Charmed Mesons and Charmonia: Three-Meson Strong Couplings. Couplings, form factors & decay constants

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1 Charmed Mesons and Charmonia: Three-Meson Strong Couplings Wolfgang Luha 1 Dmitri Melikhov 3 Hagop Sazdjian 4 and Silvano Simula 5 1 HEPHY Austrian Aademy of Sienes Vienna Austria SINP Mosow State University Russia 3 Faulty of Physis University of Vienna Austria 4 IPN CNRS/INP3 Universitè Paris-Sud Orsay Frane 5 INFN Sezione di Roma Tre Roma Italy Couplings form fators & deay onstants By use of a relativisti dispersion approah relying on the onstituent-quark modelwestudythestrongouplingsg PP V andg PV V forvetormesonsv withpolarizationvetorsε µ andpseudosalarmesonspintheamplitudes P (p )V(q) P(p 1 ) = g PP V (p 1 +p ) µ ε µ(q) V (p )V(q) P(p 1 ) = g PV V ǫ ε (q)ε (p )p 1 p q p 1 p. Ofpartiularinteresttouswillbetheasewhenthereisatleastoneofthe harmoniaj/ψorη amongthismesontriple.thesestrongouplingsenter intotheresiduesofpolesinorrespondingtransitionformfatorsforq > 0 arising from intermediate meson states. The form fators relevant for us are P (p ) q γ µ q 1 P(p 1 ) = F P P + (q )(p 1 +p ) µ + V(p ) q γ µ q 1 P(p 1 ) = V P V (q ) M P +M V ǫ µε (p )p 1 p V(p ) q γ µ γ 5 q 1 P(p 1 ) = iq µ (ε (p )p 1 ) M V q A P V 0 (q )+. 1

2 Withthedeayonstantsofthevetorandpseudosalarmesonsf V andf P defined in terms of the meson-to-vauum transition amplitude of the vetor quarkurrent q γ µ q 1 oraxial-vetorquarkurrent q γ µ γ 5 q 1 aordingto 0 q 1 γ µ q V(q) = f V M V ε µ (q) 0 q 1 γ µ γ 5 q P(q) = if P q µ thepolesatpseudosalarandvetorresonanesp R andv R areoftheform F+ P P (q ) = g PP V R f VR 1 + M VR 1 q MV R V P V (q ) = (M V +M P )g PVVR f VR M VR 1 1 q M V R + A P V 0 (q ) = g PP R V f PR M V 1 1 q M P R +. Quark-model-based dispersion approah WeomputetheformfatorsF P P + (q )V P V (q )anda P V 0 (q )within the framework of a relativisti onstituent-quark piture[1]. To this end we must relate the urrents defining the form fators to their onstituent-quark (Q) ounterparts: this task is easily aomplished for heavy-quark urrents q 1 γ µ q = g V Q1 γ µ Q + q 1 γ µ γ 5 q = g A Q1 γ µ γ 5 Q + byintroduingformfatorsg V andg A butnotsosimpleforlightquarks[]. Numerially we employ[3] for the onstituent-quark masses and ouplings m d = m u = 0.3 GeVm s = 0.35 GeVm = 1.45 GeVandg V = g A = 1. The relativisti dispersion formalism allows us to represent the quantities of interest by integrals over invariant masses of intermediate quark antiquark states of spetral densities derived from(one-loop) Feynman diagrams and wave funtions of the involved pseudosalar or vetor mesons[4] of the form φ PV (s) = π s (m 1 m ) w PV (k ) s (m1 m) s 3/4 k = (s m 1 m ) 4m 1 m 4s dkk w PV (k ) = 1.

3 Fortheradialmesonwavefuntionsw PV (k )weassumegaussianshapes: ( ) w PV (k ) exp k βpv. Thedeayonstantsf PV beomespetralintegralsofdensitiesρ PV (s)and theformfatorsf+ P P (q )V P V (q )anda P V 0 (q )generiallylabelled F(q )doubledispersionintegralsofdoublespetraldensities (s 1 s q ): f PV = dsφ PV (s)ρ PV (s) F(q ) = ds 1 φ 1 (s 1 ) ds φ (s ) (s 1 s q ). D ( ) (s) and mesonmassesmdeayonstantsf andslopeparametersβ[5]: Meson D D D s D s η J/ψ M (GeV) f (MeV) 06±8 60±10 48±.5 311± ±.4 405±7 β (GeV) Feynmangraphsfortransitionsinduedbyquarkvetorurrents Q 1 γ µ Q : η γµ η J/ ψ D γµ d DD * η J/ψ γµ d d DD* All form fators F omputed off their resonanes R an be interpolated by F(q ) = ( 1 q M R F(0) )( 1 σ 1q ) ResF() = + σ q 4 4 F(0) 1 σ 1 +σ. From the residues involving the meson masses deay onstants and strong ouplings the latter are derived by ombined fits if present more than one. 3

