Quasibound states in the continuum: A dynamical coupled-channel calculationof axial-vector charmed mesons

Susana Coito* and George Rupp†

Centro de Fısica das Interaccoes Fundamentais, Instituto Superior Tecnico, Technical University of Lisbon,P-1049-001 Lisboa, Portugal

Eef van Beveren‡

Centro de Fısica Computacional, Departamento de Fısica, Universidade de Coimbra, P-3004-516 Coimbra, Portugal(Received 14 June 2011; revised manuscript received 7 October 2011; published 17 November 2011)

Masses and widths of the axial-vector charmed mesons D1ð2420Þ, D1ð2430Þ,Ds1ð2536Þ, and Ds1ð2460Þare calculated nonperturbatively in the resonance-spectrum-expansion model, by coupling various open

and closed meson-meson channels to the bare JP ¼ 1þ c �q (q ¼ u, d) and c�s states. The coupling to two-

meson channels dynamically mixes and lifts the mass degeneracy of the spectroscopic 3P1 and1P1 states,

as an alternative to the usual spin-orbit splitting. Of the two resulting S-matrix poles in either case, one

stays very close to the energy of the bare state, as a quasibound state in the continuum, whereas the other

shifts considerably. This is in agreement with the experimental observation that the D1ð2420Þ and

Ds1ð2536Þ have much smaller widths than one would naively expect. The whole pattern of masses and

widths of the axial-vector charmed mesons can thus be quite well reproduced with only two free

parameters, one of which being already strongly constrained by previous model calculations. Finally,

predictions for pole positions of radially excited axial-vector charmed mesons are presented.

DOI: 10.1103/PhysRevD.84.094020 PACS numbers: 14.40.Lb, 11.55.Ds, 11.80.Gw, 13.25.Ft

I. INTRODUCTION

The axial-vector (AV) charmed mesons D1ð2420Þ andDs1ð2536Þ [1] have the puzzling feature that their decaywidths are much smaller than one would expect on thebasis of their principal S-wave decay modes. Namely, theD1ð2420Þ decays to D�� (possibly also in a D wave), witha phase space of more than 270 MeV, but has a total widthof only 20–25 MeV [1]. On the other hand, the Ds1ð2536Þdecays toD�K in S andDwavewith a phase space of about30 MeV, resulting in an unknown tiny width <2:3 MeV,limited by the experimental resolution [1]. The discoveryof the missing two AV charmed mesons, namely, the verynarrow Ds1ð2460Þ and the very broad D1ð2430Þ, first ob-served by CLEO [2] and Belle [3] Collaborations, respec-tively, completed an even more confusing picture. Whilethe tiny width of the Ds1ð2460Þ can be easily understood,since this meson lies underneath its lowest Okubo-Zweig-Iizuka–allowed (OZIA) and isospin-conserving decaythreshold, the huge D1ð2430Þ width, in D��, is in sharpcontrast with that of the D1ð2420Þ. Moreover, theDs1ð2460Þ lies 76 MeV below the Ds1ð2536Þ, whereas theD1ð2420Þ and D1ð2430Þ are almost degenerate in mass, ifone takes the central value of the latter resonance.

Quark potential models, with standard spin-orbit split-tings, fail dramatically in reproducing this pattern ofmasses. For instance, in the relativized quark model [4]

the c�s state that is mainly 3P1 comes out at 2.57 GeV,

assuming the already then well-established Ds1ð2536Þ tobe mostly 1P1, though with a very large mixing between3P1 and

1P1. Reference [4] similarly predicted a too high

mass for the dominantly 3P1 state in the c �q (q ¼ u, d)sector, viz. 2.49 GeV. In the chiral quark model for heavy-light systems of Ref. [5], the result for the mainly 3P1 c �qstate is also 2.49 GeV, while the discrepancy is even worsein the c�s sector, with a prediction of 2.605 GeV for themostly 3P1 state, now with a small mixing in both sectors.More recently and after the discovery of the Ds1ð2460Þ

