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STRONG AND RADIATIVE MESON DECAYS IN A

arXiv:hep-ph/9302245v1 10 Feb 1993STRONG AND RADIATIVE MESON DECAYS IN AGENERALIZED NAMBU–JONA-LASINIO MODEL 1V. Bernard†, A.H. Blin‡, B. Hiller‡, U.-G. Meißner§2 and M.C.

Ruivo‡†Physique Th´eorique, Centre de Recherches Nucl´eaires et Universit´e Louis Pasteur deStrasbourg. B.P.

20, F-67037 Strasbourg Cedex 2, France‡Centro de F´ısica Te´orica (INIC), Departamento de F´ısica da Universidade, P-3000Coimbra, Portugal§Institut f¨ur Theoretische Physik, Universit¨at Bern, Sidlerstrasse 5, CH-3012 Bern,SwitzerlandWe investigate strong and radiative meson decays in a generalized Nambu–Jona-Lasiniomodel. The one loop order calculation provides a satisfactory agreement with the datafor the mesonic spectrum and for radiative decays.

Higher order effects for strong decaysof ρ and K∗are estimated to be large. We also discuss the role of the flavour mixingdeterminantal interaction.1Work supported in part by Deutsche Forschungsgemeinschaft, Schweizerischer Nationalfonds, JNICTno.

PMCT/C/CEN/72/90, GTAE and CERN no. PCERN-FAE-74-91.2Heisenberg fellow1

1IntroductionThe Nambu–Jona-Lasinio model [1] and generalizations thereof [2,3,4] have been usedextensively to study the properties of mesons in free space and at finite temperaturesand densities. It is an effective field theory of non-linearly interacting quarks which ex-hibits spontaneous and explicit dynamical chiral symmetry breaking.

In the case of threeflavours, it is mandatory to incorporate the ’t Hooft six-fermion interaction to describethe breaking of the axial U(1) symmetry [3]. Mesons are bound quark- antiquark pairsin this approach and their properties can readily be calculated by solving the pertinentBethe-Salpeter equations.

However, no systematic study of three-point functions like thestrong and radiative meson decays, has been performed so far. Previous attempts werelimited to the leading term in the momentum expansion of the underlying quark-mesonvertices [5] and thus do not account for the full dynamics of the model.

Furthermore,these decays are also a good testing ground to find out the limitations of the model. Thiswill be discussed in some detail later on.In what follows, we will work in flavour SU(3) and use the following LagrangianL =G1[( ¯ψλiψ)2 + ( ¯ψiγ5λiψ)2]+G2[( ¯ψλaγµψ)2 + ( ¯ψλaγµγ5ψ)2]+K[det{ ¯ψ(1 + γ5)ψ} + det{ ¯ψ(1 −γ5)ψ}](1)where the flavour index i runs from 0 to 8 with λ0 =q2/3 1, the λa are color matrices(a = 1, ..., 8).

As it stands, the Lagrangian is characterized by a few parameters: The twofour–fermion coupling constants G1 and G2, the six–fermion coupling K and the cutoffΛ,which is necessary to regularize the divergences. We will use a covariant four-momentumcutoffΛ = 1 GeV.

Furthermore, to account for the explicit symmetry breaking, a quarkmass term has to be added. We work in the isospin limit mu = md and will use thecurrent quark masses to fit the meson spectrum.

Clearly, the coupling G1 is related to theproperties of the pseudoscalar Goldstone bosons, G2 can be fixed from the ρ-meson massand K is necessary to give the ηη′ mass splitting. This completely specifies the modeland we are now at a point to consider its dynamical content.2FormalismThe basic object to consider is the triangle diagram which describes the coupling of thedecaying meson (M1) into the other mesons (M2,3) or another meson (M2) and a photon γor two photons γ1, γ2 via the quark loop.

