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16 December 1999

.Physics Letters B 470 1999 2026

Radiative decay of r 0 and f mesons in a chiral unitary approachE. Marco a,b, S. Hirenzaki c, E. Oset a,b, H. Toki a

a Research Center for Nuclear Physics, Osaka Uniersity, Ibaraki, Osaka 567-0047, Japanb ( )Departamento de Fsica Teorica and IFIC, Centro Mixto Uniersidad de Valencia-CSIC, 46100 Burjassot Valencia , Spain

c Department of Physics, Nara Womens Uniersity, Nara 630-8506, JapanReceived 24 March 1999; received in revised form 24 September 1999; accepted 12 October 1999

Editor: J.-P. Blaizot

Abstract

We study the r 0 and f decays into pqpyg , p 0p 0g and f into p 0hg using a chiral unitary approach to deal with thefinal state interaction of the MM system. The final state interaction modifies only moderately the large momenta tail of thephoton spectrum of the r 0pqpyg decay. In the case of f decay the contribution to pqpyg and p 0p 0g decay

.proceeds via kaonic loops and gives a distribution of pp invariant masses in which the f 980 resonance shows up with a0very distinct peak. The spectrum found for fp 0p 0g decay agrees with the recent experimental results obtained at

0 .Novosibirsk. The branching ratio for fp hg , dominated by the a 980 , is also in agreement with recent Novosibirsk0results. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 13.25.Jx; 12.39.Fe; 13.40.Hq

In this work we investigate the reactions rpqpyg , p 0p 0g and fpqpyg , p 0p 0g , p 0hg ,treating the final state interaction of the two mesonswith techniques of chiral unitary theory recentlydeveloped. The energies of the two meson systemare too big in both the r and f decay to be treated

w xwith standard chiral perturbation theory, x PT 1 .However, a unitary coupled channels method, whichmakes use of the standard chiral Lagrangians to-gether with an expansion of Re Ty1 instead of the Tmatrix, has proved to be very efficient in describingthe meson meson interactions in all channels up to

w xenergies around 1.2 GeV 24 . The method is anal-ogous to the effective range expansion in Quantum

w xMechanics. The work of 4 establishes a directconnection with x PT at low energies and gives the

w xsame numerical results as the work of 3 where

tadpoles and loops in the crossed channels are notevaluated but are reabsorbed into the L coefficientsiof the second order Lagrangian of x PT. A techni-

w xcally much simpler approach is done in 2 where,only for Ls0, it is shown that the effect of thesecond order Lagrangian can be suitably incorpo-rated by means of the Bethe-Salpeter equation usingthe lowest order Lagrangian as a source of thepotential and a suitable cut off, of the order of 1GeV, to regularize the loops. This latter approachwill be the one used here, where the two pionsinteract in s-wave.

The diagrammatic description for the rpqpygdecay is shown in Fig. 1

In Fig. 1 the intermediate states in the loopsattached to the photon, l, can be KqKy or pqpy.However, the other loops involving only the meson

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. .PII: S0370-2693 99 01205-8

( )E. Marco et al.rPhysics Letters B 470 1999 2026 21

Fig. 1. Diagrams for the decay rpqpyg .

0 0 0 0meson interaction can be also K K or p p in thew xcoupled channel approach of 2 .

For the case of p 0p 0g decay only the diagrams . . .with at least one loop contribute, d , e , f ,

. .g , h , . . . in Fig. 1.The case of the f decay is analogous to the

0 0 . . .rp p g decay. Indeed, the terms a , b , c ofFig. 1 do not contribute since we do not have directfpp coupling. Furthermore, there is anothernovelty since only KqKy contributes to the loopwith a photon attached.

The procedure followed here in the cases of p 0p 0and p 0h production is analogous to the one used inw x5 . Depending on the renormalization scheme cho-

w xsen, other diagrams can appear 5 but the whole setis calculated using gauge invariant arguments, asdone here, with the same result. The novelty in thepresent work is that the strong interaction MMM XM X is evaluated using the unitary chiral ampli-

w xtudes instead of the lowest order used in 5 .We shall make use of the chiral Lagrangians for

w x w xvector mesons of 6 and follow the lines of Ref. 7in the treatment of the radiative rho decay. TheLagrangian coupling vector mesons to pseudoscalarmesons and photons is given by

F iGV Vyy mn m n : :LL V 1 s V f q V u u .2 mn q mn 2 2 21 .

where V is a 3=3 matrix of antisymmetric tensormnfields representing the octet of vector mesons, K ) ,

.r, v . All magnitudes involved in Eq. 1 are de-8

w xfined in 6 . The coupling G is deduced from theVrpqpy decay and the F coupling from rV

q y w xe e . We take the values chosen in 7 , G s67VMeV, F s153 MeV. The f meson is introduced inVthe scheme by means of a singlet, v , going from1

. .SU 3 to U 3 through the substitution V V q Imn mn 3v1 ,mn= , with I the 3=3 diagonal matrix. Then,33

assuming ideal mixing for the f and v mesons

1 1 22 v q v v , v y v f 2( .1 8 1 83 3 3 6

.one obtains the Lagrangian of Eq. 1 substitutingV by V , given bymn mn

1 10 q )qr q v r Kmn mn mn mn 2 2

1 1V y 0 ) 0mn r y r q v Kmn mn mn mn 2 2 0

)y ) 0K K fmn mn mn

3 .