(and D1ð2430Þ), chiral Lagrangians for heavy-light sys-tems (see e.g. Refs. [6–9]) have been employed in orderto understand the masses of the AV charmed mesons, inparticular, the mass splittings with respect to the vector (V)mesons with charm D�

s and D�, respectively. Reference [7]

analyzed in detail the curious experimental [1] observationthat the AV-V mass difference is considerably larger in thecharm-nonstrange sector than in the charm-strange one,which is not predicted by typical quark potential models[4,5]. The same discrepancy applies to the scalar-pseudoscalar mass difference in either sector [1,7]. InRef. [7], the problem was tackled by calculating chiralloop corrections, but the result turned out to be exactlythe opposite of what is needed to remove or alleviate thediscrepancy.An alternative approach to the AV charmed mesons is by

trying to generate them as dynamical resonances in chiralunitary theory [10]. Indeed, in the latter paper, describingAV mesons in other flavor sectors as well, several charmedresonances were predicted, including the D1ð2420Þ,

*susana.coito@ist.utl.pt†george@ist.utl.pt‡eef@teor.fis.uc.pt

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D1ð2430Þ, Ds1ð2536Þ, and Ds1ð2460Þ, with reasonable re-sults, though the c �q states came out about 100 MeV off.However, dynamical generation of mesonic resonances,including the ones that are commonly thought to be of anormal quark-antiquark type, may give rise to interpreta-tional difficulties, besides predicting several genuinelyexotic and so far unobserved states [10]. Dynamicallygenerated AV charmed as well as bottom mesons can befound in Ref. [11], too.

Finally, in Ref. [12] a coupled-channel calculation ofpositive-parity c�s and b�s was carried out in a chiral quarkmodel, similar to our approach in its philosophy, and withresults for the Ds1ð2536Þ and Ds1ð2460Þ close to thepresent ones (also see below).

II. RESONANCE-SPECTRUM EXPANSION

In the present paper, we employ the resonance-spectrumexpansion (RSE) to describe the AV charmed mesons. TheRSE model has been developed for meson-meson (MM)scattering in nonexotic channels, whereby the intermediatestate is described via an infinite tower of s-channel q �qstates [13]. For the spectrum of the latter, in principle anyconfinement potential can be employed, but in practicalapplications, a harmonic oscillator (HO) with constantfrequency has been used, with excellent results [13].Recent applications of the RSE concern the �ð2170Þ [14]and Xð3872Þ [15] resonances.

In order to account for the two possible spectroscopicchannels 3P1 and

1P1 contributing to a JP ¼ 1þ state withundefined C-parity, we couple both q �q channels to themost important meson-meson channels. The resulting fullyoff-energy-shell RSE T matrix reads [13–15]

TðLi;LjÞij ðpi; p

0j;EÞ ¼ �2�2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�ipir0p

jiLiðpir0Þ

� XN

m¼1

Rimf½1��R��1gmjjjLjðp0

jr0Þ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�jp

0jr0

q; (1)

with the diagonal loop function

�ijðkjÞ ¼ �2i�2�jkjr0jjLjðkjr0Þhð1ÞjLj

ðkjr0Þ�ij; (2)

and the RSE propagator

R ijðEÞ ¼X

S¼0;1

X1

n¼0

giðS;nÞgjðS;nÞ

E� EðSÞn

: (3)

Here, � is an overall coupling, r0 is the average distance

for decay via 3P0 quark-pair creation, EðSÞn is the discrete

energy of the n-th recurrence in the q �q channel with spin S,giðS;nÞ is the corresponding coupling to the i-th MM chan-

nel, �i the reduced mass for this channel, pi the off-shellrelative momentum, Li the orbital angular momentum,

and jiLiand hð1ÞjLj

ðkjr0Þ the spherical Bessel and Hankel

functions of the first kind, respectively. Note that�i, pi, and the on-energy-shell relative momentum ki aredefined relativistically. Also notice that the infinitesum over the higher recurrences converges very fast, sothat it can be truncated after 20 terms in practical calcu-

lations. The S matrix is finally given by SðLi;LjÞij ðEÞ ¼ 1þ

2iTðLi;LjÞij ðki; kj;EÞ.