4 Strong ouplings: η η J/ψ and η J/ψJ/ψ As one example representative for the general situation onsider the strong ouplingg η η ψ oftwopseudosalarη mesonsandonevetorj/ψ meson[6] by realizing that this quantity enters in and therefore an be extrated from the residue ResF η η + (Mψ ) of the pole at q = Mψ of the form fator F η η + (q )forthe( elasti )transitionη η enabledbytheurrent γ µ ouplingwithdeayonstantf ψ tothevetorresonanej/ψand the residue ResA η ψ 0 (Mη ) of the pole at q = Mη of the form fator A η ψ 0 (q )forthetransitionη J/ψinduedbytheurrent γ µ γ 5 thatoupleswithdeayonstantf η tothepseudosalarresonaneη : ResF η η + (Mψ) f ψ = g η η ψ ResA η ψ 0 (M f η η M ) = g η η ψ. ψ M ψ Upon determination of the meson wave-funtion parameters β by requiring thedispersionrepresentationofthedeayonstantsf PV toreproduetheir known values the strong ouplings may be alulated individually for eah transition of interest from the spetral representation of its assoiated form fator: our ouplings off-resonane behaviour exhibits exellent agreement. Behaviouroftheoff-shellstrongouplingg η η ψ(x)asafuntionofx q R = J/ψη fromthetwotransitionsη η (red)andη J/ψ (blue): g Η Η Ψ x F Η Η A 0 Η Ψ x Q M R Aombinedfitwiththefourparametersg η η ψa η ψ 0 (0)σ F 1 thenyields[6] g η η ψ = 5.8±1.7. 4

5 Along a similar route we find[6] for the strong oupling of one pseudosalar η mesonandtwovetorj/ψmesonsg η ψψenteringinonlyonetransition g η ψψ = (10.6±1.5) GeV 1. Strong ouplings: η and J/ψ to D and D Allowing also for urrents with a d or s quark(and merging strong-oupling multiple ourrenes all of them showing nearly perfet onord) we get[6] forthestrongj/ψorη ouplingstonon-strangeharmedmesonsd ( ) g DDψ = 6.04±1.43 g DD ψ = (10.7±0.4) GeV 1 g DD η = 15.51±0.45 g D D η = (9.76±0.3) GeV 1 andforthestrongj/ψorη ouplingstostrangeharmedmesonsd ( ) s g Ds D s ψ = 3.83±0.78 g Ds D s ψ = (9.6±0.8) GeV 1 g Ds D sη = 14.15±0.5 g D s D s η = (8.7±0.37) GeV 1. Dependene of the off-shell strong ouplings g D ˆDψ (x) = M ψ ˆxA D ψ f D 0 (q ) (left blue) g DDˆψ(x) = M ψ ˆxF D D f ψ + (q ) (left red) ˆx 1 x g D ˆD ψ (x) (rightred)g DD ˆψ(x)(rightblue)andgˆDD q ψ (x)(rightgreen)onx : g DDΨ x A 0 D Ψ F D D x Q M R g DD Η x A 0 Η D F Η D A 0 D D x Q M R 5

6 Observations omparison and onlusions Suessful extrapolations of all interpolated results for strong ouplings derivedatq < 0onfirmthepreseneofthepolesexpetedforq > 0. ConerningSU(3)breakingthenetoutomeofreplaingadquarkby thesquarkisaredutionoftheaffetedstrongouplingbyabout10%. DespiteundeniablesimilaritiesoftheapproahesourD ( ) (s) ouplings[6] aremorethantwieaslargeastheresultsarisingfromqcdsumrules. Comparison of our strong-oupling preditions with QCD sum-rule results: Coupling g DDψ g DD ψ (GeV 1 ) g Ds D s ψ g Ds D sψ (GeV 1 ) This work 6.04± ± ± ±0.8 Sum rules 11.6±1.8 [7] 4.0±0.6 [7] 11.96±1.34 [8] 4.30±1.53 [9] Referenes [1] W. Luha F. F. Shöberl & D. Gromes Phys. Rep. 00 (1991) 17; F. Cardarelli E. Pae G. Salmè&S.SimulaPhys. Lett.B357(1995)67arXiv:nul-th/ ;R.N.Faustov&V.O. Galkin Z. Phys. C 66 (1995) 119. [] D. Melikhov & B. Steh Phys. Rev. D 74 (006) 0340 arxiv:hep-ph/060603; W. Luha D. Melikhov & S. Simula Phys. Rev. D 74 (006) arxiv:hep-ph/ [3] D. Melikhov & B. Steh Phys. Rev. D 6 (000) arxiv:hep-ph/ [4] D. Melikhov Eur. Phys. J. diret C4 (00) arxiv:hep-ph/ [5] C. T. H. Davies et al. Phys. Rev. D 8 (010) arxiv: [hep-lat]; W. Luha D. Melikhov & S. Simula Phys. Lett. B 701 (011) 8 arxiv: [hep-ph]; D. Bečirević et al. JHEP 10 (01) 04 arxiv: [hep-lat]; G. C. Donald et al. Phys. Rev. D 86 (01) arxiv: [hep-lat]; W. Luha D. Melikhov& S. Simula Phys. Lett. B 735 (014) 1 arxiv: [hep-ph]; K. A. Olive et al. (PDG) Chin. Phys. C 38 (014) [6] W. Luha D. Melikhov H. Sazdjian & S. Simula Phys. Rev. D 93 (016) (016) 01990(E) arxiv: [hep-ph]. [7] R. D. Matheus et al. Int. J. Mod. Phys. E 14 (005) 555. [8] B. Osório Rodrigues M. E. Brao & M. Chiapparini Nul. Phys. A 99 (014) 143; arxiv: [hep-ph]. [9] B. Osório Rodrigues et al. Eur. Phys. J. A 51 (015) 8; arxiv: [hep-ph]. 6