[17] Y. B. Dai, C. S. Huang, C. Liu and S. L. Zhu, Phys. Rev. D , 114011 (2003)[arXiv:hep-ph/0306274].[18] Fayyazuddin and Riazuddin, Phys. Rev. D , 114008 (2004)[arXiv:hep-ph/0309283].[19] J. Lu, X. L. Chen, W. Z. Deng and S. L. Zhu, Phys. Rev. D , 054012 (2006)[arXiv:hep-ph/0602167].[20] T. Matsuki, T. Morii and K. Sudoh, arXiv:hep-ph/0605019.[21] J. Vijande, F. Fernandez and A. Valcarce, Phys. Rev. D , 034002 (2006)[Erratum-ibid. D , 059903 (2006)] [arXiv:hep-ph/0601143].[22] E. van Beveren and G. Rupp, Phys. Rev. Lett. , 012003 (2003)[arXiv:hep-ph/0305035].[23] T. E. Browder, S. Pakvasa and A. A. Petrov, Phys. Lett. B , 365 (2004)[arXiv:hep-ph/0307054].[24] S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett. , 262001 (2003)[arXiv:hep-ex/0309032].[25] D. Acosta et al. [CDF II Collaboration], Phys. Rev. Lett. , 072001 (2004)[arXiv:hep-ex/0312021].[26] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. , 162002 (2004)[arXiv:hep-ex/0405004].[27] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D , 071103 (2005)[arXiv:hep-ex/0406022].[28] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D , 014028(2005) [arXiv:hep-ph/0412098].[29] B. A. Li, Phys. Lett. B , 306 (2005) [arXiv:hep-ph/0410264].[30] F. E. Close and P. R. Page, Phys. Lett. B , 119 (2004)[arXiv:hep-ph/0309253].[31] C. Y. Wong, Phys. Rev. C , 055202 (2004) [arXiv:hep-ph/0311088].[32] E. S. Swanson, Phys. Lett. B , 189 (2004) [arXiv:hep-ph/0311229].[33] E. S. Swanson, Phys. Rept. , 243 (2006) [arXiv:hep-ph/0601110].[34] D. Gamermann, E. Oset, D. Strottman and M. J. V. Vacas,arXiv:hep-ph/0612179. In print Phys. Rev. D[35] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A , 392 (2004)[arXiv:nucl-th/0307039].
36] L. Roca, E. Oset and J. Singh, Phys. Rev. D (2005) 014002[arXiv:hep-ph/0503273].[37] J. A. Oller and E. Oset, Nucl. Phys. A (1997) 438 [Erratum-ibid. A (1999) 407] [arXiv:hep-ph/9702314].[38] J. A. Oller and E. Oset, Phys. Rev. D (1999) 074023 [arXiv:hep-ph/9809337].[39] E. Oset and A. Ramos, Nucl. Phys. A , 99 (1998) [arXiv:nucl-th/9711022].[40] J. A. Oller and U. G. Meissner, Phys. Lett. B , 263 (2001)[arXiv:hep-ph/0011146].[41] T. Inoue, E. Oset and M. J. Vicente Vacas, Phys. Rev. C , 035204 (2002)[arXiv:hep-ph/0110333].[42] L. Roca, S. Sarkar, V. K. Magas and E. Oset, Phys. Rev. C , 045208 (2006)[arXiv:hep-ph/0603222].[43] S. Eidelman et al. [Particle Data Group], Phys. Lett. B (2004) 1.[44] J. J. de Swart, Rev. Mod. Phys. , 916 (1963).[45] T. A. Kaeding, arXiv:nucl-th/9502037.[46] L. S. Geng, E. Oset, L. Roca and J. A. Oller, Phys. Rev. D , 014017 (2007)[arXiv:hep-ph/0610217].[47] K. Abe et al. , arXiv:hep-ex/0505037.[48] K. Terasaki, arXiv:0706.3944 [hep-ph].[49] A. Abulencia et al. [CDF Collaboration], Phys. Rev. Lett. , 102002 (2006)[arXiv:hep-ex/0512074].[50] G. Gokhroo et al. , Phys. Rev. Lett. , 162002 (2006) [arXiv:hep-ex/0606055].[51] G.Majumder, ICHEP2006 talk, http://belle.kek.jp/belle/talks/ICHEP2006/Majumber.ppt.[52] C. Hanhart, Yu. S. Kalashnikova, A. E. Kudryavtsev and A. V. Nefediev,arXiv:0704.0605 [hep-ph]., 162002 (2006) [arXiv:hep-ex/0606055].[51] G.Majumder, ICHEP2006 talk, http://belle.kek.jp/belle/talks/ICHEP2006/Majumber.ppt.[52] C. Hanhart, Yu. S. Kalashnikova, A. E. Kudryavtsev and A. V. Nefediev,arXiv:0704.0605 [hep-ph].