III. OZI-ALLOWED CHANNELSFOR AV CHARMED MESONS

Now we describe the physical AV charmed resonancesby coupling bare 3P1 and

1P1 c �q, c�s channels to all OZI-

allowed ground-state pseudoscalar-vector (PV) and vector-vector (VV) channels. It is true that there are also relevantpseudoscalar-scalar channels (in P-wave), most notablyDf0ð600Þ and D�

0ð2400Þ� [1] in the AV c �q case, and

DK�0ð800Þ for c�s. These will contribute to the observed

[1] D��, and D�K decay modes, respectively. Now, wehave recently developed [15] an algebraic procedure todeal with resonances in asymptotic states while preservingunitarity. However, the huge widths of the D�

0ð2400Þ,f0ð600Þ, and K�

0ð800Þ resonances may lead to fine sensi-

tivities that will tend to obscure the point we want to make,apart from the fact that there will also be nonresonantcontributions to the D�� and D�K final states. So we

TABLE I. Included meson-meson channels for D1ð2420Þ andD1ð2430Þ, with ground-state couplings squared [16] , orbitalangular momenta, and thresholds in MeV. For � and �0, apseudoscalar mixing angle of 37.3� [14] is used.

Channel ð~giðS¼1;n¼0ÞÞ2 ð~giðS¼0;n¼0ÞÞ2 L Threshold

D�� 0.027 78 0.013 89 0 2146

D�� 0.034 72 0.069 44 2 2146

D�� 0.005 24 0.002 62 0 2556

D�� 0.006 55 0.013 10 2 2556

D�sK 0.018 52 0.009 26 0 2608

D�sK 0.023 15 0.046 30 2 2608

D� 0.027 78 0.013 89 0 2643

D� 0.034 72 0.069 44 2 2643

D! 0.009 26 0.004 63 0 2650

D! 0.011 57 0.023 15 2 2650

D�� 0 0.013 89 0 2784

D�� 0.010 42 0.069 44 2 2784

D�! 0 0.004 63 0 2791

D�! 0.034 72 0.023 15 2 2791

DsK� 0.018 52 0.009 26 0 2862

DsK� 0.023 15 0.046 30 2 2862

D��0 0.004 02 0.002 01 0 2996

D��0 0.005 02 0.010 04 2 2996

D�sK

� 0 0.009 26 0 3006

D�sK

� 0.069 44 0.046 30 2 3006

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restrict ourselves to the open and closed PV and VVchannels in the present investigation, but we shall furtherdiscuss this issue below. The here included channels forc �q and c�s are given in Tables I and II, respectively,together with the corresponding orbital angular momenta,threshold energies, and ground-state couplings squaredð~giðS¼1ð0Þ;n¼0Þ2, where S ¼ 1ð0Þ refers to the 3P1 (1P1)

quark-antiquark component. In the Appendix, we show inmore detail how the ground-state coupling constants inTables I and II depend on the isospin and JPC quantumnumbers of the various meson-meson channels.

The latter squared couplings, computed in the verygeneral framework of Ref. [16], must be multiplied byðnþ 1Þ=4n for L ¼ 0 and by ð2n=5þ 1Þ=4n for L ¼ 2,so as to obtain the couplings for the radial recurrences n inthe RSE sum of Eq. (3). Note that the scheme of Ref. [16]employs overlaps of HO wave functions for the original q �qpair, the 3P0 q �q pair created out of the vacuum, and the

outgoing mesons. This allows one to rigorously calculatethe coupling constants of all excited states as well, incontrast with approaches using combinations of Clebsch-Gordan coefficients only. Nevertheless, our ground-statecouplings are identical to the usual ones in practically allsituations, including the present one. Finally, a subthres-hold suppression of closed channels is used just as inRef. [14].

The energies of the bare AV c �q and c�s states we deter-mine, as in previous work (see e.g. Refs. [14,15]), froman HO spectrum. The corresponding constant oscillatorfrequency and the constituent masses of the charmed,strange, and nonstrange quarks are also kept completelyunchanged at the values ! ¼ 190 MeV, mc ¼ 1562 MeV,ms ¼ 508 MeV, and mn ¼ 406 MeV [14,15]. This yieldsmasses of 2443MeVand 2545MeV for the bare AV c �qandc �s states, respectively, which are very close to values foundin typical single-channel quark models [4,5].

