We summarize some features of the vector meson dominance model which was
recently proposed for studying radiative decays involving the scalar mesons.
Using the experimental values of $\Gamma(a_0 \to \gamma\gamma)$, $\Gamma(f_0
\to \gamma\gamma)$ and $\Gamma(\phi \to a_0 \gamma)$ as inputs, we show that
the model predicts a large hierarchy between $\Gamma(a_0 \to \omega \gamma)$
and $\Gamma(a_0 \to \rho \gamma)$ as well as between $\Gamma(f_0 \to \omega
\gamma)$ and $\Gamma(f_0 \to \rho \gamma)$.Comment: 7pages, Talk given by M. Harada at International Symposium on Hadron
  Spectroscopy, Chiral Symmetry and Relativistic Description of Bound Systems,
  February 24-26, 2003, Nihon University, Tokyo, Japa

Black, Deirdre

Harada, Masayasu

Schechter, Joseph

English

arXiv

We summarize some features of the vector meson dominance model which was recently proposed for studying radiative decays involving the scalar mesons. Using the experimental values of \Gamma(a_0 \to \gamma\gamma), \Gamma(f_0 \to \gamma\gamma) and \Gamma(\phi \to a_0 \gamma) as inputs, we show that the model predicts a large hierarchy between \Gamma(a_0 \to \omega \gamma) and \Gamma(a_0 \to \rho \gamma) as well as between \Gamma(f_0 \to \omega \gamma) and \Gamma(f_0 \to \rho \gamma)

