contribution of the direct decay φ→π+π−γ to the process e+e−→π+π−γ at daΦne

23 March 2000

Ž .Physics Letters B 477 2000 114–124

Contribution of the direct decay f™pqpyg

to the process eqey™pqpyg at DAF NE

Kirill Melnikov a,1, Federico Nguyen b,2, Barbara Valeriani c,3, Graziano Venanzoni d,4

a Stanford Linear Accelerator Center, Stanford UniÕersity, Stanford, CA 94309, USAb Dipartimento di Fisica dell’UniÕersita and INFN, sezione di Roma Tre, Via della Vasca NaÕale 84, I-00146 Rome, Italy`

c ( )Dipartimento di Fisica dell’UniÕersita and INFN, sezione di Pisa, Via LiÕornese 1291, I-56010 S. Piero a Grado PI , Italy`d Institut fur Experimentelle Kernphysik, UniÕersitat Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany¨ ¨

Received 19 January 2000; accepted 15 February 2000Editor: P.V. Landshoff

Abstract

The potential of DAF NE to explore direct radiative decay f™pqpyg is studied in detail. Predictions of differenttheoretical models for this decay are compared. We find that it should be possible to discriminate between these models atDAF NE in one year, even assuming a relatively low luminosity LLs1031 cmy2 sy1. The influence of the decay

q y Ž q y .f™p p g on the measurement of total cross section s e e ™hadrons by tagging a photon in the reactioneqey™pqpyg is also discussed. q 2000 Published by Elsevier Science B.V. All rights reserved.

PACS: 13.65.q i; 13.25.Jx; 12.39.Fe; 13.40.Gp

1. Introduction

Investigation of CP violation is the most impor-tant physical goal of DAF NE, a high luminosity

q y Ž .e e collider which operates on the f 1020 reso-nance. However, thanks to high luminosity, there

1 E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

will be a substantial amount of data which may beused to advance our knowledge on low energy hadrondynamics and even contribute to precision elec-

w xtroweak measurements 1 .w xRecently it was suggested 2 , that the annihilation

Ž q y .cross section s e e ™hadrons at energies belowthe mass of the f resonance may be studied atDAF NE using reaction eqey

™hadronsqg . Bytagging the photon it is possible to determine thepion form factor at the momentum transfer below the

w xmass of the f meson 2 . There are several possibili-w xties to improve further the analysis of Ref. 2 and in

this paper we consider the contribution of the directrare decay f™pqpyg to the reaction eqey

™

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00209-4

( )K. MelnikoÕ et al.rPhysics Letters B 477 2000 114–124 115

q y w xp p g . Using the terminology of Ref. 2 , thedirect decay contributes to the final state radiationwhich, for the purpose of the cross section measure-ment, has to be suppressed by an appropriate choiceof cuts on the photon and pion angles and energies.One of the aims of the present paper is to find outhow the contribution of the direct decay affects the

w xanalysis of Ref. 2 .Besides that, the rare decay f™pqpyg is an

interesting process by itself. As one deals here withthe low energy limit of QCD, the first principlescalculations are not possible and one has to resort to

w xvarious models 3–8 . Since the number of models isflourishing, we think that the experiments shoulddistinguish between them. In principle, that can beachieved by studying the low energy region of thephoton spectrum in the reaction eqey

™pqpyg

w x9,10 , but it is not an easy task. The reason is thatthe relative phases of the direct decay f™pqpyg

Ž Ž .and the pure QED processes initial ISR and finalŽ . .FSR state radiation are not predicted by thesemodels. As a consequence, the sign of the interfer-ence term 5 appears to be to a large extent arbitrary.If one assumes that the interference is destructive,the branching ratio of the direct decay becomes very

Ž q y . y5small, BR f™p p g f4=10 . Under suchcircumstances, a detailed study of the decay f™

pqpyg will be rather difficult, requiring high statis-tics and a careful control over efficiencies in order todiscriminate between different models by fitting thephoton spectrum.