IV. QUASIBOUND STATES IN THE CONTINUUMAND OTHER POLES

Next we search for poles in the S matrix. Starting withthe c �q case, we choose r in the range 3:2–3:5 GeV�1

(0.64–0.70 fm), which is in between the values of2:0 GeV�1 [15] for an AV c �c system and 4:0 GeV�1

[14] for vector s�s states. In Fig. 1, we plot several poletrajectories in the complex E plane as a function ofthe overall coupling �. We see that this pole rapidlyacquires a large imaginary part, whereas the real partchanges considerably less, especially in the range r0 ¼3:3–3:5 GeV�1, making it a good candidate for the broadD1ð2430Þ resonance. For � ¼ 1:30 and r0 ¼ 3:40 GeV�1,the pole comes out at ð2430� i� 191Þ MeV, being thusfine-tuned to the experimental mass and width [1]. How-ever, there should be another pole in the S matrix, sincethere are two quark-antiquark channels and more thantwo MM channels. From the structure of the T matrix inEqs. (1)–(3), one can algebraically show that the number ofpoles for each bare state is equal to minðNq �q; NMMÞ, be-sides possible poles of a purely dynamical nature. Indeed,another pole originating from the bare c �q state is encoun-tered, with its trajectories depicted in Fig. 2. Quite remark-ably, this pole moves very little, acquiring an imaginarypart that is a factor 55 smaller than in the D1ð2430Þ case,for the values � ¼ 1:30 and r0 ¼ 3:40 GeV�1 (see solidlines and bullets in both figures). So this resonance, with apole position of ð2439� i� 3:5Þ MeV, almost decouplesfrom the only open OZIA MM channel [17], viz. D��,representing a quasibound state in the continuum (QBSC)[11]. Moreover, it is a good candidate for the D1ð2420Þ,though its width of roughly 7 MeV is somewhat too smalland its mass 16 MeV too high. These minor discrepancies

TABLE II. As Table I, but now for Ds1ð2536Þ and Ds1ð2460Þ.Channel ð~giðS¼1;n¼0ÞÞ2 ð~giðS¼0;n¼0ÞÞ2 L Threshold

D�K 0.037 04 0.018 52 0 2504

D�K 0.046 30 0.092 59 2 2504

D�s� 0.008 03 0.004 02 0 2660

D�s� 0.010 04 0.020 09 2 2660

DK� 0.037 04 0.018 52 0 2761

DK� 0.046 30 0.092 59 2 2761

D�K� 0 0.018 52 0 2902

D�K� 0.013 89 0.092 59 2 2902

Ds� 0.018 52 0.009 26 0 2988

Ds� 0.023 15 0.046 30 2 2988

D�s�

0 0.010 48 0.005 24 0 3069

D�s�

0 0.013 10 0.026 21 2 3069

D�s� 0 0.009 26 0 3132

D�s� 0.069 44 0.046 30 2 3132

FIG. 1. D1ð2430Þ pole trajectories as a function of �, for r0 ¼3:2–3:5 GeV�1 (left to right). Solid curve and bullets correspondto r0 ¼ 3:40 GeV�1 and � ¼ 1:30, respectively.

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may be due to the neglect of the PS channels, with broadresonances in the final states, as suggested above.

Nevertheless, these encouraging results might be partlydue to a fortuitous choice of the parameters � and r0.Therefore, we now check the c�s system, thereby scalingr0 and � with the square root of the reduced quark mass(see Ref. [17], Eq. (13)), so as to respect the flavor inde-pendence of our equations, which yields the c�s values r0 ¼3:12 GeV�1 and � ¼ 1:19. The ensuing c �s pole trajecto-ries are depicted in Fig. 3, but now for r0 ¼ 3:12 GeV�1

only. Thus, for � ¼ 1:19, the strongly coupling state comesout at 2452 MeV, i.e., only 7.5 MeV below the Ds1ð2460Þmass, with a vanishing width, as the pole ends up be-low the lowest OZIA channel. As for the c�s QBSC, itindeed shifts very little from the bare state, settling at

ð2540� i� 0:7Þ MeV, i.e., only 5 MeV above theDs1ð2536Þ mass, and having a width fully compatiblewith experiment [1].Besides the above ground-state AV charmed mesons, the

present model of course also predicts higher recurrences ofthese resonances. However, due caution is necessary so asto account for the most relevant open and closed decaychannels at the relevant energy scales. Now, the first radi-ally excited HO levels of the 3P1=

