Interpreting the peak structures around 1800 MeV in the BES data on J/Ψ→ϕ π + π − , J/Ψ→γωϕ
K. P. Khemchandani, A. Martinez Torres, M. Nielsen, F. S. Navarra, D. Jido, A. Hosaka, E. Oset
aa r X i v : . [ h e p - ph ] N ov Interpreting the peak structures around 1800 MeV in theBES data on J / Y → fp + p − , J / Y → gwf K. P. Khemchandani ∗ , A. Martínez Torres ∗ , M. Nielsen ∗ , F. S. Navarra ∗ , D. Jido † , A.Hosaka ∗∗ and E. Oset ‡ ∗ Instituto de Física, Universidade de São Paulo, C.P 66318, 05314-970 São Paulo, SP, Brazil. † Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan. ∗∗ Research Center for Nuclear Physics (RCNP), Mihogaoka 10-1, Ibaraki 567-0047, Japan. ‡ Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos deInvestigación de Paterna, Aptd. 22085, 46071 Valencia, Spain.
Abstract.
In this talk we present an interpretation for the experimental data available on two different processes, namely, J / Y → fp + p − , J / Y → gwf , which seem to indicate existence of two new resonances with the same quantum numbers( J p c = ++ , I =
0) and very similar mass ( 1800 MeV) but with very different decay properties. However, our studies showthat the peak structure found in the wf invariant mass, in J / Y → gwf , is a manifestation of the well known f ( ) whilethe cross section enhancement found in J / Y → fp + p − is indeed a new f resonance with mass near 1800 MeV. We presentan explanation for the different decay properties of these two scalar resonances. Keywords:
Scalar resonances, Bethe-Salpeter equation, Fadeev equations
PACS:
INTRODUCTION
A peak structure has been found in the fw invariant mass spectrum in a recent experimental study of the J / Y → gfw process made by the BES collaboration [1]. This peak has been associated to a scalar, isoscalar resonance with mass M = ± + − MeV and width G = ± + − MeV in Ref. [1]. We shall refer to this state as f ( ) as donein Ref. [1]. No such resonance is present in the spectrum of the f states in the particle data book (PDB) [2]. Theknown state with the nearest possible mass is f ( ) and one could wonder if the peak seen in the fw spectrumcan be explained with this known resonance. In fact, an attempt to explain the BES data on the fw invariant massspectrum with f ( ) was made in Ref. [3] and indeed a peak structure was found near the threshold although witha strength much weaker than the experimental data. Anyhow, no strong claims were made in Ref. [3] partly due to aparameterized treatment of the interaction of vector mesons ( V ) and VV f ( ) coupling.Interestingly, the finding of yet another f resonance with mass also around 1800 MeV was earlier reported in adifferent process: J / Y → fp + p − [4]. This resonance was found in the pp invariant mass spectrum and was named f ( ) . This state was found to possess a very curious decay property which is a suppressed decay to the KK channel. This property makes f ( ) undoubtedly distinct to the known f ( ) which, completely contrarily,decays to KK with a large branching ratio and suppressively to pp .Now, the f ( ) found in Ref. [1] must also be distinct to f ( ) of Ref. [4] since f ( ) has been found inthe fw system which must unavoidably couple, and decay, to KK (see Fig. 1). Thus, from Refs. [1, 4] it seems that FIGURE 1.
