Manifestation of Kaonium in the $ E^+ E^-\To K^+ K^-$ Process

$Manifestation of Kaonium in the $ E^+ E^-\To K^+ K^-$ Process$

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(1) due can electric Equation together opposite kaonium held between system way, attraction (a Coulombic simplest atom hydrogenlike the a ered In found. 2p the and na h osiunso inu n anu loin- Krewald also kaonium force. and strong pionium via of teract constituents the onia, htkoimbnsmr togy( strongly more binds kaonium that ue nices nbudsaeeeg en rpin a drop Zhang As a contrary, means the eV.” On energy hundred energy. bound-state few binding in a increase by an shifted value rule, is that Coulomb atom the kaonium the above for energy state ground − atydcy lcrmgeial notopoos In of exchange photons. two the into addition, electromagnetically decays partly interaction. Coulomb to corresponds si ofriywt h re fmgiueestimate magnitude of order the with 10 conformity in is on h eutn ieieo (2 and of lifetime theory resulting perturbation the chiral found amplitudes order interaction leading from meson-meson taken used [12] Lemmer the + The . C P ≈ oeprmna vdneo anu a e been yet has kaonium of evidence experimental No anu sntsal.Gon-tt 1)kaonium (1s) Ground-state stable. not is Kaonium − e − 3 π e pkoimi the in kaonium 2p g 6 + sotie yDlff[13]. Deloff by obtained fs 1 = + . olsos h olmi idn nryo the of energy binding Coulombic The collisions. 7kV nietehdoe tmadleptonic and atom hydrogen the Unlike keV. 57 π h togitrcina dominant a as interaction strong the o B e − A − −− B , rrtda ro fteeitneof existence the of proof as erpreted → AR π → hrfr,te a epoue nthe in produced be can they Therefore, . hsc n aaProcessing, Data and Physics 0 stasto oiae [9]. dominates transition 1s τ π olbrto aao h same the on data Collaboration 2 0 p K nthe on and , e 0 = + + e − K . 45 ηπ niiainpoess hymust they processes, annihilation e e − + − +1 + e π 0 e + 0 e 0 − . . process − e 08 30 π ea oe.Keasyand Klevansky modes. decay K − 0 niiaininto annihilation → × ∗ → and π 10 ewe an generates kaons between . 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The Coulombic binding energy of the ﬁrst excited state (n = 2) of kaonium is 1.64 keV. We will concentrate on the 2p state, which can be produced in the e+e− experiments. Its quantum numbers J PC =1−− forbid decays to the C=1 π0π0 and ηπ0 states. The 2p kaonium width is thus determined by the decay rate into π+π− only [14]. Its lifetime should be at least 2 times higher than that of the ground-state kaonium. The corresponding decay width is around 0.1 keV, about 4 orders of magnitude smaller than the decay width of the φ(1020), which is the dominant object in the process we are going to in- vestigate. For our purposes, we can thus neglect the 2p kaonium decay width and consider it a stable particle.