1P1 c �n and c �s states lie at2823 MeV and 2925 MeV, respectively, which allows thecorresponding resonances to be reasonably described bythe channels included in Tables I and II. Thus, we findagain four poles, tabulated in Table III, together with thoseof the ground-state AV charmed mesons. For the radiallyexcited states, we observe a similar pattern as for theground states, namely, two poles that remain close to thebare HO levels, whereas two other poles shift considerably.Note, however, that the difference is not as dramatic as inthe n ¼ 0 case. This may be due to the fact that severaldecay channels are open now. As for a possible observationof the here predicted 2P1 states, no experimental candi-dates have been reported so far. Namely, in the nearby c �qmass region, the two listed [1] resonances Dð2600Þ andDð2750Þ [1] both decay toD�� andD�, which excludes anAV assignment.Concerning the c�s sector, the only listed [1] state around

2.8–2.9 GeV is the D�sJð2860Þ [18], with natural parity and

so not an AV, decaying to D�K and DK, which makes it agood candidate for the 23P2 state, possibly overlapped by

the 23P0 [19]. Note that the lower of our two predicted 2P1

resonances also practically coincides with the D�sJð2860Þ,

both in mass and width. This may be a further indicationthat the D�

sJð2860Þ structure corresponds to more than one

resonance only.To conclude this section, we study—for the c �q system—

the dependence of the lowest-lying poles on the number ofincluded quark-antiquark and MM channels. In Table IV,besides the D1ð2420Þ and D1ð2430Þ poles resulting fromthe full calculation, with the 20 MM channels from Table I,we first give the pole positions for the cases that only two(D��, L ¼ 0; 2) or one (D��, L ¼ 0) MM channels are

FIG. 3. Ds1ð2460Þ (dashed) and Ds1ð2536Þ (solid) pole trajec-tories as a function of �, for r0 ¼ 3:12 GeV�1. Bullets corre-spond to � ¼ 1:19; vertical line shows the D�K threshold.

FIG. 2. D1ð2420Þ pole trajectories as a function of �, for r0 ¼3:3–3:5 GeV�1 (left to right). Solid curve and bullets correspondto r0 ¼ 3:40 GeV�1 and � ¼ 1:30, respectively.

TABLE III. Poles of ground-state (n ¼ 0) and first radiallyexcited (n ¼ 1) AV charmed mesons. Parameters: � ¼ 1:30(1.19) and r0 ¼ 3:40ð3:12Þ GeV�1, for c �q (c�s) states.

Quark content Radial excitation Pole in MeV

c �q 0 2439� i� 3:5c �q 0 2430� i� 191c�s 0 2540� i� 0:7c�s 0 2452� i� 0:0c �q 1 2814� i� 7:8c �q 1 2754� i� 47:2c�s 1 2915� i� 6:7c�s 1 2862� i� 25:7

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included. The last two poles then correspond to calcula-tions with the full 20 MM channels but only one quark-antiquark channel, viz. 3P1 or 1P1. Notice that only onepole is found when the number of quark-antiquark or MMchannels is equal to 1. This confirms our above conjecturethat the number of poles for each bare HO level is given byminðNq �q; NMM).

V. SUMMARYAND CONCLUSIONS

In the foregoing, we have managed to rather accuratelyreproduce the masses and widths of the D1ð2420Þ,D1ð2430Þ, Ds1ð2536Þ, and Ds1ð2460Þ with only two freeparameters, one of which is already constrained by pre-vious model calculations, as well as by reasonable esti-mates for the size of these mesons. Crucial is theapproximate decoupling from the continuum of one com-bination of 3P1 and

1P1 components, which amounts to amixing angle close to 35�. Namely, if we express a QBSCas jQBSCi ¼ � sin�j3P1i þ cos�j1P1i, it decouples fromthe L ¼ 0D�� channel (for c �q) orD�K channel (for c �s), if

� ¼ arccosffiffiffiffiffiffiffiffi2=3

p � 35:26� (see Tables I and II). Inclusionof the other, practically all closed, channels apparentlychanges the picture only slightly in our formalism. Thisresult is in full agreement with the findings in Ref. [12].However, in the present approach this particular mixing[20] comes out as a completely dynamical result, and is notchosen by us beforehand. Moreover, the bare-mass degen-eracy of 3P1 and 1P1 states is adequately lifted via thedecay couplings in Tables I and II, dispensing with the

usual ~S � ~L splitting. Also note that the occurrence of(approximate) bound states in the continuum for AVcharmed mesons had already been conjectured by two ofus [17], based on more general arguments.

