Transcript
Page 1: Radiative decay of ρ0 and φ mesons in a chiral unitary approach

16 December 1999

Ž .Physics Letters B 470 1999 20–26

Radiative decay of r 0 and f mesons in a chiral unitary approach

E. Marco a,b, S. Hirenzaki c, E. Oset a,b, H. Toki a

a Research Center for Nuclear Physics, Osaka UniÕersity, Ibaraki, Osaka 567-0047, Japanb ( )Departamento de Fısica Teorica and IFIC, Centro Mixto UniÕersidad de Valencia-CSIC, 46100 Burjassot Valencia , Spain´ ´

c Department of Physics, Nara Women’s UniÕersity, Nara 630-8506, Japan

Received 24 March 1999; received in revised form 24 September 1999; accepted 12 October 1999Editor: 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 0

™pqpyg decay. In the case of f decay the contribution to pqpyg and p 0p 0g decayŽ .proceeds via kaonic loops and gives a distribution of pp invariant masses in which the f 980 resonance shows up with a0

very distinct peak. The spectrum found for f™p 0p 0g decay agrees with the recent experimental results obtained at0 Ž .Novosibirsk. The branching ratio for f™p hg , dominated by the a 980 , is also in agreement with recent Novosibirsk0

results. 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 r™

pqpyg , p 0p 0g and f™pqpyg , 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 2–4 . 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 coefficientsi

of 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 r™pqpyg

decay is shown in Fig. 1In Fig. 1 the intermediate states in the loops

attached 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.Ž .PII: S0370-2693 99 01205-8

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( )E. Marco et al.rPhysics Letters B 470 1999 20–26 21

Fig. 1. Diagrams for the decay r™pqpyg .

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 diagramsŽ . Ž . Ž .with at least one loop contribute, d , e , f ,

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

0 0 Ž . Ž . Ž .r™p p g decay. Indeed, the terms a , b , c ofFig. 1 do not contribute since we do not have directf™pp 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 0

and 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 MM™M 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 n² : ² :LL V 1 s V f q V u uŽ .2 mn q mn' '2 2 21Ž .

where V is a 3=3 matrix of antisymmetric tensormn

fields representing the octet of vector mesons, K ) ,Ž .r, v . All magnitudes involved in Eq. 1 are de-8

w xfined in 6 . The coupling G is deduced from theV

r™pqpy decay and the F coupling from r™Vq y w xe e . We take the values chosen in 7 , G s67V

MeV, F s153 MeV. The f meson is introduced inV

the scheme by means of a singlet, v , going from1

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

assuming ideal mixing for the f and v mesons

1 1 22

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

Ž .one obtains the Lagrangian of Eq. 1 substituting˜V 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

3Ž .

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

VPPg with the G term or the VPPg with the FV V

term.The basic couplings needed to evaluate the dia-

grams of Fig. 1 are

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

G MV r nq yt s2 e e r e gŽ . Ž .rgp p n2f

2 e FVq yG P e rŽ .V m n2 ž /2M fr

= m n n mk e g yk e g ,Ž . Ž .

t q ys2 ep e m g 4Ž . Ž .gp p m

with p , pX the pq,py momenta, P , k the r andm m m m

photon momenta and f the pion decay constantwhich we take as f s93 MeV.p

Ž .The vertices of Eq. 4 are easily generalized toq y Ž .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

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( )E. Marco et al.rPhysics Letters B 470 1999 20–2622

fKqKy and fg KqKy which are needed for the f

Ž .decay are like the two first couplings of Eq. 4mŽ . mŽ .substituting M by M , e r by e f and multi-r f'plying by y1r 2 . In addition we shall take the

values G s55 MeV and F s165 MeV which areV V

suited to the f™KqKy and f™eqey decaywidths respectively.

