Towards two-body strong decay behavior of higher ρ and ρ 3 mesons
aa r X i v : . [ h e p - ph ] J u l Towards two-body strong decay behavior of higher ρ and ρ mesons Li-Ping He , , ∗ Xiao Wang , , † and Xiang Liu , ‡§ School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China Research Center for Hadron and CSR Physics, Lanzhou University & Institute of Modern Physics of CAS, Lanzhou 730000, China
In this work, we systematically study the two-body strong decay of the ρ/ρ states, which are observed andgrouped into the ρ/ρ meson family. By performing the phenomenological analysis, the underlying propertiesof these states are obtained and tested. What is more important is that abundant information of their two-bodystrong decays is predicted, which will be helpful to further and experimentally study these states. PACS numbers: 14.40.Be, 12.38.Lg, 13.25.Jx
I. INTRODUCTION
There is abundant information on ρ/ρ states collected inParticle Data Group (PDG) [1], which provides that their spin-parity J P could be 1 − / − and all of them are isovector. InTable I, we briefly review the resonance parameters of theobserved ρ/ρ states. As the total angular momentum J in-creases, the number of these states decreases.The experimental status of these states stimulates our in-terest in revealing their underlying structures, since at presentthe properties of ρ/ρ are still in chaos. First of all, we need toexamine whether these ρ/ρ can be categorized into the con-ventional meson family. Besides the study of mass spectrum,their Okubo-Zweig-Iizuka (OZI) allowed two-body strong de-cay behaviors can reflect important information on their struc-tures. Thus, in this work we pay more e ff orts to systemati-cally calculate the OZI-allowed strong decays of ρ/ρ , wherethe quark pair creation (QPC) model will be applied to calcu-lation. Before carrying out calculation, we need to determinethe corresponding radial, orbital and spin quantum numbers tothese ρ/ρ , where we can refer to the analysis of mass spec-trum, which will be summarized in the following section. Bycomparing our results with the experimental data, the mesonassignment to these observed ρ/ρ should be examined. Ad-ditionally, our obtained OZI-allowed two-body strong decaybehaviors will provide valuable information of further experi-mental study on ρ/ρ .As mentioned above, this phenomenological study on ρ/ρ can be applied to distinguish their possible meson assign-ments. In addition, owing to this work, we can learn how astate has not a suitable interpretation as a conventional me-son state. Thus, our study may provide important judgmentwhether these studies are relevant to exotic hadron configura-tion or new novel mechanism.This paper is organized as follows. After introduction, webriefly review the present research status of these ρ/ρ . InSec. III, we discuss the possible meson assignment to thesestates using the mass spectrum analysis and introduce the QPCmodel. The allowed decay channels are also selected. In ‡ Corresponding author ∗ Electronic address: [email protected] † Electronic address: [email protected] § Electronic address: [email protected]
Sect. IV, we perform the phenomenological analysis of ρ/ρ .The last section is devoted to a short summary. TABLE I: The experimental information of the observed ρ/ρ states.Here, the masses and widths (in units of MeV) are average valuestaken from PDG [1]. The states omitted from the summary table ofPDG are marked by a superscript ♭ while these state as further statein PDG are distinguished by a superscript ♮ . J P = − State Mass Width ρ (770) 775 . ± .
34 146 . ± . ρ (1450) 1465 ±
25 400 ± ρ (1570) ♭ ± ±
62 144 ± ± ρ (1700) 1720 ±
20 250 ± ρ (1900) ♭ [2] 1909 ± ±
25 48 ± ± ρ (2150) ♭ ±
17 359 ± ρ (2000) ♮ [3–6] 2000 ±
30 260 ± ρ (2270) ♮ [3–6] 2265 ±
40 325 ± J P = − State Mass Width ρ (1690) 1688 . ± . ± ρ (1990) ♭ [3] 1982 ±
14 188 ± ρ (2250) ♭ [7] ∼ ∼ II. REVIEW OF RESEARCH STATUS
As shown in Table I, there are many ρ/ρ states observed byexperiments. Among these states, ρ (770) [8] is established tobe the ground state with n S + L J = S with very broad fullwidth. Thus, we will not include ρ (770) when briefly review-ing the research status of the ρ/ρ states. In the following, weintroduce the experimental and theoretical status of ρ/ρ . ρ (1600) was omitted in the 1988 edition of PDG [9] andreplaced by ρ (1450) and ρ (1700), which is due to many theo-retical and experimental studies [10–27].In the past decades, many e ff orts have been made to explainthe structure of ρ (1450). However, its property is still unclearat present. Although the study of the mass spectrum supports ρ (1450) as a 2 S state [28], the decay behavior is hard to beunderstood. The calculation in Ref. [29] shows that the ππ and ωπ channels are dominant in the ρ (1450) decays. Usingthe nonlocal Nambu-Jona-Lasinio model, the calculated par-tial widths of ρ (1450) → ππ and ρ (1450) → πω are also com-parable with the experimental values [30, 31]. On the otherhand, the theoretical decay widths of ρ (1450) → a (1260) π and h (1170) π become small [29]. However, the experi-mental result indicates that ρ (1450) mainly decays into 4 π [1, 17, 23, 32]. To alleviate the discrepancy between the ex-perimental and theoretical results of the 4 π channel, ρ (1450)as a mixture of 2 S ρ state and hybrid was introduced in Ref.