Let us first consider the strong decays. Droppingall prefactors, the transition amplitude for the process M1 →M2M3 can be evaluated byworking out (cf.

fig.1)Γ(M1 →M2M3) = Tr(ΓM1SFΓM2SFΓM3SF)(2)2

where ΓMi gives the ith meson-quark-antiquark vertex and SF the propagator of the con-stituent quarks. The latter follows from minimizing the effective potential to one loop.The Bethe-Salpeter vertex functions relevant for our considerations are of scalar, pseu-doscalar and vector typeΓS = gS1 ⊗IΓP = gP(1 + hP ̸ p)γ5 ⊗I(3)ΓV = gV γµ ⊗IHere, I is a generic symbol for the isospin structure and we have introduced scalar,pseudoscalar and vector Mq¯q-couplings.

The coupling hp stems from the pseudoscalar-axial vector meson mixing. This is discussed in more detal in refs.

[2,4,6,7]. Since wework in the isospin limit, no scalar-vector mixing arises in the SU(2) subgroup.

In thescalar and pseudoscalar channels, a further complication is due to the λ0λ8 mixing whichhas already been discussed in ref. [8] in some detail.

The solution of the correspondingBethe-Salpeter equations is standard and we do not exhibit any details here.Let us briefly elaborate on the connection between the various transition amplitudesand meson-meson coupling constants. Consider first the decay of a scalar (S) into twopseudoscalars (P).

The transition amplitude is a purely scalar function, called TS, andwe haveΓ(S →PP) = |⃗pc||TS|28πE2SG2SP P = |TS|24π(4)with |⃗pc| the momentum of an outgoing particle in the rest frame of the decaying particle,|⃗pc| = ((s −(m1 + m2)2)(s −(m1 −m2)2)/4s)1/2 and s = m2(M1). For the decay of avector (V ) into two pseudoscalars, one hasTµ(V →PP) = (p1 −p2)µGV P PΓ(V →PP) = |⃗pc|3G2V P P6πm2V(5)Finally, for the reaction V →˜V P we findTµν(V →˜V P) = ǫαβµνpα˜V qβFΓ(V →˜V P) = |⃗pc|33F 24π(6)In the case of the radiative decays, one can use the same formalism since the photonbehaves much like a vector particle.

Since our Lagrangian contains no tensor interactionterm, the pertinent photon-quark-antiquark vertex takes the minimal formΓγ = e2(λ3 + 1√3λ8)(7)For the decay of a pseudoscalar into two photons, we have a structure similar to the onefor V →˜V P, the only difference being that in the formula for the width Γ(P →γγ) onehas a factor 1/2 instead of 1/3 which is the reduction in plolarization degrees of freedomfor a massless particle.3

3Estimates of higher order effectsThe description of mesons as q¯q pairs has been proven to be quite successful in thecalculation of the meson mass spectrum. However it may not account for other propertiesof some mesonic resonances, such as their decays.

We anticipate that this is indeed thecase for the strong decays ρ →ππ and K∗→Kπ, which come out very small in theone loop calculation. The insufficiency of a q¯q description of the mesons was alreadyemphasized by Krewald et al [9] in the context of the pion electromagnetic form factor.Due to the failure of the one loop approximation for the calculation of those decays,it is necessary to estimate the magnitude of higher order effects, such as the two loopcorrections.

The full calculation is, for the moment, out of the scope of the present work.A simple estimate of such effects for the mesonic decays can be obtained by calculating adressed meson propagator, as shown in fig.2. The dashed loop refers to ππ or Kπ states,in the case of the ρ and K∗propagators, respectively.

By bare propagator we denote themeson described as a q¯q state.The full propagator readsGα β = G0α β + G0α λ Σλ µ Gµ β(8)where the bare propagator isG0α β = gα β −ˆqα ˆqβq2 −m2V=Tα βq2 −m2V(9)and Σλ µ is the meson loop given byΣλ µ = 8 G2V P PZΛ1d4k(2π)4 ((2k −q)λ) ((2k −q)µ) S(k, mP 1) S(k −q, mP 2)(10)with S(k, mP i) the propagators of the mesons obtained in the one loop order, mP 1 and mP 2being their masses. The remaining factors in the integrand correspond to the coupling ofthe vector mesons to the pseudoscalars in the first order calculation (5).

The covariantcutoffΛ1 needs not to be the same as Λ. One obtains:Gα β =Tα βq2 −m2V −C(11)with C ≡C ( G2V P P , q2 , mP 1 , mP 2 , Λ1 ).