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

w x w xhas been done before 8–10 and in 7 following thepresent formalism. We rewrite the results in a conve-nient way for our purposes

dG 1 1 1r2r 2 2 2 2s m yM M y4mŽ .Ž .r I I p3 3dM 16m2pŽ .I r

=1

B1 2< <dcosu t 5Ž .Ý ÝH2 y1

whereB 2 8 2< < w xt s e I q I q I 6Ž .Ý Ý 1 2 33

Ž .In Eq. 5 , M is the invariant mass of the two pI

system and u the angle between the pq meson andthe photon in the frame where the pqpy system isat rest. The quantity I stands for the contribution of1

Ž .the first diagram alone, Fig. 1 a , I for the second3Ž . Ž .and third b , c and I for the interference between2

the first diagram and the other two. They are givenby

2M G K Fr V V

I s q yG ,1 V2 2 ž /2f f

M Gr V 2I s2 p D qDŽ .2 1 22f

=M G K Fr V V2sin u q yG ,V2 2½ 5ž /2f f

I s2 p 2 D qDŽ .3 1 2

=sin2u D qD p 2Ž .� 1 2

< < < <q D yD p k cosuŽ . 41 2

=

2M Gr V7Ž .2ž /f

where K is the photon momentum in the r restframe and p,k are the momenta of the meson and

the photon in the rest frame of the pqpy system,Ž . Ž .and D , D the meson propagators in the b , c1 2

Bremsstrahlung diagrams, conveniently written interms of M and u .I

The first term of the contact term, t q y, in Eq.rgp p

Ž .4 is not gauge invariant. It requires the addition ofŽ . Ž .the diagrams b and c of Fig. 1 to have a gauge

invariant set. On the other hand the second term inŽ .the contact term F r2yG part is gauge invari-V V

ant by itself. When considering final state interactionof the mesons this means that the G part of theV

Ž .contact term, diagram d , must be complemented byŽ . Ž . Ž .diagrams e , f , g to form the gauge invariant

set. On the other hand the F r2yG part of theV VŽ .contact term appears in the d diagram which is

gauge invariant by itself.The technology to introduce the final state interac-

0 0tions is available from the study of f™K K g inw x11 . There it was shown that the strong t matrix forthe M M ™M X M X transition factorizes with their1 2 1 2

on shell values in the loops with a photon attached.The same was proved for the loops of the Bethe-Salpeter equation in the meson meson interaction

w xdescription of 2 . On the other hand the sum of theŽ . Ž . Ž . Ž .diagrams d , e , f , g , which appears now with

Ž Ž ..the G part of the contact term diagram a , couldV

be done using arguments of gauge invariance whichled to a finite contribution for the sum of the loopsw x5,12,13 . A sketch of the procedure is given here.The r™pqpyg amplitude can be written asŽ . Ž . mne r e g T and the structure of the loops in Fig.m n

1 is such that

T mn sa g mn qb Q mQn qc Q mK n qd QnK m

qe K mK n 8Ž .

where Q, K are the r meson and photon momentaŽ mn .respectively. Gauge invariance T K s0 forcesn

Ž .bs0 and dsyar QPK . Furthermore, in themn Ž .Coulomb gauge only the g term of Eq. 8 con-

tributes and the coefficient a is calculated from the dŽ . Ž .coefficient, to which only the diagrams e , f , of

Fig. 1 contribute. For dimensional reasons the loopintegral contains two powers less in the internalvariables than the pieces contributing to the g mn

term from these diagrams, since the product QnK m

is factorized out of the integral. This makes the dcoefficient finite. Furthermore, the MM™MM ver-

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( )E. Marco et al.rPhysics Letters B 470 1999 20–26 23

tices appearing there have the structure a sqbÝ p2i i

2 ŽqgÝ m , which can be recast as a sq bqi i. 2 Ž 2 2 .g Ý m qbÝ p ym . The first two terms in thei i i i i

sum 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-

Ž .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 ,Ž . Ž .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

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

Ž .son interaction, h , . . . , is shown to have the sameŽ .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

Ž .contact term by the factor F M , M1 r I

˜ q y q y q yF M , M s1qG tŽ .1 r I p p p p ,p p

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

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

given by

1G M , M s ayb I a,bŽ . Ž .Ž .M M r I 21 2 8p

M 2 M 2r I

as , bs 10Ž .2 2M MM M1 1

Ž . w xwith I a,b a function given analytically in 11 . TheŽ .F r2yG part of the contact term is iterated byV V