In this paper we report on the implementation ofthe direct decay f™pqpyg into the Monte Carloevent generator for pure QED process eqey

™

q y w xp p g described in Ref. 2 . Our implementationpermits to choose between different models for thedecay f™pqpyg . A clear advantage of having aMonte Carlo event generator for these studies is thatit allows to keep control over efficiencies and resolu-tion of the detector, fine tuning of the parameters andalso provides for the possibility to generate realistic

5 Let us note that for symmetric cuts on the pions angles theinitial state radiation does not contribute to the interference termbecause of charge parity conservation.

distributions where the reaction eqey™pqpyg is

accompanied by radiation of photons collinear tow xelectrons and positrons 2 .

2. The matrix element for p Hp Ig final state

The matrix element for the direct decay f™

pqpyg is parameterized as:

MM syief Q2 ,Qq e md e a ,Ž .0 f f ma

d s Qq g yq Q , 1Ž . Ž .ma ma m a

where e and e are polarizations of the f mesonf

and the photon, respectively; q is the momentum ofthe photon and Q is the momentum of the f. The

Ž 2 . Ž .function f Q ,Qq in Eq. 1 is the form factor forf

the direct decay. Its exact form depends on thechosen model.

Considering production of the f meson in eqey

collision with the center of mass energy squaredssQ2 and its subsequent decay to pqpyg finalstate, we find:

yie32 aMM s F Q ,Qq Õ p g u p d e , 2Ž . Ž . Ž .Ž .f f 2 m 1 ma2Q

where the form factor F is defined as:f

g ffg fF s . 3Ž .f 2 2Q yM q iM Gf f f

The coupling constant g describes the mixing offg

the photon and the f meson and can be determinedfrom the decay width of the f meson into electronpositron pair. Using

4p g 2 a 2fgq yG f™e e s 4Ž . Ž .33 Mf

Ž q y. y6and G f™ e e s 1.3246 P 10 GeV, M sf

1.0194 GeV, one obtains g s7.929P10y2 GeV 2.fg

( )K. MelnikoÕ et al.rPhysics Letters B 477 2000 114–124116

yŽ . qŽ .Consider now the QED process e p qe p1 2qŽ . yŽ . Ž .™p p qp p qg q . The initial state radi-1 2

ation amplitude reads:

yie3F Q2Ž .p 1MM s Õ pŽ .isr 22Q1

=g p yq g g yp qq gˆ ˆ ˆ ˆŽ . Ž .m 1 a a 2 m

qy2 p q y2 p q1 2

=u p p me a , 5Ž . Ž .1

where Q sp qp and psp yp .1 1 2 1 2

For the amplitude of the final state radiation weobtain:

ie3F Q2Ž .pMM s Õ p g u pŽ . Ž .fsr 2 m 12Q

=

a a2p qq2 m myp qqŽ .2p q2

a ay2p yqŽ .1 m m amq yp yq y2 g e .Ž . a2p q1

6Ž .

Then, the differential cross section for eqey™

pqpyg can be written as:

ds sds qds , 7Ž .total QED f

where ds is the contribution considered in Ref.QEDw x2

< < 2ds ; MM qMM , 8Ž .QED isr fsr

and

2) )< <ds ; MM q2Re MM MM q2Re MM MM� 4 � 4f f isr f fsr f

9Ž .

includes the amplitude of the direct decay. One seesthat ds contains different interference terms. Forf

this reason one might expect a significant depen-dence of the f™pqpyg signal on the relativephases of MM and MM qMM . We will show belowf isr fsr

that this is indeed the case. We now describe threedifferent models for the direct decay f™pqpyg

that are implemented in our event generator.

w x1. ‘‘No structure’’ model 3 . In this case thedecay f™pqpyg occurs through two subsequenttransitions f™ f g™pqpyg . The form factor f0 f

Ž .in Eq. 1 becomes:

g g q yf f g f p p0 0no str .f s . 10Ž .f 2 2m yQ y im Gf 1 f f0 0 0

The coupling constants in the above equation canbe estimated by using the information on the branch-

Ž .ing ratio Br f™ f g and on the branching ratio0Ž q y. w xBr f ™p p . One obtains 3 :0

< <q yg g s 144"15 Br f™ f g .(Ž . Ž .f f g f p p 00 0

Needless to say, that the region of applicability ofthis model is restricted to relatively soft photons,when the f meson in the intermediate state is not0

too far off shell. For this reason, when implementingthis model into the event generator, we have intro-duced an additional exponential damping factorwhich suppresses the emission rate for high energy

w xphotons 9 :

