3 August 2000

Ž .Physics Letters B 486 2000 406–413www.elsevier.nlrlocaternpe

q y 0 0The ratio F™K K rK K

A. Bramon a, R. Escribano b, J.L. Lucio M. b,c, G. Pancheri b

a ( )Departament de Fısica, UniÕersitat Autonoma de Barcelona, E-08193 Bellaterra Barcelona , Spain´ `b INFN-Laboratori Nazionali di Frascati, P.O.Box 13, I-00044 Frascati, Italy

c Instituto de Fısica, UniÕersidad de Guanajuato, Lomas del Bosque a 103, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico´ ´

Received 29 March 2000; received in revised form 14 June 2000; accepted 20 June 2000Editor: R. Gatto

Abstract

q y 0 0The ratio F™K K rK K is discussed and its present experimental value is compared with theoretical expectations.A difference larger than two standard deviations is observed. We critically examine a number of mechanisms that couldaccount for this discrepancy, which remains unexplained. Measurements at DAF NE at the level of the per mille accuracycan clarify whether there exist any anomaly. q 2000 Published by Elsevier Science B.V.

1. Introduction

The f-meson was discovered many years ago as aw xKK resonance 1 . Its decay is dominated by the two

KK decay modes which proceed through Zweig-ruleallowed strong interactions. The ratio R'f™

q y 0 0K K rK K has been measured in a variety ofindependent experiments using different f-produc-tion mechanisms. Among these, the cleanest one iselectron-positron annihilation around the f reso-nance peak, i.e. the reactions eqey

™ f™q y 0 0K K rK K , which have been accurately mea-

w xsured at Novosibirsk quite recently 2 and are theobject of intense investigation at the Frascati F-fac-

w x 6tory 3 . With as much as 8=10 f ’s on tape, theKLOE experiment at DAF NE can be expected tomeasure the above ratio R with a statistical accuracyof the order of the per mille. In view of this, we wish

to discuss the theoretical expectations and comparethem with the most recent determinations for thisratio.

In the following we shall first review the presentexperimental situation, then compare it with the naıve¨expectations from isospin symmetry and phase spaceconsiderations thus observing that a disagreementseems to exist. Contributions arising from electro-magnetic radiative corrections and m ym isospinu d

breaking effects are analyzed and shown to bring theobserved discrepancy to be more than three standarddeviations. Various additional theoretical improve-ments on our analysis, such as the use of vector-me-son dominated electromagnetic form-factors, themodification of the strong vertices and the inclusionof rescattering effects through the scalar resonancesŽ . Ž .f 980 and a 980 using the charged kaon loop0 0

model, are also examined and shown not to change

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .PII: S0370-2693 00 00770-X

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413 407

in any substantial way our results which imply aclear discrepancy between theory and the availabledata.

The first combined measurement of the four ma-jor f decay modes in a single eqey dedicatedexperiment has been performed quite recently withthe general purpose detector CMD-2 at the upgraded

q y w xe e collider VEPP-2M at Novosibirsk 2 . Havinga single experiment normalized to almost 100% ofdecay modes implies a reduction of systematic er-

Ž .rors, and the following branching ratios BR andw xerrors from VEPP-2M 2 are quoted:

BR f™KqKy s 49.2"1.2 % ,Ž . Ž .

0 0BR f™K K s 33.5"1.0 % , 1Ž . Ž .Ž .leading to

BR f™KqKyŽ .R ' s1.47"0.06 . 2Ž .exp 0 0BR f™K KŽ .

All these results were in agreement with the averagevalues quoted in the then available PDG 1994 com-

w xpilation 4 :

BR f™KqKy s 49.1"0.9 %Ž . Ž .0 0 5BR f™K K s 34.3"0.7 %Ž .Ž .

´R s1.43"0.04 . 3Ž .exp

w xThe current PDG edition 5 , now including theabove VEPP-2M data, quotes

BR f™KqKy s 49.1"0.8 %Ž . Ž .0 0 5BR f™K K s 34.1"0.6 %Ž .Ž .

´R s1.44"0.04 , 4Ž .exp

as a result of a global fit, which appears as a verystable result, established with a 3% error. In thesame PDG edition, one can also find R s1.35direct

"0.06, as the averaged result of the various experi-q y 0 0ments measuring the ratio f™K K rK K di-

rectly. A reduction of these errors can be expectedfrom DAF NE, where the KLOE experiment hasalready collected 8=106 f-mesons. Like in the case

of the CMD-2 detector, all the main decay modes ofthe f will be measured by the same apparatus andthis could bring the systematic errors to a minimum,while the statistics will allow to bring the statisticalerror well below the 1% level. Our discussion cen-ters around this ratio R and the possible interest instudying it with a much reduced experimental error.