The quantity C has a cut for q2 > (mP 1 +mP 2 )2 and one rewrites Gα β asGα β = Tα βZq2 −˜m2V −i Im C ( ˜m2V ) Z(12)provided that the renormalization factor Z is roughly constant around the physical mass˜mV and where the renormalization factor is:Z = ( 1 −dd q2 Re C )−1q2 = ˜m2V(13)The decay width of the vector meson is finally4

Γ (V →PP) = Im C ( ˜m2V ) Z˜mV. (14)Using this scheme the amplitudes for radiative decays of ρ and K∗have then to be alsomultiplied by√Z (wave function renormalization).4Results and discussionAt the one loop level we use the meson spectrum and decay constants to fix the param-eters of the model.

For Λ = 1 GeV, G1Λ2 = 3.95, G2Λ2 = 5.43, KΛ5 = 42, mu = md = 4MeV and ms = 115 MeV, we find the following meson masses (the experimental valuesare given in parentheses for comparison): Mπ = 136.5(139.6), MK = 497.5(497.7), Mη =549 (548.8), Mη′ = 936 (957.5), Mρ = 775 (768.3), Mω = 764 (781.95), MK∗= 898 (891.6),MΦ = 990 (1019.41) and Ma0 = 970 (983.3) (all in MeV). For the pion and kaon decayconstants, we have Fπ = 93.9 MeV and FK = 96.6 MeV, i.e.

the ratio FK/Fπ is too small,which is a common feature in this kind of models. We find an overall satisfactory descrip-tion of the meson spectrum together with reasonable values for the vacuum expectationvalues of the scalar quark densities ¯uu and ¯ss, −< ¯uu >1/3= 272 MeV (225 ± 35) and< ¯ss > / < ¯uu >= 0.74(0.8 ± 0.2).In table 1, we show the results for the strong decays in comparison to the empiricalvalues.

Obviously, for states in the quark-antiquark continuum the results are not reliableas indicated by the decay Φ →πρ. Also, for our set of parameters the decay Φ →¯KK iskinematically forbidden.

The large width of the Φ →πρ decay is due to the too strongflavour mixing induced by the six-fermion interaction proportional to K.This couldpresumably be cured by including more terms in the Lagrangian like e.g. in ref.

[6].As for the strong decays of ρ and K∗the one loop order calculation is clearly insufficientto account for the respective empirical widths. Using the simple approximation scheme,described in the previous section, to include the second order effects and approximatingthe meson propagators in the loop by propagators of structureless particles, the resultsimprove by about a factor of 2.

We think, therefore, that it is mandatory to consider amore complex multiquark structure for the ρ and K∗mesons. The strong ρ decay widthincluding second order effects is still quite small as compared to the experimental one, butthis number should be understood only as a guide for the order of magnitude of higherloop corrections.

We notice that the parameters could have been adjusted in order tohave the correct decay width for the ρ and the KSFR relation fulfilled, but at the cost ofhaving bad values for the radiative decays.Let us now turn to the radiative decays. In table 2, our results are summarized.

Wefind an overall satisfactory description of the data, the main exceptions being the widthsfor Φ →ηγ and for the K∗→Kπ. In the first case this is, again, an artifact of theinteraction Lagrangian used and also, since the Φ lies in the unphysical quark-antiquarkcontinuum, should not be considered significantly troublesome.

The fact that the ratiosΓcalcV P P/ΓexpV P P and ΓcalcV P γ/ΓexpV P γ are larger for the K∗decays than for the ρ decays (bothstrong and radiative) is consistent with the small ratio FK/Fπ. We have also investigatedthe case K = 0 (no six-fermion term).

Reducing the strange quark mass to 80 MeV [2],which is necessary to find a decent fit to the spectrum, one is not able to get a satisfactorydescription of strong and radiative decay widths. After finishing this work, we became5

aware of a preprint by Takizawa and Krewald [10], who deal with the radiative decaysπ0 →γγ and η →γγ in a similar model. Their Lagrangian contains the four–fermioninteraction proportional to G1 and the six–fermion determinantal interaction.Whiletheir results are similar to ours, we disagree with their conclusions at various places.First, in the case of π0 →γγ they remove the cut–offto find agreement with the currentalgebra prediction.