II. 2p KAONIUM AS A POLE IN THE e+e− → K+K− AMPLITUDE

In this paper, we analyze the existing precise data on the e+e− K+K− process with the aim of finding a pole in the→ amplitude corresponding to a bound state. The reaction amplitude is a function of s, the invariant − energy squared. Under very general conditions, the am- FIG. 1. Cross section for the e+e annihilation into two plitude can be continued into the complex s-plane. The charged kaons measured in the CMD-3 experiment [15] and resonances are represented by the poles at imaginary s. two fits to it using Eq. (2). Their parameters are given in The (quasi) stable particle, or bound state, appears as a Table I. pole at real s below the reaction threshold. The formula for the cross section of the e+e− anni- + − NDF = 24 3 = 21 implied a confidence level (C.L.) of hilation into a K K pair based on the vector-meson- − dominance model with two resonances is zero. This fit is depicted by a dashed curve in Fig. 1. The parameters of the fit are listed in the middle col- 3 2 πα2 4m2 / umn of Table I. It must be said that the Kozyrev et σ(s)= 1 K 3s − s al. [15] achieved a better result. They used a more sophisticated parametrization of the φ(1020)-resonance shape iδ 2 R1e R2 based on an energy dependent width Γ(s) and attained 2 + 2 , (2) 2 × s M iM1Γ1 s M iM2Γ2 χ /NDF = 25/20, which gave C.L. = 20%. − 1 − − 2 − We then tried to improve our fit by considering two where M and Γ determine the position and width of the i i resonances. To our surprise, the width of the second res- ith resonance, respectively. The residuum R includes the i onance came out close to zero (0.6 1.7 MeV [18]) and product of two constants. One characterizes the coupling the resonance position was below the± threshold, which of the ith resonance to the photon (up to the elementary charge e, which is taken off to form, after squaring, an α in the prefactor) and the other is the coupling of the + − − resonance to the K K pair. The phase δ regulates the TABLE I. Parameters of the two fits to the CMD-3 K+K interference between the resonances. data [15] depicted in Fig. 1. We use the e+e− K+K− cross section data [15] ob- → One resonance Resonance and pole tained by the CMD-3 (Cryogenic Magnetic Detector) ex- 2 R1 (GeV ) 0.3679(15) 0.3759(15) periment at the VEPP-2000 e+e− collider in Novosibirsk, Russia. The advantage of the energy scan method used M1 (GeV) 1.019393(14) 1.019247(18) in this experiment is in getting high statistics data at pre- Γ1 (MeV) 4.359(32) 4.172(38) cisely known energies, to which the collider is tuned up δ 0 (fixed) 1.334(34) 2 step by step. The number of data points is 24, and they R2 (GeV ) 0 (fixed) 0.298(23) are concentrated in a narrow region around the φ(1020) b (keV) 1.64a (fixed) resonance. The data are shown in Fig. 1, together with Γ2 0 (fixed) two curves depicting our fits [16]. χ2 341.8 4.6 We first tried to fit the data assuming just one res- NDF 21 19 onance, i.e., using three free parameters [17]. The re- Confidence level 0 100% sult was disastrous: the usual χ2 was equal to 341.8, a M = 2m − b which together with the number of degrees of freedom 2 K+ 3 signalized the bound state. Inspired by that, we re- placed the second resonance with a bound-state pole by TABLE III. Parameters of the fits to the √s < 1.065 GeV subset of BABAR K+K− data [22] assuming no pole be- putting Γ2 0 and M2 = 2m + b, where b is the ≡ K − low the threshold (second column) and the 2p kaonium with 2p kaonium binding energy. We took b = 1.64 keV as a the Coulombic binding energy b (third column) or with b = first try. We attained perfect agreement with the data 10 MeV (fourth column). (χ2/NDF = 4.6/19, C.L. = 100%), depicted by the full curve in Fig. 1. All parameters of the fit are listed in No kaonium b = 1.64 keV b = 10 MeV 2 the rightmost column of Table I. Minimizing the χ 2 R1 (GeV ) 0.3577(17) 0.3575(17) 0.3572(18) with respect to the binding energy gives an estimate of +14.8 M1 (GeV) 1.019144(17) 1.019120(17) 1.019034(19) b = (10.8−9.1 ) MeV [19]. Let us mention that in 1961 Γ1 (MeV) 4.461(37) 4.455(37) 4.300(40) Uretsky and Palfrey [20] assumed a mesonium binding 2 energy of 10 MeV. R2 (GeV ) 0.0140(13) 0.184(18) A binding energy much larger that its Coulombic value M2 2mK+ b 2mK+ b + − − of 1.64 keV would mean that the strong force between K Γ2 0(fixed) 0(fixed) − and K is attractive and responsible for keeping kaonium δ2 –0.07(18) –1.070(56) together. But our result is still not conclusive. The parameters of the fit with b = 10.8 MeV (χ2/NDF = 2.4/19, C.L. = 100%) are only marginally better than those with are given in Table III. the Coulombic binding energy shown in Table I. To get more insight into this problem, we explored two other datasets. III. 2p KAONIUM IN THE e+e− → π+π− DATA The e+e− K+K− data obtained by the CMD-2 ex- → + − periment at the VEPP-2000 e e collider in Novosibirsk, There is also another possible way of seeing the 2p Russia, were published in 2008 [21]. They contain 21 kaonium. Thanks to its π+π− decay mode, the 2p kao- points in an energy range narrower than that of the later nium may, in principle, reveal itself as an irregularity CMD-3 data [15] explored above. We have performed the in the e+e− π+π− excitation function slightly below fits to data [21] under three different assumptions; the re- → √s = 2mK+ . The contemporary experimental data do sults are shown in the middle panel of Table II. The one- not show any anomaly that could be interpreted as a resonance fit is already perfect. Adding kaonium with manifestation of 2p kaonium. The invariant energy re- either Coulombic binding energy or the Uretsky-Palfrey 2 gion in question has been ignored by most experiments, value of 10 MeV further decreases χ . Owing to per- which have concentrated on the measurement in the ρ/ω fect fit without kaonium, this cannot be considered an region or at energies higher than 1 GeV. The important unambiguous proof of kaonium’s existence. exception is the BABAR experiment [23], which covered B B The A AR data [22] from 2013 extend up to 5 GeV. energies ranging from 0.3 to 3 GeV by using the initial We have used only 48 energies up to 1.065 GeV, which state radiation method. Their data in the presumed sig- is the highest energy in the CMD-3 [15] data. The nal region are shown in Fig. 2, together with our “con- BABAR data also cover the low-energy region and pro- + − servative” fit that does not include kaonium. There is an vide 13 data points between the K K threshold and insignificant excess at √s 973 MeV, but no firm con- the lowest CMD-3 [15] energy. They are therefore more clusion can be drawn yet.≈ More precise and denser data sensitive to the subthreshold behavior [given mainly by would be necessary to confirm or reject the presence of the position(s) and residuum (residua) of the pole(s)] of 2p kaonium in the e+e− π+π− data. According to the reaction amplitude. Thanks to that, we have gotten [24], we may expect new data→ from the CMD-3 experi- the following results (see the rightmost panel of Table II): ment soon. Figure 4 in Ref. [24] hints at their covering the “no kaonium” and “Coulombic kaonium” hypotheses energies to almost 1 GeV. are rejected, and the kaonium with a binding energy of 10 MeV gives an acceptable fit. The details of the fits IV. 2p K0-ONIUM AS A POLE IN THE + − 0 0 e e → KS KL AMPLITUDE TABLE II. Evidence for 2p kaonium from two other experi- 2 ments. The last two rows show results of the fits (χ /NDF, The result we obtained when exploring the K+K− C.L.) with kaonium considered in addition to the resonance. BABAR data [22] indicates that the strong interaction CMD-2 [21] BABAR [22] between a kaon and its antiparticle is attractive. One 0 ¯ 0 Energy range (GeV) 1.01136–1.03406 0.985–1.065 may therefore speculate about the existence of the K K No. of points 21 48 bound state. This would be the first (and probably only) onium composed of neutral particles. To pursue this idea, One-resonance fit 7.5/18 98.5% 244.4/45 0% + − 0 0 we use the high-precision data on the e e KSKL Kaonium b = 1.64 keV 2.2/16 100% 184.5/43 0% process [25], again coming from the CMD-3 experiment→ Kaonium b = 10 MeV 2.3/16 100% 48.1/43 27.4% at Budker Institute of Nuclear Physics in Novosibirsk, 4