QqŽ .no str . no str .f ™1.625= f exp y , 11Ž .f f 2½ 5D

w xwith Ds0.3 GeV 9 .q y w x2. K K model 5,4 . In this model one also has

a two step transition, similar to ‘‘no structure’’model. However, the f™ f g decay amplitude is0

generated dynamically through the loop of chargedŽ .kaons. The form factor f in Eq. 1 reads:f

g q yg q yg q y2 2m Qq y fK K f p p f K K f 10 0K Kf s I , .f 2 22 2 2 2 ž /m m2p m m yQ y im GŽ . K KK f 1 f f0 0 0

12Ž .

The coupling constants g q y, g q y, g q y canfK K f p p f K K0 0

be estimated by using the information on correspond-w xing decay rates 5 :

2 g 2q yq yg f p pf K K 0s1.66, s0.105,24p 4p mf 0

g 2q yf K K0 2s0.6 GeV , 13Ž .

4p


Ž .and I a,b is the function known in the analyticw xform 5,4,11 :

1I a,b sŽ .

2 aybŽ .2

y1 y1y f b y f aŽ . Ž .2aybŽ .a

y1 y1q g b yg a . 14Ž . Ž . Ž .2aybŽ .Ž . Ž .The functions f x and g x are given by:

1°12yarcsin x) ,4'2 x~f x sŽ . 2hq1 1ln y ip x- ,4 4¢ hy

15Ž .1°

1'4 xy1 arcsin x) ,4'2 x~g x sŽ .hq1 1'1y4 x ln y ip x- ,2 4¢ hy

'Ž . Ž .with h s 1" 1y4 x r 2 x ."

( ) w x3. Chiral unitary approach Ux PT 8,11 . In thiscase the decay f™pqpyg occurs through a loopof charged kaons that subsequently annihilate intopqpyg . The f resonance is generated dynamically0

by unitarizing the one-loop amplitude. Using nota-w x Ž .tions of Ref. 8 , the form factor f in Eq. 1 reads:f

G M m2 Q2V f f 1Ux P Tf s t I ,f ch 2 22 2 2½ ž /' m mf 2 2 p m K Kp K

'2 FVq yq yG G , 16Ž .V K K2 ž / 52M ff p

where the coupling G and F are related to theV V

decays f™KqKy and f™eqey, respectively, fpŽ .is the pion decay constant and I a,b is the function

Ž . q ygiven in Eq. 14 . G is defined by the integral:K K

1q yG sK K 22p

=q2dqqmax

.H2 2 2 2 20 (q qm Q y4 q qm q ieŽ .Ž .K 1 K

17Ž .

Ž .In Eq. 16 t is the strong scattering amplitudech

1Is0t s t . 18Ž .ch K K ,pp'3

Is0The scattering amplitude t is determined byK K ,pp

Ž w x.using chiral perturbation theory see Ref. 7 . Wehave used the following values for the above con-stants: G s 0.055 GeV, F s 0.165 GeV, f sV V p

0.093 GeV, q s0.9 GeV.max

In Fig. 1 we present a comparison of the photonspectrum obtained using the event generator and

< < 2retaining only the term MM in the cross sectionf

Ž Ž ..cf. Eq. 9 , with the analytic expressions from Refs.w x3,5,8 . One sees a good agreement between theMonte Carlo simulation and the analytic results.Note also, that different models predict differentshapes of the photon spectrum.

3. Studying the direct decay f™p Hp Ig atDAF NE

We now address the question of whether precisionstudies of the direct decay f™pqpyg are possibleat DAF NE. While writing the general formula forthe process eqey

™pqpyg , we have pointed outthat the observable signal of f™pqpyg mightstrongly depend on the interference with the FSR.The f signal may be enhanced if the sign between0< < 2 ) ŽMM and 2Re MM MM is the same constructive� 4f fsr f

.interference or may be reduced in the opposite caseŽ .destructive interference .