We shall approach this discussion by starting withthe most naıve result for the above ratio R, i.e.¨Rs1, which follows from assuming that these f™

KK decay modes proceed exclusively via the stronginteraction dynamics in the good isospin limit m su

m and ignoring phase space differences. The massd

difference between neutral and charged kaons –Žwhich includes both isospin breaking effects m /u

. Ž .m and electromagnetic photonic contributions –d

considerably increases this too-naıve prediction via¨Ž .the purely kinematical phase-space factor. Assum-

ing now perfect isospin symmetry only for the strongŽ q yinteraction dynamics equal couplings for fK K

0 0.and fK K and knowing that f™KK are P-wavedecay modes of a narrow resonance, one necessarilyhas

3r22 2

q1y4m Mž /K f

Rs s1.528 , 5Ž .3r22 2

01y4m Mž /K f

with negligible errors coming from the mass valuesquoted in the PDG. The phase-space correction thuspushes the ratio R two standard deviations above its

Ž .experimental value 4 . This kinematical correctionis exceptionally large because of the vicinity of thef mass to the KK thresholds, which translates intoconsiderably large differences between the charged

Žand neutral kaon momenta or velocities, Õ rÕ sq 0.0.249r0.216s1.152 , a difference which is further

increased to its third power in such P-wave decaymodes.

This two-s discrepancy between experiments andthe theoretical tree level predictions obviously claimsfor further corrections. The most immediate of suchcorrections is due to electromagnetic radiative effectson the ratio R, which affect the numerator but notthe denominator, and which will be discussed in thenext section.

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413408

2. Electromagnetic radiative corrections

Electromagnetic radiative corrections are fre-quently ignored when dealing with strong decays. Inour case, they could be relevant since, althoughsmall, they affect the charged decay mode but notthe neutral one, and, in order to solve the discrep-ancy in the ratio R under consideration, only a fewper cent correction is needed. Many years ago theywere already considered by Cremmer and Gourdinw x6 who found a positive correction of the order of

Ž .4% to the prediction in Eq. 5 , thus enlarging thatdiscrepancy. The dominant contribution was found toarise from the so-called Coulomb term which ispositive for f™KqKy and rather large because ofthe small kaon velocities Õ s0.249. A similar in-"

Ž .crease of the ratio R some 5% by radiative correc-tions is expected by the experimentalists at VEPP-2Mw x2 , whose quoted result is inclusive of any vertexcorrection. If we include this correction in the theo-retically predicted ratio, the final result for the radia-

w xtively corrected ratio is then R,1.59 6 , in agree-ment with still another independent analysis by

w xPilkuhn leading to R in the range 1.52–1.61 7 . Tobetter qualify these statements, we shall now exam-ine in detail the contribution of such corrections tothe ratio R.

We have recalculated the electromagnetic radia-tive corrections to f™KqKy along the lines of

w xRef. 6 . For the charged amplitude we start with theŽusual and simplest tree level expression A f™0

q y. Ž . mK K sg e p yp , where g is the uncor-0 m q y 0

rected strong coupling constant for fKK , e is them

f polarization and p are the kaon four-momenta."

As is well known, the various contributions to theradiative corrections can be grouped in two parts.The first part comprises one-loop corrections to the

Ž q y.uncorrected amplitude A f™K K . This part0

contains three vertex diagrams with one virtual pho-ton exchanged between the two charged-kaons orbetween the fKqKy vertex and each charged-kaon.In addition, it also contains wave-function renormal-ization of external kaon lines that render the wholeamplitude ultraviolet finite1. The second part is

1 Ž . w xNotice that Eq. 19 in Ref. 6 contains a small imaginarypart while it is supposed to be the real part of the one-loopamplitude.

needed to cancel the infrared divergence. It containsthree real-photon emission diagrams which are order'a . Adding these two parts we find the completeorder a corrective factor to the f™KqKy decaywidth

2 DE1qC qb logf f

qmK

a 1qÕ2 2 DE2'1q p y2 1q log½ ž /qp 2Õ mK

1qÕ2 1yÕ 1 1yÕ= 1q log y logž /2Õ 1qÕ Õ 1qÕ

1qÕ2 1yÕ 1yÕ2

y log log4Õ 1qÕ 4

21qÕ 2Õ y2Õy Li yLi2 2ž / ž /2Õ 1qÕ 1yÕ

21qÕ 1qÕ 1yÕq Li yLi2 2ž / ž /2Õ 2 2

1qÕ2

y Li Õ yLi yÕ , 6Ž . Ž . Ž .2 2 5Õ

2 2qwhere Õs 1y4m rM is the kaon velocity and( K f

DE stays for the photon energy resolution. For DEŽ .s1 MeV the correction 6 amounts to a 4.2%

increase. Taking for DE the maximal available pho-Žton energy 32.1 MeV, not far from the energy

resolution in the KLOE detector at DAF NE, which.is f 20 MeV makes no substantial difference as

the main contribution comes from the Coulomb term,the first one inside the brackets.

The above discussion ignores the fact that what isactually measured at VEPP-2M and at DAF NE isthe ratio

s eqey™f™KqKyŽ .

q yR ' , 7Ž .e e q y 0 0s e e ™f™K KŽ .and that radiative corrections to R correspond toconsider the ratio of the radiatively corrected cross-sections which appear at the numerator and denomi-

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413 409

nator of R q y. In addition to consider both initiale e

and final state corrections, a complete treatment alsorequires to discuss the presence of the f resonance

w xand the associated distortion of the cross-sections 8 .At the numerator, radiative corrections include vir-tual corrections as well as emission of soft unob-served photons, both from the initial and final states,with no interference between initial and final state

Žradiation for an inclusive measurement i.e. in ameasurement that does not distinguish the charges of

. w xthe kaons 9 . For the cross-section at the denomina-tor, there are only initial state radiative correctionssince the final kaons are neutral. In the absence offinal state radiation, the presence of a narrow reso-nance like the f in the intermediate state introduceslarge double logarithms which can be resummedw x8,10 and factorized in an expression like

b iGf

1qC , 8Ž . Ž .iž /Mf

2a swhere b s log y1 is the initial state radia-2Ž .i p me

tion factor and C is the finite part of the initiali

virtual and soft photon corrections, which survivesafter the cancellation of the infrared divergence andthe exponentiation of the large resonant dependentfactors. The same factor for initial state radiationappears both at numerator and denominator, andsince there is no interference between initial andfinal state radiation, the real soft-photon radiativecorrections to the initial state cancel out in the ratioŽ .7 . In principle, one should also resum the contribu-tions coming from final state radiation but the finalstate radiative factor

2a 1qÕ2 1qÕy4b s log y1 ,3.9=10f ž /p 2Õ 1yÕ

is very small and resummation in this case is irrele-vant. One then obtains the following expression for

Ž .q ythe ratio R as defined in Eq. 7 :e e

G f™KqKyŽ .q yR se e 0 0G f™K KŽ .

=

2 DE1qC qC qb logi f f

qmK. 9Ž .

1qCi

Since

a s p 23 y2C f log q y2 ,5.6=10i 2 2ž /p 3me

w x Ž .11 , one can expand the denominator in Eq. 9 ,canceling the C term and remaining with the finali

state correction terms C and b given explicitly inf fŽ .Eq. 6 . We thus conclude that one is justified in

using the expressions as above and that the conven-tional treatment of radiative corrections increases theprevious two-s discrepancy between experiment andtheory for the ratio R to the level of three standarddeviations.

( )3. SU 2 -breaking in f KK vertices

q y 0 0The fK K and fK K vertices are not equalŽ . Ž .and thus do not cancel in the ratio R once SU 2 -breaking effects are taken into account. The way

Ž .SU 2 -breaking is usually introduced in the effectiveŽ .lagrangians is the same as for SU 3 -breaking,

namely, via quark mass differences. In the latterŽ .SU 3 case, an improved description of the vector-

meson couplings to two pseudoscalar-mesons caneasily be achieved as shown, for example, in Refs.w x12,13 . But the situation is by far less convincing

Ž .when turning to the much smaller SU 2 -breakingw xeffects 14 . The essential feature – common to most

models – is that the dynamics of these flavoursymmetry breakings suppress the creation of heavier

q y 0 0qq pairs. In the fK K and fK K vertices, oneneeds to produce a uu and a dd pair, respectively.