ble IV) provides χ2/NDF = 36.7/22, C.L. = 2.5% [26]. The sophisticated fit by experimentalists [25] is much better (χ2/NDF = 15/21, C.L. = 82%). To explore the one- resonance–one-bound state scenario, we choose again the Uretsky-Palfrey value of 10 MeV for the binding energy b. The result of the fit is χ2/NDF = 7.5/20, C.L. = 99.5%, and all parameters are listed in the rightmost column in Table IV. Keeping in mind a good fit by the exper- imenters themselves, we can say that the K0-onium is not required by the CMD-3 [25] data. When looking for K0-onium in the π+π− data, the signal region should be shifted upward by about 7.9 MeV.

V. CONCLUSIONS

+ − FIG. 2. Cross section for the e e annihilation into two To conclude, we have found an indication of the exis- charged pions measured in the BABAR experiment [23]. The 2 tence of kaonium made of charged kaons in the 2p state curve is our fit to data from 0.3 to 1.6 GeV (χ /NDF = + − + − + − by analyzing the data on the e e K K process. 257.6/303, C.L. = 97.2%). The K K threshold is marked + − → with an arrow. A similar analysis of the e e KSKL data has pro- vided somewhat weaker evidence→ for the 2p K0-onium. To this point, we have ignored the fact that both 2p 0 0 TABLE IV. Parameters of the two fits to the CMD-3 KS KL kaonia have the same quantum numbers. As a conse- data [25]. quence, the K+K− kaonium should be present in the + − e e KSKL amplitude as a bound-state pole, and One resonance Resonance and pole the kaonium→ composed of neutral kaons should reveal it- 2 + − + − R1 (GeV ) 0.3488(14) 0.3500(39) self in the e e K K amplitude as a bound-state → M1 (GeV) 1.019221(14) 1.019263(16) pole or a resonance, depending on its bounding energy. Γ1 (MeV) 4.144(28) 4.168(29) Both 2p kaonia decay into two charged pions. To get a clear and consistent picture of kaonia, the information δ 0 (fixed) 3.2 2.0 + − 2 ± should be combined from the e e annihilation data to R2 (GeV ) 0 (fixed) 0.0289(53) + − + − a the K K , KSKL, and π π final states. Work in this b (MeV) 10.0 (fixed) direction is in progress. Γ2 0 (fixed) χ2 36.7 7.5 NDF 22 20 ACKNOWLEDGMENTS Conf. level 2.5% 99.5% a M2 = 2mK0 − b I thank Filip Blaschke, Martin Blaschke, and Josef Juráˇnfor the useful discussions. Russia, and comprising of 25 data points in a narrow This work was partly supported by Ministry of Edu- interval around the φ(1020) resonance. cation, Youth and Sports of the Czech Republic Inter- Our simple one-resonance fit (the middle column in Ta- Excellence Projects No. LTI17018 and No. LTT17018.

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