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 6-7-2003 Radiative Decays Involving a0(980) and f0(980) in the Vector Meson Dominance Model Joseph Schechter Department of Physics, Syracuse University, Syracuse, NY Deirdre Black Jefferson Lab Masayasu Harada Nagoya University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Schechter, Joseph; Black, Deirdre; and Harada, Masayasu, "Radiative Decays Involving a0(980) and f0(980) in the Vector Meson Dominance Model" (2003). Physics. 278. https://surface.syr.edu/phy/278 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-ph/0306065v1  7 Jun 20031DPNU-03-16Radiative Decays involving a0(980) and f0(980)in the Vector Meson Dominance Model ∗)Deirdre Black(a), Masayasu Harada(b) and Joseph Schechter(c)(a) Theory Group, Jefferson Lab, 12000 Jefferson Ave., Newport News, VA 23606,USA(b) Department of Physics, Nagoya University, Nagoya, 464-8602, Japan(c) Department of Physics, Syracuse University Syracuse, NY 13244-1130, USA(Received February 1, 2008)We summarize some features of the vector meson dominance model which was recentlyproposed for studying radiative decays involving the scalar mesons. Using the experimentalvalues of Γ (a0 → γγ), Γ (f0 → γγ) and Γ (φ → a0γ) as inputs, we show that the modelpredicts a large hierarchy between Γ (a0 → ωγ) and Γ (a0 → ργ) as well as between Γ (f0 →ωγ) and Γ (f0 → ργ).§1. IntroductionAccording to the recent theoretical and experimental analysis, there is a pos-sibility that nine light scalar mesons exist below 1 GeV, and they form a scalarnonet.1) In addition to the well established f0(980) and a0(980) evidence of bothexperimental and theoretical nature for a very broad σ (≃ 560) and a very broadκ (≃ 900) has been presented. However, the properties of the nonet members suchas quark structure and interactions with other mesons are not well known. It isinteresting to study the properties of these light scalar mesons, which would be ofgreat importance for a full understanding of QCD in its nonperturbative low energyregime.In particular, the reactions φ → f0γ and φ → a0γ have recently been observed2)with good accuracy and are considered as useful probes of scalar properties. Thetheoretical analysis was initiated by Achasov and Ivanchenko3) and followed up bymany others.4) The models employed are essentially variants of the single K mesonloop diagram to which a φ-type vector meson, a photon and two pseudoscalars or ascalar are attached.In this paper we summarize some features of the vector meson dominance modelwhich was recently proposed in Ref. 5) for studying radiative decays involving thescalar mesons. Our model predicts that a large hierarchy between Γ (a0 → ωγ) andΓ (a0 → ργ) as well as between Γ (f0 → ωγ) and Γ (f0 → ργ).∗) Talk given by M. Harada at International Symposium on Hadron Spectroscopy, Chiral Sym-metry and Relativistic Description of Bound Systems, February 24-26, 2003, Nihon University,Tokyo, Japan.2§2. Vector Meson Dominance ModelFramework of the model proposed in Ref. 5) is that of a standard non-linearchiral Lagrangian containing, in addition to the pseudoscalar nonet matrix field φ,the vector meson nonet matrix ρµ and a scalar nonet matrix field denoted N . Underchiral unitary transformations of the three light quarks; qL,R → UL,R ·qL,R, the chiralmatrix U = exp(2iφ/Fπ), where Fπ ≃ 0.131GeV, transforms as U → UL · U · U †R.The convenient matrix K(UL, UR, φ)6) is defined by the following transformationproperty of ξ (U = ξ2): ξ → UL ·ξ ·K† = K ·ξ ·U †R, and specifies the transformationsof “constituent-type” objects. The fields we need transform asN → K · N · K† ,ρµ → K · ρµ · K† +ig̃K · ∂µK† ,Fµν(ρ) = ∂µρν − ∂νρµ − ig̃ [ρµ , ρν ] → K · Fµν · K† , (2.1)where the coupling constant g̃ is about 4.04. One may refer to Ref. 7) for ourtreatment of the pseudoscalar-vector Lagrangian and to Ref. 8) for the scalar addi-tion. The entire Lagrangian is chiral invariant (modulo the quark mass term inducedsymmetry breaking pieces) and, when electromagnetism is added, gauge invariant.In Ref. 5), the strong trilinear scalar-vector-vector terms were included into theeffective Lagrangian asLSV V = βA ǫabcǫa′b′c′ [Fµν(ρ)]aa′ [Fµν(ρ)]bb′ Ncc′+ βB Tr [N ] Tr [Fµν(ρ)Fµν(ρ)]+ βC Tr [NFµν(ρ)] Tr [Fµν(ρ)]+ βD Tr [N ] Tr [Fµν(ρ)] Tr [Fµν(ρ)] . (2.2)Chiral invariance is evident from (2.1) and the four flavor-invariants are needed forgenerality. (A term ∼ Tr(FFN) is linearly dependent on the four shown). Actu-ally the βD term will not contribute in our model so there are only three relevantparameters βA, βB and βC . Equation (2.2) is analogous to the PV V interaction∗)which was originally introduced as a πρω coupling a long time ago.10) One can nowcompute the amplitudes for S → γγ and V → Sγ according to the diagrams ofFig. 1.The decay matrix element for S → γγ is written as (e2/g̃2)XS ×(k1 · k2 ǫ1 · ǫ2 −k1 · ǫ2 k2 · ǫ1)where ǫµ stands for the photon polarization vector. It is related to thewidth byΓ (S → γγ) = α2 π4m3S∣∣∣∣XSg̃2∣∣∣∣2, (2.3)∗) It was shown9) that the complete vector meson dominance (VMD) is violated in the ω →π0π+π− decay which is expressed by PV V interactions. However, since the VMD is satisfied inother processes such π0 → γγ∗ as well as in the electromagnetic form factor of pion, we here assumethat it holds in the processes related to SV V interactions.3(b)21γ (    , ε  )kγ(  ,ε)vV(p,   )ε2kγ(   , ε  )100ρ ,ω,φρ ,ω,φSk0ρ ,ω,φ(a)SFig. 1. Feynman diagrams for (a) S → γγ and (b) V → Sγ.and XS takes on the specific forms:Xσ =49βA(√2s − 4c)+83βB(c −√2s),Xf0 = −49βA(√2c + 4s)+83βB(√2c + s),Xa0 =4√23βA . (2.4)Here α = e2/(4π), s = sin θS and c = cos θS where the scalar mixing angle, θS isdefined from5)(σf0)=(c −ss c)(N33(N11 + N22 )/√2). (2.5)Furthermore ideal mixing for the vectors, with ρ0 = (ρ11−ρ22)/√2, ω = (ρ11 +ρ22)/√2,φ = ρ33, was assumed for simplicity.Similarly, the decay matrix element for V → Sγ is written as (e/g̃)CSV ×[p · kǫV · ǫ − p · ǫk · ǫV ]. It is related to the width byΓ (V → Sγ) = α3∣∣kSV∣∣3∣∣∣∣CSVg̃∣∣∣∣2, (2.6)where kSV = (m2V − m2S)/(2mV ) is the photon momentum in the V rest frame. Forthe energetically allowed V → Sγ processes we haveCf0φ =2√23βAc −43βB(√2c + s)+√23βC(c −√2s),Cσφ = −2√23βAs −43βB(c −√2s)− 23βC(c +1√2s),Ca0φ =√2 (βC − 2βA) ,Cσω =2√23βA(c +√2s)+2√23βB(c −√2s)− 23βC(√2c + s),Cσρ0 = −2√2βAc + 2√2βB(c −√2s). (2.7)In addition, the same model predicts amplitudes for the energetically allowedS → V γ processes: f0 → ωγ, f0 → ρ0γ, a00 → ωγ, a00 → ρ0γ and, if κ0 is sufficiently4heavy κ0 → K∗0γ. The corresponding width isΓ (S → V γ) = α∣∣kVS∣∣3∣∣∣∣DVSg̃∣∣∣∣2, (2.8)where kVS = (m2S − m2V )/(2mS) andDωf0 =23βA(−2c +√2s)+23βB(2c +√2s)+23βC(c −√2s),Dρ0f0= −2√2βAs + 2βB(2c +√2s),Dωa0 = 2βC ,Dρ0a0=43βA ,DK∗0κ0 = −83βA . (2.9)§3. ResultsLet us summarize main points of the results obtained in Ref. 5) together withnew results from our recent analysis.11)We should stress again that all the different decay amplitudes are described bythe parameters βA, βB and βC . Below we shall first illustrate the procedure for themodel of a single putative scalar nonet.8)We determine the value of βA from the a0 → γγ process. Substituting Γexp(a0 →γγ) = (0.28 ± 0.09) keV (obtained using12) B(a0 → KK̄)/B(a0 → ηπ) = 0.177 ±0.024) into Eqs. (2.3) and (2.4) yieldsβA = (0.72 ± 0.12)GeV , (3.1)where we assumed positive