In Fig. 2 we present the spectrum of photons inthe reaction eqey

™pqpyg in the situation whenthe invariant mass of two pions is close to the massof the f meson. We consider both constructive and0

destructive interference and generate events with andwithout collinear radiation, but initial and final stateradiation is always kept. One sees from Fig. 2 thatthe collinear radiation results in the reduction of thesignal. However, if the tagged photon is emitted at arelatively large angle, the effect of collinear radiationcan be partially removed by combining the informa-tion on the position of the neutral cluster in thecalorimeter with the directions of the charged pionsdetermined with the drift chamber. In this case thekinematics of the reaction becomes over-constrained


Fig. 1. The energy spectrum of photons produced in the direct decay f™pqpyg without initial and final state radiation. The results of theŽ . w xMonte Carlo simulation are compared with the results of analytical calculations solid curves for different models 3,5,8 . We useŽ . y4 Žm s0.976 GeV, G s34 MeV. For the ‘‘no structure’’ model the branching ratio Br f™ f g s10 has been used as an input thisf f 00 0.branching ratio is of the same order of magnitude as is predicted by the other two models . For the purpose of comparison, no exponential

damping has been applied to the ‘‘no structure’’ model and the events were generated without collinear radiation. We have applied thefollowing cuts on the polar angle and the energy of the photons: 108-u -1708, 20 MeV -E -100 MeV.g g

and it is possible to restore the ‘‘actual’’ center ofmass energy for any given event. This will require adedicated analysis, however.

Fig. 3 shows the signal-to-background ratio

dstotalSrBs 19Ž .

dsQED

for different models. As expected, the sign of theinterference affects not only the magnitude of the

decay f™pqpyg , but also the shape of the distri-bution. The models where the structure of the f0

meson is assumed show a broader signal for theconstructive interference 6 than in the opposite case.

6 The excess of events is significant in the region of photonenergies 20MeV- E -100MeV.g


q y q y < < < <"Fig. 2. The spectrum of photons in the reaction e e ™p p g . The following cuts were applied: cosu -0.9, cosu -0.9,g p

< < qcosu -0.9, where u and u are the polar angles of photon and pions, respectively, and u is the angle between the photon and the ppg g p pg

momenta in the center of mass frame of the two pions. The exponential damping was applied to the ‘no structure‘’’ model and the branchingŽ . y4ratio Br f™ f g s2.5=10 has been chosen. Different models are distinguished by different hatching. ‘‘Pure QED’’ means that only0

the contribution due to ds is considered.QED

In addition, the ‘‘no structure’’ model does not showa clear peak in the case of destructive interference

and the Ux PT model in the case of constructiveinterference.


Ž . Ž . < <Fig. 3. The signal-to-background ratio ds rd E r ds rd E as a function of photon energy. The cuts are cosu -0.9,total g QED g g

< < < <"cosu -0.9, cosu -0.9. See text for more details.p pg

The number of events required to separate thesignal from the background can be obtained by

estimating the necessary number of events in theenergy region around the f peak. We require the0


Table 1The number of events and the integrated luminosity required to observe the direct decay f™pqpyg

Collinear radiation Models Constructive interference Destructive interferencey1 y1Ž . Ž .j aevents LL pb j aevents LL pb

a awithout no structure 0.35 1100 1 0.01 1000000 ;2000q y a bK K 0.22 ;2100 2 0.07 22000 40

a bUx PT 0.1 110000 ;11 0.15 5100 9a awith no structure 0.25 ;2000 ;2 0.01 1000000 ;2000

q y a bK K 0.16 ;4500 ;4 0.04 65000 118a bUx PT 0.08 ;17000 16 0.1 11000 20

a 20 MeV-E -100 MeV. b 20 MeV-E -50 MeV.g g

statistical error to be smaller than 10% of the signalitself. Hence,

DNdNF , 20Ž .

10

where DN is the number of events due to directdecay of the f meson:

DNsN yN . 21Ž .total QED

If we introduce a parameter j such that

N s 1qj N , 22Ž . Ž .total QED

Ž .Eq. 20 takes the form:

j NQEDdNF . 23Ž .