0 0Since the latter is heavier, the f™K K decay isfurther suppressed and then the ratio R is furtherincreased. To be somewhat more precise, we willconsider two recent and independent models dealing

w xquite explicitly with such kind of effects 12,15 .Ž .In the SU 3 -breaking treatment of VP P ver-1 2

w xtices by Bijnens et al. 15 , these decays proceedthrough two independent terms containing the rele-

Ž .vant vector and pseudoscalar masses M and mV 1,2

and thus incorporating quark-mass breaking effects.w xIn the notation of Ref. 15 , to which we refer for

details, these VP P couplings are then proportional1 2

to2 2m qm1 22 'M g q2 2 f . 10Ž .V V x 2ž /MV

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413410

q y 0 0For the fK K and fK K coupling constants,the uncorrected strong coupling constant g be-0

comes, respectively,

2 2q 0M f mf x K , K'1q4 2 , 11Ž .22' ž /g M2 2 g f V f0

with the pion decay constant f,92 MeV. One thenobtains the ratio

2 2 <q y q 0g f m ym m / mf K K x K K u d',1q4 2 ,1.01 ,20 0g g MfK K V f

12Ž .2 2 < y3

q 0where we have used m ym ,y6 10m / mK K u d

GeV 2 for the non-photonic kaon mass difference1w x16 and the estimate f g ,y obtained in Ref.x V 3

w x )15 when fitting the r™pp and K ™Kp decaywidths.

Ž .Similarly, in the independent treatment of SU 3w xsymmetry breaking 12 , some relevant VP P cou-1 2

plings are given by

'g s 2 g ,rpp

q y 0 0g sg syg 1q2c 1yc , 13Ž . Ž . Ž .fK K f K K V A

Ž w xwith c ,0.28 and c ,0.36 see Ref. 12 forV A. Ž .notation and details mimicking the SU 3 mass

difference effects discussed in the previous approachw x Ž . Ž .15 . The transition from SU 3 - to SU 2 -breakingoffers no difficulties. One now obtains

2 2 <q y q 0g m ym m / mf K K K K u d,1y c ,1.01 . 14Ž .A2 20 0g m ymfK K K p

w x Ž .As in the approach of Ref. 15 , these SU 2 -break-ing corrections work in the undesired direction andthe discrepancy between theory and experiment forthe ratio R increases by an additional 2%.

Ž .An independent SU 2 -breaking effect can arisefrom r–f mixing. This is both isospin and Zweig-rule violating, and should therefore lead to rathertiny corrections. Indeed, in this context one canimmediately obtain the following relation among

2q y 0 0 q ycoupling constants : g y g s g ,fK K f K K fp p

2 Ž .Notice that this isospin relation not only accounts for r 770 –f mixing effects but also for those between f and any otherhigher mass isovector r-like resonance.

with a small value for the g q y coupling comingfp p

from the observed smallness of the f™pqpy

Ž Ž y4 . w x.branching ratio OO 10 5 in spite of its muchlarger phase space. A more quantitative estimate isnow possible thanks to the recent data on eqey

™fq y w x™p p coming from VEPP-2M 17 . These data

describe the pion form factor around the f peak,Ž 2 .F s,M , in terms of the complex parameter Z byf

the expression

ZM Gf fF s 1y . 15Ž . Ž .2ž /M ysy iM Gf f f

This Z, in turn, can be easily related to e , thefr

complex parameter describing the amount of r-like'Ž Ž . .or uuydd r 2 contamination in the f wave

function. One finds

f Gf f 2e ,y F ssM Z , 16Ž .Ž .fr ff Mr f

3where the first coefficient f f ,y is the well-'f r 2

known ratio of f–g to r–g couplings. One finallyobtains

q ygfK K ',1y 2 R e ,1.001 , 17Ž . Ž .fr0 0gfK K

w xwhere an average of the values for Z in Ref. 17 andŽ 2 . w xthe parametrization of F ssM from Ref. 18f

have been used in the final step. This time thecorrection is tiny and the accuracy of our estimate israther rough, but again it tends to increase the dis-crepancy on the ratio R.