[29] since Close et al. indicated that the vector hybrid withmass about 1.5 GeV can strongly couple with a (1260) π [33].Other theoretical studies [33, 34, 36, 37] also support this mix-ture.Besides these two explanations for ρ (1450) as mentionedabove, explanation of ρ (1450) as a 1 D state was proposed inRefs. [38, 39] using the chiral symmetry method. If consider-ing the mass spectrum analysis on the ρ meson family, we no-tice that the mass of the 1 D ρ meson should be 1600 − ρ (1450) as a 1 D ρ meson. ρ (1700) is a good candidate of the 1 D ρ meson. Boththe analysis of the branching ratio of ρ (1700) → π, π [32]and the study of e + e − → ωπ via the nonrelativistic P quarkmodel [41] show that ρ (1700) is a 1 D state.There are many experiments relevant to ρ (1900). The DM2Collaboration once reported a dip around 1.9 GeV by analyz-ing the e + e − → π process [21]. Later, the FENICE Collab-oration observed a dip around 1.9 GeV in the R value mea-surement, which can be produced by the interference of a res-onance with one of these broad vector mesons [42]. In 2001,the E687 Collaboration at Fermilab found a narrow dip struc-ture at 1.9 GeV through the 3 π + π − di ff ractive photoproduc-tion [43] . If this dip is due to a destructive interference ofa resonance with a continuum background, the resonance pa-rameters can be extracted as m = (1 . ± . ± . Γ = (29 ± ±
4) MeV. By refitting their data, theE687 Collaboration indicated that the interference e ff ect of anarrow resonance with known vector mesons (such as a broad ρ (1700)) can result in a dip [44]. In both of e + e − → π + π − and e + e − → π + π − π processes, the BaBar Collaborationannounced the observation of a structure around 1.9 GeV [45],which was confirmed by BaBar in the e + e − → φπ process[2]. The CMD3 Collaboration observed a peak near the p ¯ p threshold, which can be identified as ρ (1900) [46]. In Ref.[40], Bugg indicated that this CMD3’s observation can be ex-plained to be a S state captured by the very strong pp S-wave or to be a non-resonant cusp e ff ect.Analyzing the data of the 6 π mass spectrum from the e + e − annihilation [47] and the di ff ractive photoproduction [48],Clegg and Donnachie indicated the existence of ρ (2150) [49].Later, Biagini et al. [50] suggested that there exists the third radial excitation of ρ (770) by phenomenologically fitting thepion form factor [51], and gave the corresponding resonantparameters m ≃ Γ ≃
320 MeV, which is con-sistent with the result in Ref. [49]. In addition, the GAMSCollaboration also confirmed the observation of ρ (2150) in π − p → ωπ n [52, 53]. In Refs. [3, 4, 6, 54], the Crystal Bar-rel data was analyzed, where a 1 −− resonance with the mass2.15 GeV can be as the evidence of ρ (2150). In 2007, BaBarobserved ρ (2150) in the new process e + e − → η ′ (958) π + π − and f (1285) π + π − [55].Godfrey and Isgur has predicted a 2 D state with mass2.15 GeV [28], which can correspond to ρ (2150). However,there exists another explanation to ρ (2150), i.e., Anisovich etal. suggested ρ (2150) to be a 4 S state [56], which was con-firmed in Ref. [40, 57, 58].In Table. I, there are two more states of 1 −− listed in PDG[1], which are ρ (2000) and ρ (2270). In the p ¯ p → ππ reaction,a resonance around 1988 MeV was found [7]. Later, Aniso-vich et al. obtained a J PC = −− state at 2000 MeV in thesame reaction [4], which also appears in the p ¯ p → ωηπ and ωπ processes [3, 5, 6]. ρ (2000) was suggested as the radialexcitation of ρ (1700) [3]. In Ref. [59], Bugg concluded that ρ (2000) can be a mixed state with a significant D compo-nent.In the reaction γ p → ωπ + π − π , a resonance at 2280 ± ρ (2270)is important to fit the ωηπ data, and can be ignored to describethe ωπ data [3]. The Regge trajectory analysis shows that ρ (2270) can be a 3 D state, i.e., the second radial excitationof ρ (1700).In PDG [1], there are three ρ states. ρ (1690) was firstobserved in Refs. [60, 61], which was once regarded as a π + π − resonance. At present, ρ (1690) is established to be a D state, which can decay into 2 π , K ¯ K , K ¯ K π , 4 π , ωπ , and ηπ + π − as shown in PDG [1]. Additionally, two more new de-cay modes a (1320) π and ρη were reported in Ref. [62]. Be-sides the D explanation for ρ (1690), it could be interpretedas a three-rho meson molecular state in Ref. [63].As a 3 −− state, ρ (1990) with m ∼ Γ ∼