10

The value of j can be estimated from the SrB ratioshown in Fig. 3. Using the standard formula forstatistical fluctuation, dNs N , we estimate the( total

number of events required to separate the contribu-tion of f™pqpyg from the QED background:

210 NQEDN G 1qj , LLG . 24Ž . Ž .QED ž /j esQED

Here e is the overall detector efficiency for pqpyg

events and LL is the required integrated luminosity.The results are summarized in Table 1. We use

< <es0.5 and s s2.1 nb with the cuts cosu -0.9,QED g

< < < < Ž"cosu -0.9, cosu -0.9 without collinear ra-p pg

.diation . When the collinear radiation is included,s is reduced to approximately 2 nb.QED

In a similar way we make a rough estimate of thenumber of events and the luminosity required todiscriminate between different models for the directdecay. We obtain:

210N G , 25Ž .QED ž /j 12

with

j sj yj , 26Ž .12 1 2

and j are the j-parameters for two models under1,2

consideration. In Table 2 we summarize the results.One can see that the required number of events canbe accumulated at DAF NE in less than one yearassuming the luminosity LLs1031 cmy2 sy1. Therequired luminosity is, however, only indicative.

4. Direct decay of the f meson and the measure-ment of the electron positron annihilation crosssection at DAF NE

w xIt was suggested in Ref. 2 that the measurementŽ q y .of s e e ™hadrons at DAF NE for different val-

Table 2The number of events and the integrated luminosity required to distinguish between two different models for the rare decay f™pqpyg

Interference Models Without collinear radiation With collinear radiationy1 y1Ž . Ž .j aevents LL pb j aevents LL pb12 12

q y 3 3constructive no structure versus K K 0.13 6=10 ;5 0.09 ;11=10 ;11q y 3 3destructive K K versus Ux PT 0.08 16=10 29 0.06 26=10 47


ues of the center of mass energy can be performedby analyzing events with additional hard photon

Ž .emitted at a relatively large angle u )78 . Theg

difficulties of this approach are related to the obvi-

ous fact that the hard photon can be emitted fromboth initial and final state of the process. If the ISRtakes place, the total energy of the collision is re-duced and such events can be used to measure

Ž . Ž .Fig. 4. Ratio ds rd E r ds rd E as a function of the energy of the photon. The cuts are 78-u -208, 308-u -1508, and thetotal g ISR g g p2 2 Ž . Ž .q yinvariant mass of detected particles in the final state Q )0.9 GeV . ‘‘Pure QED’’ ratio is defined as ds rd E r ds rd E .p p g ISRqFSR g ISR g

Different pictures correspond to different models for the f resonance. The cases of constructive and destructive interference are considered.0

See text for more details.


Ž q y .s e e ™hadrons at different energies. In contrastto that, the photons caused by the FSR represent abackground that must be suppressed by applying

w xsuitable cuts. Since, to be competitive 1,2 , theq y 'Ž .measurement of s e e ™hadrons for s -1 GeV

has to be performed at the one percent level, thepractical realization of this idea is a non-trivial ex-

w xperimental task. In Ref. 2 only the QED processwas studied. Here we would like to add the directdecay f™pqpyg which also contributes to theFSR and therefore increases the background. In Fig.

Ž . Ž .4 we show the values of ds rd E r ds rd Etotal g ISR g

with and without the contribution of the direct decay.w xThe ‘‘pure QED’’ case was studied in Ref. 2 . The

photon energies 20 MeV-E -100 MeV are con-g

sidered, which corresponds to 0.836 GeV 2 -Q q y2p p

-0.996 GeV 2. These invariant masses of two pionsinclude the contribution of the f resonance and for0

this reason the largest contribution of the directdecay is expected in this region. The cuts reduce theFSR considerably; nevertheless, its contribution closeto f peak is significant.0

w xAs discussed in Ref. 2 , even ‘‘pure QED’’theoretical predictions for the FSR are, strictly speak-ing, model dependent. It is therefore important to get

w xa handle on it experimentally. In Ref. 2 it wassuggested to use the forward-backward asymmetryof the produced pions to control the FSR. The directdecay f™pqpyg changes the forward-backwardasymmetry in the expected manner. Since the contri-

Fig. 5. pq angular distribution. The photon angle is 608-u -1208. The invariant mass of the two pions is 0.836 GeV2 -Q q y2 -g p p

0.996 GeV2.


bution of the direct decay is significant only if theinvariant mass of the two pions is close to the massof the f meson, the forward-backward asymmetry0

integrated over large range of Q q y2 is not affectedp p

by the direct decay. Hence it can be used to controlthe models for QED-like final state radiation. On theother hand, by applying the cut 0.836 GeV 2 -Q q y

2p p

-0.996 GeV 2, we significantly enhance the contri-bution of the direct decay to forward-backwardasymmetry. This is shown in Fig. 5 where predic-tions of KqKy model are displayed for both con-structive and destructive interference.