The fw → KK process. two f resonances exist with mass around 1800 MeV but with very different decay properties and as a consequencethere must be three f ’s with mass between 1700-1800 MeV: the well known f ( ) and a twin set of new statesaround 1800 MeV. In this manuscript we discuss that this is not the case although it might appear to be so. We showhat the peak structure found in the fw spectrum in Ref. [1] is a manifestation of f ( ) while the resonance seen inthe pp system [4] is indeed a new scalar, isoscalar resonance which is undoubtedly distinct to f ( ) . In addition,we provide an explanation for the suppressed decay of f ( ) to the KK channel. f ( ) AS A pp f ( ) RESONANCE
We shall first review our study of the pp f ( ) system [5] where a scalar resonance was found with properties verysimilar to f ( ) . With the motivation of studying pseudoscalar systems with total quantum numbers of the vacuum,we first studied three pseudoscalar systems with total strangeness zero in Ref. [5], namely, p K + K − , p K K , p p h , p + K K − , p + p − h , p − K + K and p − p + h as coupled channels. The formalism of the study is based on calculationof the Faddeev equations with the input amplitudes obtained by solving the Bethe-Salpeter equation for the two-pseudoscalar subsystems in a coupled channel formalism. The Faddeev partitions, T , T and T , are written in thisformalism as T i = t i d ( ~ k ′ i − ~ k i ) + (cid:229) j = i = T i jR , i = , , , (1)with ~ k i ( ~ k ′ i ) being the initial (final) momentum of the particle i and t i the two-body t -matrix which describes theinteraction of the ( jk ) pair of the system, j = k = i = , ,
3. Using Eq. (1), the full three-body T -matrix is obtained interms of the two-body t -matrices and the T i jR partitions as T = T + T + T = (cid:229) i = t i d ( ~ k ′ i − ~ k i ) + T R (2)where we define T R ≡ (cid:229) i = (cid:229) j = i = T i jR . (3)The T i jR partitions in Eq. (1) satisfy the following set of coupled equations T i jR = t i g i j t j + t i h G i ji T jiR + G i jk T jkR i , i = j , j = k = , , . (4)where g i j is a three-body Green’s function depending on the variables of the external legs while G i jk is a loop function(see Refs. [5, 6, 7, 8, 9, 10] for more details). Studies of several three-hadron systems made of mesons and baryons haveearlier been made within this formalism [6, 7, 8, 9, 10] leading to the generation of some meson and baryon resonanceswhich, in turn, indicates that the three-body dynamics plays an important role in understanding the properties of suchstates.Going back to the study of three pseudoscalar mesons of Ref. [5], we would like to recall that the input two-bodyamplitudes were obtained following Refs. [11, 12] where dynamical generation of the light scalar mesons was found.Thus, our two-body amplitudes also contain this information. To be precise, the isoscalar KK and pp t -matricesdynamically generate the resonances f ( ) and s ( ) , while the system composed of the channels KK and ph inisospin 1 gives rise to the a ( ) state. In the strangeness + K p and K h systems the k ( ) is formed.With these inputs we solve Eq. (4) while keeping all the interactions in S-wave, which implies that the total quantumnumbers of the three-body system and, thus, the possible bound states or resonances present in it are J p = − .To identify the peaks obtained in the three-body T -matrix for the different channels with physical states we need toproject these amplitudes on an isospin basis. To do that, we consider the total isospin I of the three-body system andthe isospin of one of the two-body subsystems, which in the present case is taken as the isospin of the KK subsystemor (23) subsystem, I , and evaluate the transition amplitude h I , I | T R | I , I i . The isospin I can be 0 or 1, thus, thetotal isospin I can be 0, 1 or 2. For the cases involving the states | I = I = i , | I = I = i and | I = I = i we have not found any resonance or bound state. Although we do find one in the case of I = I = ∼ ∼
85 MeV. We find that this resonance is formed when the KK system is organized as f ( ) and, thus, its structure is dominantly p f ( ) .This state can be associated with the p ( ) listed in the PDB [2], whose mass is in the range 1300 ±
100 MeVand the width found from the different experiments listed varies between 120 to 700 MeV [2]. Using these values as aeference, the peak position obtained here is in the experimental upper limit for this state, while the width is close to thelower experimental value, thus, our findings are compatible with the known data set. Surely, for a better comparisonone needs more experiments which could help in determining the properties of this state with more precision. Thedecay modes seen for this resonance are rp and p ( pp ) Swave . The channel ppp is a three-body channel which couplesto p KK and pph . However the three pion threshold (around 410 MeV) is far away from the region in which the stateis formed, thus, it naturally is not essential in the generation of the p ( ) . However, the inclusion of channels like ppp or rp could help in increasing the width found for the state within our approach, since there is more phase spacefor the p ( ) to decay to these channels.Having obtained this information on three pseudoscalar systems we solve the Faddeev equations, once again, for the pp f ( ) system. The pp interaction is obtained from chiral Lagrangians, as earlier, while the result of the Faddeevequations solved for three-pseudoscalar system is used for the p f ( ) amplitude. Consequently, we find a resonancewith scalar, isoscalar quantum numbers and mass ∼ FIGURE 2.
The squared amplitude for the p KK channel as a function of the total energy and the invariant mass of the KK system. two pions interact in the s ( ) region and the p f ( ) subsystems are organized as p ( ) when the three-bodyresonance is formed. The important thing to notice from these results is that the scalar, isoscalar resonance found inour work decays dominantly to pp , pppp , pp KK but not to KK (as shown in Fig. 3) which is strikingly similar to thecharacteristics of the f ( ) [4]. f (1790) π (1300) π f (980) ππ f (1790) π (1300) π f (980) π π π ( ¯ K ) π ( K ) π FIGURE 3.