287 MeV was observed in the ππ invariant mass spectrum of p ¯ p → ππ [7], which was confirmed by analyzing the CrystalBarrel data [4, 5, 54], where a 3 −− resonance exists in the p ¯ p → π + π − , ωπ processes. The ωηπ decay of ρ (1990) wasreported in Ref. [6]. In Ref. [3], a combined fit to the ωπ , ωηπ and π − π + data was performed, which gives the weighedmean of mass and width of ρ (1990) as listed in Table I.There are many experimental papers relevant to ρ (2250)as shown in PDG [1]. ρ (2250) was first observed by BNLthrough studying the S-channel ¯ pN cross section [64]. Later, ρ (2250) was also found in the reactions p ¯ p → ¯ pp [65], p ¯ p → ¯ NN [66], p ¯ p → K + K − [67], and p ¯ p → ππ [7, 68–70]. In2000, the VES Collaboration reported a 3 −− resonance at 2290MeV in the reaction π − p → ηπ + π − n [62]. The analysis ofthe Crystal Barrel data for the p ¯ p → π − π + [4], p ¯ p → ωηπ [6] and p ¯ p → ωπ [5] reactions also requires the existence of ρ (2250).A plot of the Regge trajectory for the mass spectrum of the3 −− states was presented in Refs. [3–5], where ρ (1990) and ρ (2250) are treated as the 2 D and 3 D states, respectively. III. TWO-BODY STRONG DECAYS
Before carrying out the study of the two-body strong decayof these ρ/ρ states shown in Table I, we need to illustrate theanalysis of their Regge trajectory.The analysis of the Regge trajectory is an e ff ective approachto quantitatively study meson mass spectrum. In general, thereexists an expression [56, 71] M = M + ( n − µ , (1)where M is the mass of ground state and µ denotes the tra-jectory slope and n is the radial quantum number of the cor-responding meson with mass M . The relation expressed byEq. (1) is roughly satisfied by ρ/ρ states as shown in Fig. 1,which indicates:1. ρ (1450), ρ (1900) and ρ (2150) are the radial excitationsof ρ (770).2. ρ (1700), ρ (2000) and ρ (2270) can be grouped intothe n D ρ meson family. Among these three states, ρ (1700) is the ground state while ρ (2000) and ρ (2270)are the first and the second radial excitations of ρ (1700).3. ρ (1690), ρ (1990) and ρ (2250) can be as good candi-dates of the 1 D , 2 D and 3 D states, respectively. S D nn M ( G e V ) D FIG. 1: (color online). The analysis of the Regge trajestories for the ρ/ρ states. The trajectory slopes are 1.365 GeV , 1.203 GeV and1.094 GeV for the S , D and D states, respectively. ⊙ denotesthe theoretical values, while the red, blue and green dots correspondto the experimental data listed in Table I. Fig. 1 only gives a rough estimate of categorizing ρ/ρ states into the meson families. A further study of their two-body strong decay behaviors can test whether the assignment shown in Fig. 1 is reasonable. Here, the QPC model isadopted to calculate the partial decay widths of these decays.The QPC model was first proposed by Micu [72] and fur-ther developed by the Orsay group [73–77]. It is has beenwidely adopted to study the OZI-allowed strong decay ofhadrons [78–100]. For depicting a quark-antiquark pair cre-ated from the vacuum, a transition operator T is introducedby T = − γ X m h m ; 1 − m | i Z d p d p δ ( p + p ) ×Y m ( p − p χ , − m φ ω b † i ( p ) d † j ( p ) . (2)Here, p / p denotes the three momentum of quark / antiquarkcreated from the vacuum. Thus, the transition matrix elementof the A → B + C process can be expressed as h BC |T | A i = δ ( P B + P C ) M M JA M JB M JC , (3)where P B / P C is the three momentum of the final state hadron B / C in the center of mass frame of the initial state A . InEq. (3), Y ℓ m ( p ) ≡ | p | ℓ Y ℓ m ( θ p , φ p ) denotes the ℓ -th solid har-monic polynomial, χ , − m is a spin triplet state, i and j are the S U (3) color indices of the created quark pairs from the vac-uum. φ = ( u ¯ u + d ¯ d + s ¯ s ) / √ ω = δ α α / √ α = , ,
3) corresponds to color singlet.By the Jacob-Wick formula [101], the decay amplitude isexpressed as M JL ( A → BC ) = √ L + J A + X M JB , M JC h L JM J A | J A M J A i×h J B M J B ; J C M J C | JM J A iM M JA M JB M JC . Furthermore, the decay width reads as Γ A → BC = π | P B | m A X J , L |M JL | , (4)where m A is the mass of the initial state A . In the concretecalculation, the harmonic oscillator wave function Ψ n ,ℓ, m ( R , q ) = R n ,ℓ ( R , q ) Y ℓ m ( q ) (5)is applied to describe the meson wave function. In the QPCmodel, two parameters R and γ are introduced. Here, R canbe determined by reproducing the realistic root mean squareradius which is obtained by solving the Schr¨odinger equationwith the potential in Ref. [89]. Although the R values canbe obtained by the above approach in principle, these valuesare to be used for reference only. Thus, we illustrate the cal-culated partial decay widths of these ρ and ρ states in termsof parameter R within a typical range of values. γ is a di-mensionless constant for describing the strength of the quarkpair creation. By systematically fitting the experimental data, γ = . u ¯ u / d ¯ d pair creation (see Table II inRef. [100] for more details in extracting the γ value), whilethe strength of the s ¯ s pair creation satisfies γ = . / √ ρ/ρ are listed. Using the QPC model, we obtain the corre-sponding partial decay widths. In the next section, we willcompare our theoretical results with the experimental data toperform a phenomenological analysis, which will be helpfulto further reveal the underlying properties of these ρ/ρ states. IV. PHENOMENOLOGICAL ANALYSISA. n S states Assuming ρ (1450) as a 2 S isovector meson, its two-body strong decay behavior is calculated and shown in Fig. 2.Our calculation shows that ππ , π a (1260), πω and π h (1170)are its dominant decay modes, where π a (1260), πω and π h (1170) can contribute to the 4 π final state. In addition,the obtained width of ρ (1450) → ηρ is also in good agree-ment with the data in Refs. [1, 23]. The partial decay widthsof ρ (1450) into K ¯ K , K ¯ K ∗ + H . c . and ππ (1300) are small in ourcalculation. As for ρ (1450) → π a (1320), the decay widthis tiny. Thus, the experimental data listed in PDG [1] canbe quantitatively compared with our results. Given the infor-mation of partial decay widths, we obtain the total width of ρ (1450) by summing over all partial decay widths. In Fig. 2,we show the comparison of our results with the CMD-2 data[102], which indicates that there exists a common range be-tween our theoretical total width and the experimental data.Additionally, the obtained total width is also consistent withthe experimental width given in Ref. [15], and overlaps withthe measured full width listed in Refs. [23, 103], which isabout 310 MeV.Besides providing the information of the partial decaywidths of ρ (1450), in Table III several ratios Γ ππ / Γ π a (1260) , Γ π h (1170) / Γ π a (1260) and Γ π a (1260) / Γ Total are also given, whichare weakly dependent on the parameter R . Experimental mea-surement of these ratios will be a good test of the 2 S assign-ment to ρ (1450).Because of the above analysis, we conclude that it is easyto explain ρ (1450) as a 2 S state, which is also supportedby a recent work in Ref. [104] that claims there is no clearevidence for a hybrid state with J PC = −− .According to the Regge trajectory analysis, ρ (1900) is agood candidate for a 3 S state. At present, its resonanceparameters are not yet determined experimentally, i.e., dif-ferent experiments give di ff erent results as listed in PDG [1].The calculated two-body strong decays of ρ (1900) are pre-sented in Fig. 3, where the theoretical total width overlapswith the BaBar’s data [29]. In addition, the main decaymodes of ρ (1900) are ππ , π a (1260), π h (1170), ππ (1300),and πω (1420). Thus, ρ (1900) has a large 4 π branching ra-tio and the decays into ρρ , K ¯ K , and η b (1235) are sizeable.In Table. III, we also show several ratios of its partial decaywidths. These predicted decay behaviors will be helpful toexperimentally study ρ (1900) in future.As shown in Fig. 1, ρ (2150) can be a 4 S state. The OZI-allowed two-body strong decay widths are listed in Fig. 4. Theobtained total width is dependent on the R value due to the node e ff ect, where the total width is (108 − R = (4 . − .
0) GeV − . From PDG [1], we noticethat the measured total width of ρ (2150) from the e + e − inter-action is larger than that from the p ¯ p → ππ process and S -channel N ¯ N interaction. Here, the experimental total widthsof ρ (2150) are (350 [55], 389 [50], 410 [49], 310 [55]) MeV,(296 [7], 40 [105], 250 [70], 200 [69]) MeV, and (230 [3],135 [65], 98 [106], 85 [64]) MeV corresponding to the e + e − interaction, p ¯ p → ππ channel and S -channel N ¯ N process, re-spectively. Our calculation favors the data measured at the p ¯ p → ππ process and S -channel N ¯ N interaction. For exam-ple, in Fig. 4 we compare our result of the total width with thatin Ref. [3] obtained by analyzing the SPEC’s data, where thetheoretical and experimental results overlap with each otherwhen R = (4 . − .
98) GeV − . The calculation of the par-tial decay widths shows that ρ (2150) decays dominantly into ππ , π a (1260), πω and π h (1170). More information of otherpartial decay widths can be found in Fig. 4. We notice that ρ (2150) was observed in the decay channels π + π − , ωπ , η ′ ππ , f (1285) ππ , ωπη , K + K − and 6 π [1], which can be reasonablyexplained by our study. Furthermore, based on the obtainedpartial decay widths, we also give several ratios of some par-tial decay widths in Table III, which are also important to testwhether ρ (2150) is a 4 S state.The ranges of R in Figs. 2-4 needed to reproduce the ex-perimental total widths are (3 . − .
23) GeV − , (3 . − . − and (4 . − .
