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
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
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
( )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
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
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 obtained by omit-ting the terms I , I and also omitting the first term2 3 . .the unity in the definition of the F M , M ,1 r I . . .F M factors in Eqs. 9 and 11 and dividing by a2 I
factor two the width to account for the identity of theparticles.
The evaluation of the f decay is straightforwardby noting that the tree level contributions, diagrams . . .a , b , c are not present now, and that only kaonicloops attached to photons contribute in this case.Hence, the invariant mass distribution for f
( )E. Marco et al.rPhysics Letters B 470 1999 202624
q y .p p g is given in this case by Eq. 5 , changingm m , withr f
M G 1B f V2 4 2 Is0
< < q yt s e G t K K K K ,pp3 2 f 32K F 1V Is0
q yq yG G tV K K K K ,pp2 / 2f 314 .
For fp 0p 0g the cross section is the same di-vided by a factor two to account for the identity ofthe two p 0s.
For the fp 0hg case we haveM G 1
B f V2 4 2 Is1< < q yt s e G t K K K K ,ph3 2 f 2
2K F 1V Is1 q yq yG G tV K K K K ,ph2 / 2f 2
In Fig. 2 we show dGrdK for rpqpyg .decay, dG rdK s m dG rM dM . The dashed-r r r I I
dotted line shows the contribution of diagrams . . .1 a , b , c and taking F s0. The dashed lineV
shows again the contribution coming from diagrams . . .1 a , b , c but now considering also the F contri-V
butions. Finally, the solid line includes the full set of
Fig. 2. Photon distribution, dG rdK , for the process rpqpygas a function of the photon momentum. Solid line: spectrumincluding final state interaction of the two mesons and the F andVG contributions; dashed line: spectrum including only the treeV
. . .level diagrams a , b , c of Fig. 1 and the F and G contribu-V Vtions; dashed-dotted line: spectrum including only the tree level
. . .diagrams a , b , c of Fig. 1 and taking F s0. The experimen-Vw xtal data taken from 15 are normalized to our results.
diagrams in Fig. 1 to account for final state interac-tion and with the F and G contributions. TheV Vprocess is infrared divergent and we plot the distribu-tion for photons with energy bigger than 50 MeV,
w xwhere the experimental measurements exist 15 . Wew xhave also added the experimental data, given in 15
with arbitrary normalization, normalized to our re-sults.
As one can see in Fig. 2, the shape of thedistribution of photon momenta is well reproduced.For the total contribution we obtain a branching ratioto the total width of the r
B r 0pqpyg s1.18=10y2 .
for K)50 MeV 16 .
which compares favourably with the experimentalw x exp 0 q y . number 15 , B r p p g s 0.99"0.04"
. y20.15 =10 for K)50 MeV.The changes induced by the F term found hereV
w xreconfirm the findings of 7 . The effect of the finalstate interaction is small and mostly visible at highphoton energies, where it increases dGrdK by about
0 0 0 .25%. The branching ratio for B r p p g thatwe obtain is 1.4=10y5 which can be interpreted in
0 0 0. 0 0our case as r gs p p since the p p inter-action is dominated by the s pole in the energyregime where it appears here. This result is very
w xsimilar to the one obtained in 5 . In the case oneconsiders F G -0, the result obtained is 1.0=V V10y4. The measurement of this quantity may serveas a test for the sign of the F G product.V V
As for the fppg decay, as we pointed abovethe fpqpyg rate is twice the one of the fp 0p 0g . We have evaluated the invariant mass distri-bution for these decay channels and in Fig. 3 we plotthe distribution dBrdM for fp 0p 0g which al-Ilows us to see the f f g contribution since the f0 0is the important scalar resonance appearing in the
q y q y w xK K p p amplitude 2 . The solid curveshows our prediction, with F G )0, the sign pre-V V
w xdicted by vector meson dominance 6 . The dashedcurve is obtained considering F G -0. We com-V Vpare our results with the recent ones of the Novosi-
w xbirsk experiment 16 . We can see that the shape ofthe spectrum is relatively well reproduced consider-
ing statistical and systematic errors the latter ones
( )E. Marco et al.rPhysics Letters B 470 1999 2026 25
Fig. 3. Distribution dBrdM for the decay fp 0p 0g , with MI Ithe invariant mass of the p 0p 0 system. Solid line: our prediction,with F G )0. Dashed line: result taking F G -0. The dataV V V V
w xpoints are from 16 and only statistical errors are shown. Thew xsystematic errors are similar to the statistical ones 16 . The
distribution for fpqpyg is twice the results plotted there.
.not shown in the figure . The results consideringF G -0 are in complete disagreement with theV Vdata.
The finite total branching ratio which we find forfpqpyg is 1.6=10y4 and correspondingly 0.8=10y4 for the fp 0p 0g . This latter number is
w x slightly smaller than the result given in 16 , 1.14". y40.10"0.12 =10 , where the first error is statisti-
cal and the second one systematic. The result givenw x . y4in 17 is 1.08"0.17"0.09 =10 , compatible
with our prediction. The branching ratio measured inw x q y .19 for fp p g is 0.41"0.12"0.04 =10y4.
The branching ratio obtained for the case fp 0hg is 0.87 = 10y4. The results obtained at
w x . y4 w xNovosibirsk are 18 0.83"0.23 =10 and 17 . y40.90 " 0.24 " 0.10 = 10 . The spectrum, notshown, is dominated by the a contribution.0
q y.The contribution of f f p p g , obtained by0integrating dG rdM assuming an approximatef IBreit-Wigner form to the left of the f peak, gives0us a branching ratio 0.44=10y4. As argued above,the branching ratio for fp 0p 0g is one half offpqpyg , which should not be compared to the
w xone given in 16 since there the assumption that allthe strength of the spectrum is due to the f excita-0tion is done. As...