Ž . Ž .means of diagrams d , h . . . in order to accountfor 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-

Ž .plished by multiplying the F r2yG part of theV V

contact term by the factor

F M s1qG q yt q y q yŽ .2 I p p p p ,p p

1q y q y q yq G t 11Ž .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 ,Ž .p p ,p p pp ,pp I3

1Is0

q y q yt s t M 12Ž . Ž .K K ,p p K K ,pp I'3

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

w xduced, since the isospin amplitudes of 2 used in Eq.Ž .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

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

M Gr VI ™ F M , MŽ .1 1 r I2f

2K FV

q yG F M ,Ž .V 2 I2 ž /2f

M Gr V 2 2I ™2 p D qD sin uŽ .2 1 22f

=M Gr V

Re F M , MŽ .1 r I2½ f

K FVq yG F M ,Ž .V 2 I2 5ž /2f

I ™ I 13Ž .3 3

The r™p 0p 0g width is readily obtained by omit-ting the terms I , I and also omitting the first term2 3Ž . Ž .the unity in the definition of the F M , M ,1 r IŽ . Ž . Ž .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, diagramsŽ . Ž . Ž .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™

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( )E. Marco et al.rPhysics Letters B 470 1999 20–2624

q y Ž .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 tÝ Ý K K K K ,pp3 2 'f 3

2K F 1V Is0

q yq yG G tV K K K K ,pp2 ž / '2f 314Ž .

For f™p 0p 0g the cross section is the same di-vided by a factor two to account for the identity ofthe two p 0’s.

For the f™p 0hg case we have

M G 1B f V2 4 2 Is1˜< < q yt s e G tÝ Ý K K K K ,ph3 2 'f 2

2K F 1V Is1˜ q yq yG G tV K K K K ,ph2 ž / '2f 2

15Ž .

In Fig. 2 we show dGrdK for r™pqpyg

Ž .decay, dG rdK s m dG rM dM . The dashed-r r r I I

dotted line shows the contribution of diagramsŽ . Ž . Ž .1 a , b , c and taking F s0. The dashed lineV

shows again the contribution coming from diagramsŽ . Ž . Ž .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 r™pqpyg

as a function of the photon momentum. Solid line: spectrumincluding final state interaction of the two mesons and the F andV

G contributions; dashed line: spectrum including only the treeVŽ . Ž . Ž .level diagrams a , b , c of Fig. 1 and the F and G contribu-V V

tions; dashed-dotted line: spectrum including only the tree levelŽ . Ž . Ž .diagrams a , b , c of Fig. 1 and taking F s0. The experimen-V

w 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 V

process 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 0™pqpyg s1.18=10y2Ž .

for K)50 MeV 16Ž .

which compares favourably with the experimentalw x expŽ 0 q y . Žnumber 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

Ž 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 Ž 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 V

10y4. The measurement of this quantity may serveas a test for the sign of the F G product.V V

As for the f™ppg decay, as we pointed abovethe f™pqpyg rate is twice the one of the f™

p 0p 0g . We have evaluated the invariant mass distri-bution for these decay channels and in Fig. 3 we plotthe distribution dBrdM for f™p 0p 0g which al-I

lows us to see the f™ f g contribution since the f0 0

is the important scalar resonance appearing in theq y q y w xK K ™p p amplitude 2 . The solid curve

shows our prediction, with F G )0, the sign pre-V Vw xdicted by vector meson dominance 6 . The dashed

curve is obtained considering F G -0. We com-V V

pare our results with the recent ones of the Novosi-w xbirsk experiment 16 . We can see that the shape of

the spectrum is relatively well reproduced consider-Žing statistical and systematic errors the latter ones

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( )E. Marco et al.rPhysics Letters B 470 1999 20–26 25

Fig. 3. Distribution dBrdM for the decay f™p 0p 0g , with MI I

the 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 f™pqpyg is twice the results plotted there.