98) GeV − , respectively, where the exper-imental error is considered. These obtained ranges of R alsoroughly reflect a regularity, i.e., the corresponding R valuebecomes larger when the radial quantum number increases,which is consistent with the estimate of the potential model[89]. B. n D states The Regge trajectory analysis shows that ρ (1700), ρ (2000)and ρ (2270) can be categorized into the n D ρ meson family(see Fig. 1). In this subsection, we discuss their two-bodydecay behaviors.As the ground state of the D ρ meson family, ρ (1700)mainly decays into π a (1260) and π h (1170). Of course, ππ and ρρ are the important decay channels. These results areconsistent with the experimental data [1, 23], which naturallyexplains why ρ (1700) can be found in its 4 π and ρππ channels.However, the obtained total decay width is larger than most ofexperimental data listed in PDG. In Fig. 5, we give the com-parison between our result and the experimental total widthfrom Ref. [23], where the theoretical total width can overlapthe experimental result with error when R > .
55 GeV − .In addition, we find that the decay width of ρ (1700) → ωπ is always smaller than that of ρ (1700) → ππ , which doesnot depend on the R value. This conclusion is consistentwith the results in Refs. [23, 28, 107]. The obtained decaywidth of ρ (1700) → ππ is comparable with the value (39 ± ρ (1700) → ωπ is about 25 MeV, which well agrees with our calculation of TABLE II: The OZI-allowed two-body decay modes of the ρ/ρ states. Here, ω , ρ and η ′ denote ω (782), ρ (770) and η ′ (958), respectively. Theallowed two-body decays are marked by X . ππ π h (1170) ππ (1300) πω (1420) πω (1650) ρ b (1235) ρ f (1285) ω a (1260) ηρ (1450) ρ f (1420) ρ (1450) X X X ρ (1700) X X X X ρ (1900) X X X X X ρ (2000) X X X X X ρ (2150) X X X X X X X X X ρ (2270) X X X X X X X X X X ρ (1690) X X X X ρ (1990) X X X X X ρ (2250) X X X X X X X X X X πω π a (1260) π a (1320) π a (1450) πω (1670) ρ f (1270) ρη (1295) ωπ (1300) π a (2040) ρρ (1450) ρ (1450) X X X ρ (1700) X X X X ρ (1900) X X X X X ρ (2000) X X X X X ρ (2150) X X X X X X X X X ρ (2270) X X X X X X X X X X ρ (1690) X X X X ρ (1990) X X X X X ρ (2250) X X X X X X X X X X ηρ ρρ ρη ′ η b (1235) ππ (1670) ππ (1800) ω a (1320) ηρ (1690) ρη (1475) ω a (1450) ρ (1450) X ρ (1700) X X ρ (1900) X X X X X ρ (2000) X X X X X X ρ (2150) X X X X X X X ρ (2270) X X X X X X X X X X ρ (1690) X X ρ (1990) X X X X X X ρ (2250) X X X X X X X X X
KK KK ⋆ K ⋆ K ⋆ KK (1270) KK (1400) KK ⋆ (1410) KK ⋆ (1430) KK ⋆ (1680) K ⋆ K (1270) η ′ b (1235) ρ (1450) X X ρ (1700) X X ρ (1900) X X X X X ρ (2000) X X X X X X X ρ (2150) X X X X X X X ρ (2270) X X X X X X X X X X ρ (1690) X X ρ (1990) X X X X X X X ρ (2250) X X X X X X X X X X
CMD−2
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) Total width πππ a (1260) πω π h (1170) ηρ KKKK * ππ (1300) π a (1320) FIG. 2: (color online). The calculated partial and total decay widths of ρ (1450) dependent on the R value. Here, the dashed line with band isthe experimental total width from Ref. [102]. BABAR
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) π a (1260) π h (1170) πωππ (1300) πω (1420) KK (1270) ρρπ a (1320) ηρ KK η b (1235) KK * ππ (1670)K * K * πω (1650) KK (1400) ρη′ πω (1670)Total width ππ FIG. 3: (color online). The calculated partial and total decay widths of ρ (1900) dependent on the R value. Here, we do not list the decay widthof ρ (1900) → π a (1450) since this channel is tiny. The dashed line with band is the experimental total width from Babar [29]. ρ (1700) → ωπ . For the ρ (1700) → ηρ decay, the calculatedresult is comparable with that listed in Ref. [23]. In TableIII, some ratios of the partial decay widths of ρ (1700) are pre-sented.According to PDG, as for ρ (1700) the ratio Γ ππ (1300) / Γ π is 0.3 while the ratio Γ π a (1260) / Γ π is 0.16 (with a largeuncertainty) [32]. We need to emphasize that these ratios Γ ππ (1300) / Γ π and Γ π a (1260) / Γ π listed in PDG can be changedwith di ff erent considerations of fitting the experimental data (see Sec. 4.3 in Ref. [32] for more details). If adoptingthese two experimental ratios, that would mean that the de-cay into ππ (1300) should be more likely than into π a (1260).However, we get an order of magnitude larger decay rate into π a (1260). This discrepancy should be explained when as-signing ρ (1700) as a 1 D state. Introducing the exotic stateexplanation to ρ (1700) and studying the corresponding decaybehavior are an interesting topic.As the candidate of a 2 D state, the two-body decay and Total width (SPEC)
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) Total width ππρ b (1235) π π (1800) πωππ (1300) KK KK (1270) ρρπω (1420) ππ (1670) ηρρ f (1270) ηρ (1450) πω (1670)KK * (1410)KK (1400) KK * K * K * KK *2 (1430) ρη ′ (958) π a (1260) π h (1170) ωπ (1300) π a (1320) η b (1235) ρ f (1285) πω (1650) ρη (1295) ω a (1260) ω a (1320) FIG. 4: (color online). The calculated partial and total decay widths of ρ (2150) dependent on the R value. The dashed line with band is theexperimental total width taken from Ref. [3]. RVUE
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) Total width π a (1260) π h (1170) πω ρρππ ηρ ππ (1300) πω (1420)KK KK * π a (1320) FIG. 5: (color online). The partial and total decay widths of ρ (1700) dependent on the R value. Here, we do not list ρ (1700) → π a (1450) dueto its tiny decay width. The dashed line with band is the experimental total width from Ref. [23]. total decay widths of ρ (2000) are obtained in Fig. 6. The totalwidth can overlap with the Crystal Barrel result in Ref. [4]when R = (4 . − .