5. Conclusions

We have discussed the contribution of the directdecay f™pqpyg to the process eqey

™pqpyg

at DAF NE energies. To facilitate this study, threedifferent models 7 for the direct decay f™pqpyg

have been implemented into the Monte Carlo eventw xgenerator described in Ref. 2 .

The importance of this decay is twofold. First, itŽ .gives the information about the nature of the f 9800

meson. Second, it provides an additional backgroundŽ q y q y.to the measurement of s e e ™p p at differ-

ent values of the center of mass energy by taggingthe hard photon in the reaction eqey

™pqpyg .We have shown that DAF NE has a very good

potential to study the nature of f resonance. Even0

with moderate luminosity LLs1031 cmy2 sy1, it ispossible to discriminate between different models forthe decay f™pqpyg in a relatively short time.

As for the measurement of the hadronic crossq y 'Ž .section s e e ™hadrons at s -1 GeV using the

process eqey™pqpyg , we have found that the

direct decay f™ pqpyg increases the finalstate radiation by several percent in the region of

7 It is relatively straightforward to include other models for thew xdirect decay, for example the four quark model of Ref. 6 , to the

event generator. We plan to do that in the nearest future.

pion invariant masses 0.836 GeV 2 - Q q y2 -p p

0.996 GeV 2, but quickly dies out beyond this region.Finally, we note that it will also be possible to

perform a detailed study of the decay f™p 8p 8g atDAF NE. We believe that it will be quite useful tocombine these independent measurements in order tocheck the theoretical understanding of the f meson.0

Acknowledgements

We are grateful to J.H. Kuhn, W. Kluge, G.¨Pancheri, M. Greco, A. Denig and G. Cataldi foruseful discussions and E. Marco for providing uswith his code. We also thank M. Pennington fororganizing a pleasant EuroDAF NE meeting inDurham where part of this work was done. Thiswork is supported by the EU Network EURO-DAPHNE, contract FMRX-CT98-0169, by BMBFunder the contracts BMBF-06KA860 and BMBF-057KA92P, by the United States Department ofEnergy, contract DE-AC03-76SF00515, byGraduierten-kolleg ‘‘Elementarteilchenphysik anBeschleunigern’’ at the University of Karlsruhe andby the DFG Forschergruppe ‘‘Quantenfeldtheorie,Computeralgebra und Monte-Carlo-Simulation’’

References

w x Ž .1 S. Eidelman, F. Jegerlehner, Z. fur. Physik C 67 1995 585.¨w x Ž .2 S. Binner, J.H. Kuhn, K. Melnikov, Phys. Lett. B 459 1999¨

279.w x3 A. Bramon, G. Colangelo, M. Greco, Phys. Lett. B 287

Ž .1992 263.w x Ž .4 F.E. Close, N. Isgur, S. Kumano, Nucl. Phys. B 389 1993

513.w x Ž .5 J.L.M. Lucio, M. Napsuciale, Phys. Lett. B 331 1994 418.w x6 N.N. Achasov, V.V. Gubin, E.P. Solodov, Phys. Rev. D 55

Ž .1997 2672.w x Ž .7 J.A. Oller, E. Oset, Nucl. Phys. A 620 1997 438.w x8 E. Marco, S. Hirenzaki, E. Oset, H. Toki, hep-phr9903217.w x9 P.J. Franzini, W. Kim, J.L. Franzini, Phys. Lett. B 287

Ž .1992 259.w x10 R.R. Akhmetshin et al., hep-exr9907005.w x Ž .11 J.A. Oller, Phys. Lett. B 426 1998 7.

contribution of the direct decay φ→π+π−γ to the process e+e−→π+π−γ at daΦne

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