This is, however, not a consistent procedure since once the cut–offis introduced, the effective theory is defined and should not be altered in the process ofcalculating various quantities. Second, for calculating the width of η →γγ, they use theempirical η mass, which is 17 per cent larger than the value they find within the model.This, of course, alters substantially the result for this particular width.

Comparing theirresults with ours, we also find a satisfactory description for these two particular radiativedecays. It should be obvious, however, that the model is somewhat too crude too draw asfar reaching conclusions as done in ref.[10].

As long as one is not able to properly accountfor the SU(3) breaking effects in the pseudoscalar decay constants, it is doubtful that onecan make firm quantitative statements about such breaking effects in other processes.In summary, we have used the generalized three-flavour NJL model to calculate strongand radiative meson decays (three-point functions) taking the full solution to the Bethe-Salpeter equations in the one-loop approximation. A reasonable agreement to the ex-perimental data is obtained, with the exception of the ρ and K∗decays.

This seems toindicate that a more complex multiquark structure should be accounted for these mesons.This conjecture is supported by a simple estimate of the two loop order corrections. Wealso demonstrated the importance of the flavour-mixing determinantal six–fermion in-teraction.

Further studies including also the effects of isospin–breaking seem necessaryto understand some fine details of the mesonic interactions at low energies and to gaininsight into effects of SU(3) breaking on various observables.References[1] Y. Nambu and G. Jona-Lasinio, Phys.Rev.122(1961)345; 124(1961)246[2] D. Ebert and H. Reinhardt, Nucl.Phys.B271(1986)188[3] V. Bernard, R.L. Jaffe and U.-G. Meißner, Nucl.Phys.

B308(1988)753[4] M. Takizawa et al., Nucl.Phys.A507(1990)611[5] M.K. Volkov, Ann.

Phys. (NY)157(1984)282[6] S. Klimt, M. Lutz, U. Vogl and W. Weise, Nucl.Phys.

A516(1990)429; U. Vogl, M.Lutz, S. Klimt and W. Weise, Nucl.Phys.A516(1990)469[7] V. Bernard and U.-G. Meißner, Nucl.Phys.A489(1988)647[8] V. Bernard and U.-G. Meißner, Phys.Rev.D36(1987)819. [9] S. Krewald, K.Nakayama and J. Speth, Phys.

Lett. B272 (1991)190.

[10] M. Takizawa and S. Krewald, ”SU(3) symmetry breaking effects in the η →γγdecay”, J¨ulich preprint, 1992.6

TABLESρ →ππK∗+ →π+K0K∗+ →π0K+a0 →πηφ →πρNJL(I)52.018.09.074.51.5NJL(II)94.038.219.1--Exp.151.5 ± 1.238.6 ± 0.616.7 ± 0.357 ± 110.6 ± 0.3Table 1: Strong meson widths for various decays in units of MeV: (I) calculated in oneloop order; (II) with estimates of two loop order included.π0 →γγη →γγρ± →π±γρ →ηγNJL7.9 · 10−30.7760.160.4Exp. (7.7 ± 0.6) · 10−30.46 ± 0.0467.1 ± 7.657.6 ± 10.7ω →π0γω →ηγK∗+ →K+γΦ →ηγNJL7626.392.0259.0Exp.716.6 ± 43.04.0 ± 1.950.3 ± 4.656.7 ± 2.8Table 2: Anomalous and radiative meson decay widths in units of keV, calculated in theone loop order.

The second order corrected decays are Γρπγ = 63 keV and ΓK∗Kγ = 98.9keV (Λ1 = 1.3 GeV).FIGURE CAPTIONSFig. 1: Quark triangle diagram to calculate the strong meson decays.

In the caseof radiative decays, one has to substitute the third meson M3 by a photon, and for theanomalous decays M2 and M3 by two photons.Fig. 2: The dressed vector meson propagator (thick line) includes a 2-pseudoscalarexcitation (dashed line), which is a 2nd order effect.

The one loop order (q¯q state) isdenoted by the thin line, the bare meson propagator.7

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