4. Further attempts

Since the discrepancy between the theoretical andŽexperimental value for R remains or has even been

Ž .increased by some additional 2% due to the SU 2 -.breaking effects just discussed , we have tried to

improve our analysis in different aspects. First, wehave taken into account that the couplings of photons

Žto kaons, rather than being point-like as assumed in

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413 411

our previous and conventional treatment of radiative.corrections , are known to be vector-meson domi-

w xnated 19 . Accordingly, we have redone the calcula-tion performed in Section 2 including the corre-

Ž .sponding electromagnetic vector-meson dominatedkaon form-factors. Now, not only the decay mode

q y 0 0f™K K can be affected but also the f™K Kone due to the r, v and f mass differences. For thef™KqKy case, the contribution of the chargedkaon form-factor modifies the point-like result forŽ q y. y3G f™K K by f2 10 . For the case of f™0 0K K , a vanishing effect will be obtained in the

Ž .limit of exact SU 3 symmetry, and a fraction of theŽ .preceding one if SU 3 -broken masses are used. In

both cases, the effect of kaon form-factors on real-photon emission diagrams is null. So then, the addi-tional net effect of electromagnetic kaon form-fac-tors on the ratio R leads to a modification of thepoint-like radiative corrections result of Section 2 bysome per mille and is thus fully negligible.

A second and independent possibility consists inadopting a different framework for VPP decays.This is usually done in terms of more general effec-tive lagrangians with VPP vertices containing twoderivatives of the pseudoscalar fields instead of asingle one as in our previous discussion. The radia-tive decay r™pqpyg – quite similar to the pro-cesses we are considering – has been quite recently

w xanalyzed in this modern context in Ref. 20 . TheŽtwo relevant coupling constants F and G , in theV V

w x.notation of Ref. 21 and their relative sign can be' w xfixed to the canonical values F s2G s 2 f 22V V p

thanks to the experimental data for r™pqpyg andw xother r meson processes 5 . As discussed in Ref.

w x20 , a good description of these data is then achievedin terms of an amplitude that coincides with the one

w xpreviously introduced in Ref. 23 , and which origi-nated from the simple one-derivative VPP vertices

w xused by Ref. 6 as well as in our recalculation inSection 2. In other words, both types of effectivelagrangians lead to exactly the same real-photonemission amplitudes once the coupling constants areproperly fixed. This is also true for the other correc-tions concerning one-loop effects: for the canonicalvalue F s2G one reobtains precisely our previ-V V

Ž .ous expression in Eq. 6 .A third attempt includes the effect of final KK

rescattering through scalar resonances. It is well

known that the charged kaons emitted in f™KqKy

are always accompanied by soft photons. In the caseof single photon emission, the KqKy system isfound to be in a J P C s0qq or 2qq state with aninvariant mass just below the f mass. The presence

P C qq Ž .of the J s0 scalar resonances f 980 and0Ž .a 980 , with masses and decay widths that cover the0

Žinvariant mass range of interest from the KK. w xthreshold to the f mass 5 , would suggest that

rescattering effects could be important3. We havecomputed these rescattering effects through the ex-change of the f and a using the charged kaon loop0 0

w xmodel 24–26 . In this model, the f decays into aq y ŽK K system that emits a photon from the charged

q y .kaon internal lines and from the fK K vertexq y 0 0before rescattering into a final K K or K K

state through the propagation of f and a reso-0 0

nances. If the emitted soft photon is unobserved, theq yŽ . Ž . q yŽ .process f™K K g ™ f ra g ™K K g or0 0

0 0Ž .K K g contributes to the ratio R, both at thenumerator and denominator. In order to calculatethese effects, one needs an estimate of the couplingconstant g , where S is either the f or the a .SK K 0 0

Recent measurements of the f™ f g and a g decay0 0w xmodes at VEPP-2M 27 are consistent with the

predictions of the charged kaon loop model forvalues of the above couplings given by

2gf K K0 2s 1.48"0.32 GeV ,Ž .4p

2ga K K0 2s 1.5"0.5 GeV . 18Ž . Ž .4p

We have then found that the contribution of theseŽ q yŽ .. Ž y7 .kaon loops to the BR f™K K g is OO 10 ,

0 0 y9Ž Ž .. Ž .while for BR f™K K g is OO 10 . Forcharged kaons in the final state, there is an additionalcontribution from the interference between the soft-

3 Rescattering effects from 2qq states are suppressed becauseŽ . Ž .the nearest tensorial resonances, f 1270 and a 1310 , are well2 2

w xabove the f mass 5 .

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413412

bremsstrahlung and the scalar amplitudes. This con-tribution is given by

G f™KqKy gŽ .Ž .int

2 g 2q yq yg Mf K Kf K K f04sy a3 2 2

q4p 4p 2p mK

=1DE

dv v R I a,bŽ .H ž D mŽ .0 f 0

2q yg 1a K K0q 2 /q y D mg Ž .af K K 00

=1yÕ2 1yw

wq log , 19Ž .ž /2 1qw

Ž .where I a,b is the kaon loop integral defined inw x Ž 2 2 2 2 .q qRefs. 25,26 with a'M rm and b'm rm ,f K KŽ .D m are the scalar propagators and wf r a0 0

2 2q(s 1y4m rm , with msM 1y2vrM be-(K f f

ing the invariant mass of the KK system and v theŽ .photon energy. Using the values in Eq. 18 for the

scalar couplings, we find that the interference term,which contributes to R only in the numerator, is

Ž y5 .positive and OO 10 , i.e. completely negligible inspite of being the dominant one.