80) GeV − . ρ (2000) dominantly decaysinto ππ (1300), ρρ , ππ (1670) and π a (1260). The decay chan-nels of the ρ (2000) into ππ , π h (1170), π a (1320), πω (1420)and η b (1235) are also important.Fig. 7 shows the decay information of ρ (2270) from thecalculation of the QPC model. Although more decay chan-nels are open, ρ (2270) has a smaller total decay width com- pared with the former two D states. The obtained totaldecay width can overlap with the Crystal Barrel data [3] asshown in Fig. 7. The main decay modes are ππ (1300) and ππ (1800). Other important decay channels include π a (1260), ω a (1260), ρ f (1270), and ρ b (1235).At present, experiments scarcely provide information on ρ (2270). Thus, the theoretical predictions of the two-bodystrong decays of ρ (2270) shown in Fig. 7 and Table IV canprovide valuable guidance to future experimental study on RVUE
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) Total width ρρ ππ (1300) ππ (1670) π a (1260) π h (1170) ππ π a (1320) η b (1235) πω (1420)KK (1400) πω KK * πω (1650) ηρ KK KK * (1410)KK (1270) ρη ′ (958) πω (1670) ππ (1800)K * K * ηρ (1450)KK (1430) FIG. 6: (color online). The partial and total decay widths of ρ (2000) as the 2 D state dependent on the R value. The dashed line with band isthe experimental total width [4].TABLE III: The obtained ratios of the partial decay widths of the ρ states discussed in Figs. 2-5. Here, we have only listed the ratiosweakly dependent on R , which is the reason why we have not listedthe ratios of Γ ηρ / Γ ωπ and Γ KK / Γ ωπ for ρ (1450) that are strongly de-pendent on R . Γ ππ / Γ π a (1260) Γ π h (1170) / Γ π a (1260) Γ π a (1260) / Γ Total ρ (1450) 0 . − .
873 0 . − .
381 0 . − . ρ (1700) 0 . − .
178 0 . − .
696 0 . − . ρ (1900) 0 . − .
432 0 . − .
360 0 . − . ρ (2150) 2 . − .
674 1 . − .
404 0 . − . ρ (2270). TABLE IV: Several calculated branching ratios of the partial decaywidths of ρ (2000) and ρ (2270). Γ π a (1260) / Γ ππ (1300) Γ π h (1170) / Γ ππ Γ ππ (1300) / Γ Total ρ (2000) 0 . − .
997 0 . − .
714 0 . − . ρ (2270) 0 . − .
790 0 . − .
507 0 . − . In Figs. 5-7, we also notice that the corresponding R valuesfor reproducing the experimental data are within the allowed range. C. n D states If ρ (1690) is a 1 D state, the partial decay widths areshown in Fig. 8, where the decay of ρ (1690) is dominatedby the ρρ channel. The other large decay modes include πω , ππ and π h (1170). The decay modes of π a (1320), π a (1260)and ηρ are also sizeable. TABLE V: Several calculated branching ratios of ρ (1690) and theratio Γ π a (1320) / Γ ηρ . Here, we also list the corresponding experimentaldata in the third column.Ratios This work Experimental data Γ ρρ / Γ Total (49 . − . Γ ππ / Γ Total (5 . − . . ± . Γ πω / Γ Total (14 . − . ± Γ π h (1170) / Γ Total (6 . − . Γ π a (1320) / Γ ηρ . − .