From there one can obtain the couplings correspond- .ing to VPP V vector and P pseudoscalar and

VPPg with the G term or the VPPg with the FV Vterm.

The basic couplings needed to evaluate the dia-grams of Fig. 1 are

G MV r X mq yt sy p yp e r , . .rp p m m2f

G MV r nq yt s2 e e r e g . .rgp p n2f

2 e FVq yG P e r .V m n2 /2M fr= m n n mk e g yk e g , . .

t q ys2 ep e m g 4 . .gp p mwith p , pX the pq,py momenta, P , k the r andm m m mphoton momenta and f the pion decay constantwhich we take as f s93 MeV.p

.The vertices of Eq. 4 are easily generalized toq y .the case of K K . Using the Lagrangian of Eq. 1 ,

in the first two couplings one has an extra factor 1r2and the last coupling is the same. The couplings for

( )E. Marco et al.rPhysics Letters B 470 1999 202622

fKqKy and fg KqKy which are needed for the f .decay are like the two first couplings of Eq. 4

m . m .substituting M by M , e r by e f and multi-r fplying by y1r 2 . In addition we shall take thevalues G s55 MeV and F s165 MeV which areV Vsuited to the fKqKy and feqey decaywidths respectively.

The evaluation of the r width for the first three . . .diagrams a , b , c of Fig. 1 is straightforward and

w x w xhas been done before 810 and in 7 following thepresent formalism. We rewrite the results in a conve-nient way for our purposesdG 1 1 1r2r 2 2 2 2s m yM M y4m . .r I I p3 3dM 16m2p .I r

=1

B1 2<

( )E. Marco et al.rPhysics Letters B 470 1999 2026 23

tices appearing there have the structure a sqb p2i i2 qg m , which can be recast as a sq bqi i

. 2 2 2 .g m qb p ym . The first two terms in thei i i i isum give the on shell contribution and the third onethe off shell part. This latter term kills one of themeson propagators in the loops and does not con-

.tribute to the d term in Eq. 8 . Hence, the mesonmeson amplitudes factorize outside the loop integralwith their on shell values. A more detailed descrip-

w xtion, done for a similar problem, can be seen in 14 , . .following the steps from Eqs. 13 to 23 .

w xFollowing these steps, as done in 11,14 , it iseasy to include the effect of the final state interaction

. .of the mesons. The sum of the diagrams d , e , . .f , g and further iterated loops of the meson-me-

.son interaction, h , . . . , is shown to have the same .structure as the contact term of a in the Coulomb

gauge, which one chooses to evaluate the ampli-tudes. The sum of all terms including loops is readilyaccomplished by multiplying the G part of theV

.contact term by the factor F M , M1 r I q y q y q yF M , M s1qG t .1 r I p p p p ,p p

1 q y q y q yq G t 9 .K K K K ,p p2

where t X X are the strong transition matrixM M , M M1 2 1 2w xelements in s-wave evaluated in 2 and G isM M1 2

given by1

G M , M s ayb I a,b . . .M M r I 21 2 8pM 2 M 2r I

as , bs 10 .2 2M MM M1 1 . w xwith I a,b a function given analytically in 11 . The

.F r2yG part of the contact term is iterated byV V . .means of diagrams d , h . . . in order to account

for final state interaction. Here the loop function isthe ordinary two meson propagator function, G, ofthe Bethe-Salpeter equation, TsVqVGT , for themeson-meson scattering and which is regularized inw x2 by means of a cut-off in order to fit the scatteringdata. The sum of all these diagrams is readily accom-

.plished by multiplying the F r2yG part of theV Vcontact term by the factorF M s1qG q yt q y q y .2 I p p p p ,p p

1q y q y q yq G t 11 .K K K K ,p p2

By using isospin Clebsch Gordan coefficients theamplitudes t X X can be written in terms of theM M , M M1 2 1 2 w xisospin amplitudes of 2 as

2 Is0q y q yt s t M , .p p ,p p pp ,pp I3

1Is0

q y q yt s t M 12 . .K K ,p p K K ,pp I3

.neglecting the small Is2 amplitudes. In Eq. 12 ,q yone factor 2 for each p p state has been intro-

w xduced, since the isospin amplitudes of 2 used in Eq. .12 are written in a unitary normalization which

includes an extra factor 1r 2 for each pp state.The invariant mass distribution in the presence of

. .final state interaction is now given by Eqs. 5 7 .by changing in Eq. 7

M Gr VI F M , M .1 1 r I2f2K FVq yG F M , .V 2 I2 /2f

M Gr V 2 2I 2 p D qD sin u .2 1 22f

=M Gr VRe F M , M .1 r I2 f

K FVq yG F M , .V 2 I2 5 /2fI I 13 .3 3

The rp 0p 0g width is readily