in sign. By using this value, the value of βC is deter-mined from Γexp(φ → a0γ) = (0.47 ± 0.07) keV (obtained by assuming φ → ηπ0γ isdominated by φ → a0γ) and Eq. (2.7) asβC = (7.7 ± 0.5 , −4.8 ± 0.5)GeV−1 . (3.2)We stress that the values of βA and βC obtained above are independent of the mixingangle θS . It should be noticed that |βA| is almost an order of magnitude smaller than|βC |. As we can see from Eq. (2.9), the amplitude Dωa0 is given by βC while Dρ0a0 isgiven by only βA. Then, the large hierarchy between βC and βA implies that thereis a large hierarchy between Γ (a0 → ωγ) and Γ (a0 → ργ). Actually, by using thevalues of βA and βC given in Eqs. (3.1) and (3.2), they are estimated asΓ (a0 → ωγ) = (641 ± 87 , 251 ± 54) keV ,Γ (a0 → ργ) = 3.0 ± 1.0 keV . (3.3)This implies that there is a large hierarchy between Γ (a0 → ωγ) and Γ (a0 → ργ)which is caused by an order of magnitude difference between |βC | and |βA|.5We next determine the value of βB from the f0 → γγ process. Xf0 in Eq. (2.4)depends on βB as well as on βA and the mixing angle θS. Here we take the mixingangle asθS ≃ −20◦ , (3.4)which is characteristic of qqq̄q̄ type scalars.8) By using this and the value of βA inEq. (3.1), Γexp(f0 → γγ) = 0.39 ± 0.13 keV yieldsβB = (0.61 ± 0.10 , −0.62 ± 0.10)GeV−1 . (3.5)This implies that |βB | is on the order of |βA|, and almost an order of magnitudesmaller than |βC |. Equation (2.9) shows that Dωf0 includes βC while Dρf0does not.Thus, we have a large hierarchy between decay widths of f0 → ωγ and f0 → ργ:The typical predictions are given byΓ (f0 → ωγ) = (88 ± 17) keV ,Γ (f0 → ργ) = (3.3 ± 2.0) keV . (3.6)This implies that there is a large hierarchy between Γ (f0 → ωγ) and Γ (f0 → ργ)which is caused by the fact that |βC | is an order of magnitude larger than |βA| and|βB |.Let us check the dependence of the above results on the choice of the scalarmixing angle θS. In Ref. 8), the value of θS ≃ −90◦ was obtained as another solutionto reproduce the masses of the lightest scalar nonet, although the predicted value off0-π-π coupling is much larger than the value obtained in Ref. 13) by fitting to theππ scattering amplitudeAs we stressed above, the values of βA and βC are independent of the scalarmixing angle θS. The value of βB determined from Γ (f0 → γγ) becomesβB = (1.1 ± 0.1 , 0.12 ± 0.13)GeV−1 . (3.7)Then the typical predictions for Γ (f0 → ωγ) and Γ (f0 → ργ) are given byΓ (f0 → ωγ) = (86 ± 16) keV ,Γ (f0 → ργ) = (3.4 ± 3.2) keV . (3.8)These predictions are very close to the ones in Eq. (3.6). This can be understood bythe following consideration: From the expression of Dωf0 in Eq. (2.9), we can see thatit is dominated by the term including βC which is proportional to (cos θS−√2 sin θS).Then, the approximate relationcos(−20◦) −√2 sin(−20◦) ≃ cos(−90◦) −√2 sin(−90◦) ≃ 1.4 (3.9)implies that the value of Dωf0 for θS = −90◦ is close to that for θS = −20◦, and thusΓ (f0 → ωγ) for θS = −90◦ to that for θS = −20◦. As for Γ (f0 → ργ) we note thatthe following relation is satisfied for Xf0 in Eq. (2.4) and Dρ0f0in Eq. (2.9):3Xf0 − 2√2Dρ0f0= −43√2βA(c −√2s) . (3.10)6Since we use the experimental value of Γ (f0 → γγ), i.e., Xf0 as an input, this relationimplies that the predicted value of Γ (f0 → ργ) for θS = −90◦ is roughly equal tothat for θS = −20◦. From the above consideration we conclude that there is a largehierarchy between Γ (f0 → ωγ) and Γ (f0 → ργ) for θS = −20◦ and θS = −90◦:Γ (f0 → ωγ) ≫ Γ (f0 → ργ) for θS = −20◦ and θS = −90◦ . (3.11)§4. DiscussionIn this paper we showed the predictions of our model only for Γ (a0 → ωγ),Γ (a0 → ργ), Γ (f0 → ωγ) and Γ (f0 → ργ). Predictions on several other processessuch