40 5 . ± . In Table V, several branching ratios of ρ (1690) and the ra-tio Γ π a (1320) / Γ ηρ are calculated in comparison with the cor- RVUE
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) −3 R (GeV −1 ) Total width ππ (1300) ππ (1800) ω a (1260) π a (1260) ρ f (1270) ρ b (1235) ρ f (1285) ω a (1320) ππ (1670) ρρππ πω (1420) π h (1170) η b (1235) ηρ (1450) ρρ (1450) KK ωπ (1300) πω (1650)KK (1400) πω (1670) KK * (1410) πω KK (1270)KK * (1680) π a (1320) ρ f (1420) ρη (1295) η ′ b (1235)KK * ηρ ρη ′ (958) ρη (1475) K * K * KK (1430)K * K (1270) ω a (1450) π a (1450) ηρ (1690) π a (2040) FIG. 7: (color online). The partial and total decay widths of ρ (2270) dependent on the R value. The dashed line with band is the experimentaltotal width [3]. OMEG
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) Total width ρρ πππω π a (1320) π a (1260) ηρ ππ (1300)KK KK * πω (1420) π h (1170) FIG. 8: (color online). The partial and total decay widths of ρ (1690) dependent of the R value. The dashed line with band is the experimentaltotal width [109]. responding experimental values. Our branching ratios of ρ (1690) → ππ, πω and the ratio Γ π a (1320) / Γ ηρ are compara-ble with the experimental results. At present, experiments re-veal that ρ (1690) dominantly decays into 4 π with the branch-ing ratio ∼ .
1% [1], which is supported by our calculation, where the final states πω and ρρ can mainly contribute to the4 π final state.In Fig. 8, we give comparison of our results with the ex-perimental data [109]. If reproducing the experimental totalwidth, the adopted R value is about 3 GeV − , which is un-0 TABLE VI: Some typical branching ratios of ρ (1690), ρ (1990) and ρ (2250). Γ ρρ / Γ Total Γ ππ / Γ Total Γ ππ (1300) / Γ Total ρ (1690) 0 . − .
588 0 . − .
058 0 . − . ρ (1990) 0 . − .
182 0 . − .
240 0 . − . ρ (2250) 0 . − .
068 0 . − .
358 0 . − . reasonable. In addition, the obtained total decay width of ρ (1690) is larger than the data in PDG [1] when taking R around 4 GeV − [89]. This situation shows that ρ (1690) asa 1 D state seems questionable. For clarifying this point,we suggest the precise measurement of its resonance param-eters in future experiments. Of course, this discrepancy men-tioned above also provides a possibility of introducing the ex-otic state explanation to ρ (1690). We notice that a three- ρ meson molecular state was proposed in Ref. [63].The partial decay widths of ρ (1990) are predicted in Fig.9, where the mass of ρ (1990) in Table I is adopted in our cal-culation. ρ (1990) mainly decays into ρρ , ππ , πω , ππ (1300)and πω . Several typical decay branching ratios of ρ (1990)are presented in Table VI. The calculated total decay widthof ρ (1990) is compatible with the experimental data [3] asshown in Fig. 9. In addition, ρ (1990) → ππ, πω were ob-served in the experiment [1]. Our calculation shows that thedecay widths of ρ (1990) → ππ, πω are sizeable.In Fig. 10 and Table VI, the decay properties of ρ (2250) asa 3 D are illustrated. For higher ρ meson, the decay behav-ior reflects the node e ff ect, where ρ (2250) decay widths aredependent on the R value. If taking a typical value of R = . − , we can obtain the total decay width consistent with theexperimental data [3]. The corresponding main partial decaychannels are ππ , ππ (1300) and πω . Contrary to the former ρ (1690) and ρ (1900), the decay width of ρ (2250) → ρρ issmall. At present, ρ (2250) was observed in its ππ , K ¯ K , ηππ , πω and ω a (1320) decay channels. V. SUMMARY
In the past decades, many more ρ/ρ states have been ob-served in experiments. How to categorize these ρ/ρ statesinto the meson family is an intriguing research topic, whichcan improve our knowledge of light hadron spectrum. Inthis work, we systematically study the OZI-allowed two-bodystrong decay behaviors of the observed ρ/ρ states, where theQPC model [72] is applied to the concrete calculation.As shown in Fig. 1, the mass spectrum analysis can pro-vide preliminary information on these ρ/ρ states, where theirquantum numbers are assigned. Given these assignments, weperform the calculation of two-body strong decays of these ρ/ρ states listed in Table I. By comparing our theoretical re-sults with the existing experimental data, the hadron structureproperties of these ρ/ρ states can be obtained and examined.Besides getting the hadron structure properties of these ρ/ρ states, our study also provides abundant decay informa-tion of these states, which can be as a valuable guidance tofurther experimental study on light hadron spectrum. Acknowledgement