.not shown in the figure . The results consideringF G -0 are in complete disagreement with theV V

data.The finite total branching ratio which we find for

f™pqpyg is 1.6=10y4 and correspondingly 0.8=10y4 for the f™p 0p 0g . This latter number is

w x Žslightly 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 Ž . y4in 17 is 1.08"0.17"0.09 =10 , compatible

with our prediction. The branching ratio measured inw x q y Ž .19 for f™p p g is 0.41"0.12"0.04 =

10y4.The branching ratio obtained for the case f™

p 0hg is 0.87 = 10y4. The results obtained atw x Ž . y4 w xNovosibirsk are 18 0.83"0.23 =10 and 17

Ž . y40.90 " 0.24 " 0.10 = 10 . The spectrum, notshown, is dominated by the a contribution.0

Ž q y.The contribution of f™ f p p g , obtained by0

integrating dG rdM assuming an approximatef I

Breit-Wigner form to the left of the f peak, gives0

us a branching ratio 0.44=10y4. As argued above,the branching ratio for f™p 0p 0g is one half off™pqpyg , 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-0

tion is done. As one can see in Fig. 3, we find alsoan appreciable strength for f™sg .

We should also warn not to compare our pre-dicted rate for f™pqpyg directly with experi-ment. Indeed, the experiment is done using the reac-tion eqey

™f™pqpyg , which interferes with ther contribution eqey

™r™pqpyg at the tail ofw xthe r mass distribution in the f mass region 20 .

w xAlso the results in 17,19 are based on model depen-dent assumptions. For these reasons, as quoted inw x 0 017 , the p p g mode is more efficient to study thepp mass spectrum.

Our result for f™p 0p 0g is 50 % larger than thew xone obtained in 5 owed to the use of the unitary

KqKy™p 0p 0 amplitude instead of the lowest or-

der chiral one. The shape of the distribution foundhere is, however, rather different than the one ob-

w xtained in 5 , showing the important contribution ofthe f resonance which appears naturally in the0

unitary chiral approach.The f™ f g decay has been advocated as an0

important source of information on the nature of thef resonance and experiments have been conducted0

w xat Novosibirsk 21 and are also planned at Frascatiw x22 , trying to magnify the signal for f production0

through interference with initial and final state radia-q y Ž q y.tion in the e e ™ f™ f p p g reaction0

w x w20,22–24 . The completion of the experiments 16–x19 is a significant step forward.Present evaluations for f™ f g™ppg are based0

w xon models assuming a KK molecule for the f 250y5with a branching ratio 1–2=10 , a qq structure

y5 w xwith a value 5=10 25 and a qqqq structure withy4 w xa value 2.4=10 25 .

The model for f™ f g assumed in Fig. 1 is0w xsimilar to the one of 26 where the production also

proceeds via the kaonic loops. There a KK moleculeis assumed for the f resonance while here the0

w xrealistic KK™pp amplitude of 2 is used. Empha-sis is made in the importance of going beyond thezero width approximation for the resonance inw x26,27 . Our approach automatically takes this intoaccount since the KK™pp amplitude correctly

w xincorporates the width of the f resonance 2 .0

We would also like to warn that the peak of the f0

seen in Fig. 3 cannot be trivially interpreted as aresonant contribution on top of a background, sincethere are important interference effects between thef production and the s background. The strength of0

the peak comes in our case in about equal amounts

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( )E. Marco et al.rPhysics Letters B 470 1999 20–2626

from the real and the imaginary parts of the ampli-tude for the process.

The agreement found between our results for thef™p 0p 0g , f™p 0hg and experiment provides animportant endorsement for the chiral unitary ap-proach used here. Improvements in the future, reduc-ing the experimental errors, should put further con-straints on available theoretical approaches for thisreaction.

Acknowledgements

We would like to acknowledge useful commentsfrom J.A. Oller and from A. Bramon who called ourattention to the recent experimental results on f™

p 0p 0g . We are grateful to the COE Professorshipprogram of Monbusho which enabled E.O. to stay atRCNP to perform the present study. One of us, E.M.,wishes to thank the hospitality of the RCNP of theUniversity of Osaka, and acknowledges financialsupport from the Ministerio de Educacion y Cultura.´This work is partly supported by DGICYT contractno. PB 96-0753 and by the EEC-TMR ProgramContract No. ERBFMRX-CT98-0169.

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