as Γ (σ → γγ) and Γ (φ → σγ) can be seen in Ref. 5).The value of Γ (φ → f0γ) predicted in Ref. 5) is considerably smaller than theexperimental value of Γ (φ → f0γ). We may need to include the effect of K-loopwhich would give a large enhancement as shown in Ref. 3). We leave the analysiswith K-loop corrections to future publications.11)AcknowledgementsWe are happy to thank N. N. Achasov for a suggested correction to the firstversion of Ref. 5) and with A. Abdel-Rahiem and A. H. Fariborz for very helpfuldiscussions. D.B. wishes to acknowledge support from the Thomas Jefferson NationalAccelerator Facility operated by the Southeastern Universities Research Association(SURA) under DOE contract number DE-AC05-84ER40150. The work of J.S. issupported in part by DOE contract DE-FG-02-85ER40231.References1) See, e.g., the confereproceedings, S. Ishida et al, “Possible existence of the sigma mosonand its implication to hadron physics”, KEK Proceedings 2004, Soryushiron Kenkyu 102,No. 5, 2001; D. Amelin and A.M. Zaitsev, “Hadron spectroscopy”, Ninth internationalconference on hadron spectroscopy, Protvino, Russia (2001).2) M. N. Achasov et al. [SND Collaboration], Phys. Lett. B 479, 53 (2000); R. R. Akhmetshinet al. [CMD-2 Collaboration], Phys. Lett. B 462, 380 (1999); A. Aloisio et al. [KLOECollaboration], arXiv:hep-ex/0107024.3) N. N. Achasov and V. N. Ivanchenko, Nucl. Phys. B 315, 465 (1989).4) F. E. Close, N. Isgur and S. Kumano, Nucl. Phys. B 389, 513 (1993); N. N. Achasov andV. V. Gubin, Phys. Rev. D 56, 4084 (1997); Phys. Rev. D 57, 1987 (1998); N. N. Achasov,V. V. Gubin and V. I. Shevchenko, Phys. Rev. D 56, 203 (1997); J. L. Lucio Martinez andM. Napsuciale, Phys. Lett. B 454, 365 (1999); arXiv:hep-ph/0001136; A. Bramon, R. Es-cribano, J. L. Lucio M., M. Napsuciale and G. Pancheri, Phys. Lett. B 494, 221 (2000);F. E. Close and A. Kirk, Phys. Lett. B 515, 13 (2001); N. N. Achasov, S. A. Devyanin andG. N. Shestakov, Phys. Lett. B 88, 367 (1979); N. N. Achasov and A. V. Kiselev, Phys.Lett. B 534, 83 (2002).5) D. Black, M. Harada and J. Schechter, Phys. Rev. Lett. 88, 181603 (2002); D. Black,M. Harada and J. Schechter, arXiv:hep-ph/0303223.6) C. G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177, 2247 (1969).7) M. Harada and J. Schechter, Phys. Rev. D 54, 3394 (1996).8) D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D 59, 074026 (1999).9) See, e.g., M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988); M. Haradaand K. Yamawaki, to appear in Phys. Rept., arXiv:hep-ph/0302103.10) M. Gell-Mann, D. Sharp and W. G. Wagner, Phys. Rev. Lett. 8, 261 (1962). A general7formulation of the PV V interaction in a gauge invariant manner was given in: T. Fuji-wara, T. Kugo, H. Terao, S. Uehara and K. Yamawaki, Prog. Theor. Phys. 73, 926 (1985);O. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30, 594 (1984); O. Kaymak-calan and J. Schechter, Phys. Rev. D 31, 1109 (1985); P. Jain, R. Johnson, U. G. Meissner,N. W. Park and J. Schechter, Phys. Rev. D 37, 3252 (1988).11) D. Black, M. Harada and J. Schechter, in preparation.12) D. E. Groom et al. [Particle Data Group Collaboration], Eur. Phys. J. C 15, 1 (2000) and2001 partial update for edition 2002 (URL: http://pdg.lbl.gov).13) M. Harada, F. Sannino and J. Schechter, Phys. Rev. D 54, 1991 (1996).

Radiative Decays Involving a0(980) and f0(980) in the Vector Meson Dominance Model

https://surface.syr.edu/cgi/viewcontent.cgi?article=1277&amp;context=phy

Radiative Decays involving a0(980) and f0(980) in the Vector Meson Dominance Model

Abstract

Similar works

Full text

Available Versions

Syracuse University Research Facility and Collaborative Environment