The decay process of the f ( ) found in our work. MANIFESTATION OF f ( ) IN THE fw MASS SPECTRUM
The resonances found in Refs. [1, 5] can, certainly, not be related to the one found in the fw invariant mass spectrumin Ref. [4] since the latter one must unavoidably decay to KK through the mechanism shown in Fig. 1. This argumentactually leads to finding of a flaw in the interpretation of the peak seen in the fw spectrum [1] as a new f ( ) resonance since in the KK decay channel the mass of f ( ) would be very far from the KK threshold and the peakshould be clearly observable, with no ambiguities about its interpretation. Yet, in the experiment studying J / Y decayinto g KK , clear peaks are seen for the f ( ) and f ( ) but no trace is seen of any peak around 1800 MeV [13].Similarly, MARK III [14] reports a clear signal for the f ( ) in the KK spectra but no signal around 1800 MeV.n fact in Ref. [15] we showed that the peak in the fw data obtained in Ref. [1] can be interpreted as a manifestationof the f ( ) resonance produced below the fw threshold. We discuss that the presence of this resonance necessarilyleads to a peak around the fw threshold with a shape and strength compatible with experiment and that the observedpeak is not a signal of a new resonance. To do this we studied the J / Y → gfw process within a formalism where aphoton is radiated from the initial cc state (as in Refs. [16]). The cc component after the g radiation, then, decays intopairs of vectors which interact among themselves as shown in Fig. 4. J/ψ VV + + + · · · c ¯ c FIGURE 4.
Schematic representation of J / Y decay into a photon and one dynamically generated resonance. The cc can be considered as an SU(3) singlet and, thus, the pair of vector mesons produced after hadronization mustcouple to an SU(3) singlet. The vector-vector content in the SU(3) singlet can be easily obtained from the trace ofV · V VVSU(3) singlet = Tr [ V · V ] , (5)where V is the SU(3) matrix of the vector mesons V = √ r + √ w r + K ∗ + r − − √ r + √ w K ∗ K ∗− K ∗ f . (6)We, thus, find the vertexVVSU(3) singlet = r r + r + r − + r − r + + ww + K ∗ + K ∗− + K ∗ K ∗ + K ∗− K ∗ + + K ∗ K ∗ + ff . (7)It is important to note that there is no primary production of fw with the mechanism of Fig. 4. The production of fw occurs through the rescattering of the vector mesons produced primarily. The amplitude for the process shown inFig. 4 can be written as t J / Y → gfw = A (cid:229) j = w j G j t j → fw , (8)where A is an unknown constant representing the reduced matrix element for the operator responsible for the transition cc → VVSU(3) singlet, w j are the weight factors corresponding to the probability of hadronization of different vectormesons, G j the loop function for the intermediate two mesons state and t j → fw represents the transition amplitude forthe intermediate vector mesons to fw . We take the information for the G j and t i j functions from Ref. [17]. The t i → j matrices can be written as t i → j = g i g j s − M R + iM R G R (9)where g i , g j are the couplings of the resonance to the i , j channels given in Ref. [17].With the amplitude of Eq. (8), which depends on the invariant mass of fw , we can construct the fw mass distributiongiven by d G dM inv = ( p ) M J / Y p g q w | t J / Y → gfw | , (10)where p g and q w are the photon momentum in the J / Y rest frame and the w momentum in the fw rest frame,respectively.In Fig. 5 we show the fw invariant mass distribution obtained by fixing the total strength such as to reproducethe peak of the experimental data on the number of fw events per bin. In order to account for the strength of thedistribution at large values of M inv , far away from the f ( ) resonance, we allow for a small background, whichwe take as a constant amplitude for simplicity. As we can see, there is a perfect agreement between our results and .0 2.5 3.00100200 A r b i t r a r y un i t s M inv (GeV) FIGURE 5.
The invariant mass distribution d G dM inv for the process J / Y → gfw from Eq. (10). The data points, shown by filledcircles, have been taken from Ref. [ ? ]. The dotted and dashed lines represent the background and the f ( ) resonancecontribution, respectively. The solid line shows the coherent sum of the two. the experimental data. However, we must remember that we have an unknown constant A in Eq. 8. In order to make astronger claim, we have calculated the ratio R G = Z dM inv d G dM inv G J / Y → g f ( ) , (11)where the constant A gets cancelled. In our work this ratio turns out to be 0 . + . − . which is in good agreement withthe experimental value 0 . + . − . .With this we can summarize the present manuscript by mentioning that we provide evidence for existence of anew scalar, isoscalar resonance f ( ) which is distinct to the known f ( ) . We present an explanation for thesuppressed decay of this resonance to KK . We also show that the BES data on the fw spectrum can be explained interms of f ( ) and thus a new f ( ) is not required. Thus there are two f states in the 1700-1800 MeV. ACKNOWLEDGMENTS
The authors acknowledge the support from the funding agencies FAPESP and CNPq.
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