The puzzling discrepancy between the AV-V mass split-tings in the c �q and c �s sectors is resolved in our calculationby dynamical, nonperturbative coupled-channel effects. Asimilar phenomenon we have observed before [21] for theD�

0 (2300–2400) [1] resonance, and may be related to an

effective Adler-type zero [22] in theD�� andD� channelsin the AV and scalar c �n cases, respectively, owing to thesmall pion mass.

Summarizing, we have reproduced the whole pattern ofmasses and widths of the AV charmed mesons dynami-cally, by coupling the most important open and closedtwo-meson channels to bare c �q and c �s states containing

both 3P1 and 1P1 components. The dynamics of thecoupled-channel equations straightforwardly leads to onepair of strongly shifted states and another pair of QBSCs.Ironically, the state that shifts most in mass, namely, theDs1ð2460Þ, ends up as the narrowest resonance. This em-phasizes the necessity [23] to deal with unquenched mesonspectroscopy in a fully nonperturbative framework.One might argue that these conclusions will depend on

the specific model employed. Admittedly, our numericalresults could change somewhat if slightly different baremasses for the AV charmed mesons were chosen,non-S-wave decay channels were included as well, or adifferent scheme was used to calculate the decreasingcouplings of the higher recurrences. Nevertheless, we areconvinced the bulk of our results will not change, mostnotably the appearance of QBSCs and the large shifts oftheir partner states, as the almost inevitable consequence ofexact nonperturbative coupled-channel dynamics.

ACKNOWLEDGMENTS

This work was supported in part by the Fundacao para aCiencia e a Tecnologia of the Ministerio da Ciencia,Tecnologia e Ensino Superior of Portugal, under ContractNo. CERN/FP/116333/2010 and Grant No. SFA-2-91/CFIF.

APPENDIX: THREE-MESON COUPLINGS

The ground-state couplings in Tables I and II are ob-tained by multiplying the isospin recouplings given inTable V with the JPC couplings in Table VI, for an OZIA

TABLE IV. Poles of AV c �q mesons, for different sets ofincluded channels. Parameters: � ¼ 1:30, r0 ¼ 3:40 GeV�1.

c �q channels MM channels Pole 1 (MeV) Pole 2 (MeV)

3P1 þ 1P1 20 2430� i� 191 2439� i� 33P1 þ 1P1 2 2402� i� 36 2441� i� 13P1 þ 1P1 1 2431� i� 39 -3P1 20 2409� i� 65 -1P1 120 2425� i� 96 -

TABLE V. Squared isospin recouplings for the three-mesonprocess MA ! MB þMC, with MA ¼ c�s or c �q.

MA MB MC g2I

Ds1 Ds, D�s �s, � 1=3

Ds1 D, D� K, K� 2=3D1 Ds, D

�s K, K� 1=3

D1 D, D� �, � 1=2D1 D, D� �n, ! 1=6

TABLE VI. Squared ground-state coupling constants for thethree-meson process MA ! MB þMC, with JPCðMAÞ ¼ 1þ�,and MA, MB belonging to the lowest pseudoscalar or vectornonet.

JPCðMAÞ JPCðMBÞ JPCðMCÞ LMBMCSMBMC

g2ðn¼0Þ1þþ 0�þ 1�� 0 1 1=181þþ 0�þ 1�� 2 1 5=721þþ 1�� 1�� 0 1 0

1þþ 1�� 1�� 2 2 5=241þ� 0�þ 1�� 0 1 1=361þ� 0�þ 1�� 2 1 5=361þ� 1�� 1�� 0 1 1=361þ� 1�� 1�� 2 1 5=36

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process MA ! MB þMC based on 3P0 q �q creation [16].

For clarity, we represent here all couplings by rationalnumbers. Note that �n and �s in Table V stand for

the pseudoscalar I ¼ 0 states ðu �uþ d �dÞ= ffiffiffi2

pand s�s,

respectively. Then, we get the couplings to the physical� and�0 mesons by applying a mixing angle—in the flavorbasis—of 41.2�, as in Ref. [21], second paper. For the !and � we assume ideal mixing.

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