Admittedly, this estimate of the KK rescatteringeffects is model dependent and affected by largeuncertainties. Before concluding, we would thus liketo make a few comments on possible variations onthe magnitude of the scalar coupling constants and

Ž .the expressions for the scalar propagators D mf r a0 0

which enter into our evaluation in the precedingparagraph. The values of the couplings g dependSK K

on the nature of the scalar mesons, i.e. whether theyare two- or four-quark states, or KK molecules. Theresults of the KK molecule model, in addition to thecouplings g , depend upon the spatial extensionSK K

of the scalar KK bound state, and the predictions forŽ . Ž .BR f™ f ra g for the same g are always0 0 SK K

smaller than in the purely point-like case, i.e. theeffects on R tend to vanish for more extended

w xobjects 26 . The two-quark model, irrespectively of'Ž .the ss versus uuqdd r 2 quark content of theŽf , predicts too small values see, for example, Refs.0

w x. Ž .26,28 for the branching ratios BR f™ f ra g0 0

w x27 , and is unable anyway to account for the nearmass degeneracy of the isoscalar f and isovector0

a . On the other hand, such mass degeneracy is well0

understood in the four-quark model, critically reex-w xamined very recently in Refs. 29,30 . The four-quark

model also predicts values for g that seem to beSK K

in agreement with the available measurements ofŽ . w xBR f™ f ra g 27,28 . In all cases, the different0 0

possibilities are found to modify the previouslyquoted sizes of the KK rescattering effects by atmost one order of magnitude. Something similarhappens with the lack of consensus on the specificform for the scalar propagators to be used in theseestimates. Here the uncertainties arise because of theopening of the KK channels quite close to thenominal scalar masses. This translates into sharpmodifications of the conventional Breit–Wignercurves and changes the size of the KK rescatteringeffects again by one order of magnitude. Althoughaffected by large uncertainties, the contributionscoming from final-state KK rescattering are thusfound to be negligible and their effects on the ratio Rirrelevant.

5. Conclusions

In this letter, we have performed a discussion ofq y 0 0the ratio R'f™K K r K K . From the experi-

mental point of view, the value R s1.44"0.04expw xseems to be firmly established 5 . However, in our

present theoretical analysis of this ratio R we havefailed to reproduce the value R quoted above. In aexp

first and conservative attempt, including isospinsymmetry for the strong vertices and the appropriatephase-space factor, one obtains Rs1.53 which istwo s ’s above R . In a second step, we have alsoexp

included conventional electromagnetic radiative cor-rections to order a , thus obtaining Rs1.59 andincreasing the previous discrepancy up to three s ’s.This value confirms some existing results and hasbeen checked to be quite independent from the de-tails of the relevant vertices. In a third step, we havetried to correct our predictions for R introducingvarious isospin breaking corrections to the fKKcoupling constants. As a result, the ratio R is foundto be further increased by some 2%, an estimate

( )A. Bramon et al.rPhysics Letters B 486 2000 406–413 413

affected by rather large errors reflecting our poorŽ .knowledge on the SU 2 -breaking details. In view of

all this, we have introduced final-state rescatteringeffects which should be dominated by the almost

Ž . Ž .on-shell formation of the f 980 and a 980 reso-0 0

nances in the S-wave KK channel. The controversialnature of these scalar resonances allows for quitedisparate estimates of their effects, but one cansafely conclude that they are well below those previ-ously mentioned. The disagreement on the ratio R

Ž .persists well above two experimental standard de-viations. Higher statistics from DAF NE are ex-pected in order to settle definitively whether thediscrepancy on R is a real problem, or final agree-ment between theory and experimental data can beachieved.

Acknowledgements

We would like to thank J. Bijnens, M. Block, J.Gasser, E. Oset, M.R. Pennington and E.P. Solodovfor helpful comments and clarifying discussions.Work partly supported by the EEC, TMR-CT98-0169, EURODAPHNE network. J.L. Lucio M. ac-knowledges partial financial support from CONA-CyT and CONCyTEG.

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