We would like to thank Qiang Zhao for useful discussions.X.L. also would like to thank Takayuki Matsuki for readingour manuscript carefully. This project is supported by theNational Natural Science Foundation of China under Grants11222547, 11175073, 11035006, the Ministry of Educationof China (FANEDD under Grant No. 200924, SRFDP un-der Grant No. 20120211110002, NCET, the Fundamental Re-search Funds for the Central Universities), the Fok Ying-TongEducation Foundation (No. 131006). [1] J. Beringer et al. [Particle Data Group Collaboration], Phys.Rev. D , 010001 (2012).[2] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D ,092002 (2008) [arXiv:0710.4451 [hep-ex]].[3] A. V. Anisovich, C. A. Baker, C. J. Batty, D. V. Bugg, L. Mon-tanet, V. A. Nikonov, A. V. Sarantsev and V. V. Sarantsev et al. ,Phys. Lett. B , 8 (2002) [arXiv:1109.5247 [hep-ex]].[4] A. V. Anisovich, C. A. Baker, C. J. Batty, D. V. Bugg, C. Hodd,H. C. Lu, V. A. Nikonov and A. V. Sarantsev et al. , Phys. Lett.B , 47 (2000) [arXiv:1109.0883 [hep-ex]].[5] A. V. Anisovich, C. A. Baker, C. J. Batty, D. V. Bugg,V. A. Nikonov, A. V. Sarantsev, V. V. Sarantsev and B. S. Zou,Phys. Lett. B , 6 (2001).[6] A. V. Anisovich, C. A. Baker, C. J. Batty, D. V. Bugg,V. A. Nikonov, A. V. Sarantsev, V. V. Sarantsev and B. S. Zou,Phys. Lett. B , 281 (2001).[7] A. Hasan and D. V. Bugg, Phys. Lett. B , 215 (1994).[8] M. A. Abolins, R. L. Lander, W. A. W. Mehlhop, N. huu Xuong and P. M. Yager, Phys. Rev. Lett. , 381 (1963).[9] G. P. Yost et al. [Particle Data Group Collaboration], Phys.Lett. B , 1 (1988).[10] G. Cosme, B. Dudelzak, B. Grelaud, B. Jean-Marie, S. Jullian,D. Lalanne, F. Laplanche and V. Lepeltier et al. , Nucl. Phys.B , 215 (1979).[11] C. Bacci, G. De Zorzi, G. Penso, B. Stella, R. Baldini Celio,G. Battistoni, G. Capon and R. Del Fabbro et al. , Nucl. Phys.B , 31 (1981).[12] L. M. Barkov, A. G. Chilingarov, S. I. Eidelman, B. I. Khazin,M. Y. .Lelchuk, V. S. Okhapkin, E. V. Pakhtusova andS. I. Redin et al. , Nucl. Phys. B , 365 (1985).[13] S. I. Dolinsky, V. P. Druzhinin, M. S. Dubrovin, S. I. Ei-delman, V. B. Golubev, V. N. Ivanchenko, I. A. Koop andA. A. Mikhailichenko et al. , Phys. Lett. B , 453 (1986).[14] C. Erkal and M. G. Olsson, Z. Phys. C , 615 (1986).[15] A. Donnachie and H. Mirzaie, Z. Phys. C , 407 (1987).[16] A. Donnachie and A. B. Clegg, Z. Phys. C , 257 (1987). SPEC
R (GeV −1 ) R (GeV −1 ) −4 −3 R (GeV −1 ) Total width ππρρ ππ (1300) πωπ a (1320) π a (1260) π h (1170) πω (1420) ππ (1670) πω (1670) ηρη b (1235)KK (1270) K * K * KK KK * ρη ′ (958)KK (1430) KK * (1410) πω (1650)KK (1400) ππ (1800) FIG. 9: (color online). The partial and total decay widths of ρ (1990) dependent on the the R value. Here, the π a (1450) channel is not listed.The dashed line with band is the experimental total width [3]. SPEC
R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) R (GeV −1 ) Total width ππρ b (1235) ω a (1320) πω (1420) ππ (1300) πωρρω a (1260) ππ (1670) ωπ (1300) π a (1320) π h (1170) ππ (1800) ρ f (1285) ρη (1295) π a (1260) ηρηρ (1450)KK * (1410) KK ρ f (1270) η b (1235) KK * π a (2040) K * K * πω (1650) ρη ′ (958) ρ f (1420)KK (1430) KK (1400)KK * (1680) η ′ b (1235) πω (1670) KK (1270) FIG. 10: (color online). The partial and total decay widths of ρ (2250) dependent on the the R value. Since the decay widths of the π a (1450), ρρ (1450), K ∗ K (1270) and ρη (1475) channels are much smaller than that of the K ¯ K ∗ (1680) channel, we do not list these channels here. Thedashed line with band is the experimental data [3]. [17] A. B. Clegg and A. Donnachie, Z. Phys. C , 313 (1988).[18] S. Fukui, N. Horikawa, S. Inaba, T. Inagaki, Y. Inagaki,Y. Ishizaki, T. Iwata and T. Kinashi et al. , Phys. Lett. B ,441 (1988).[19] A. Antonelli et al. [DM2 Collaboration], Phys. Lett. B ,133 (1988).[20] D. Bisello et al. [DM2 Collaboration], Phys. Lett. B , 321(1989).[21] A. Castro et al. [DM2 Collaboration], LAL-88-58.[22] S. I. Dolinsky, V. P. Druzhinin, M. S. Dubrovin, V. B. Gol-ubev, V. N. Ivanchenko, E. V. Pakhtusova, A. N. Peryshkinand S. I. Serednyakov et al. , Phys. Rept. , 99 (1991).[23] A. B. Clegg and A. Donnachie, Z. Phys. C , 455 (1994).[24] A. Donnachie and A. B. Clegg, Phys. Rev. D , 4979 (1995).[25] N. N. Achasov and A. A. Kozhevnikov, Phys. Rev. D , 2663(1997) [hep-ph / ,117503 (2000) [hep-ph / et al. , arXiv:1303.5198 [hep-ex].[28] S. Godfrey and N. Isgur, Phys. Rev. D , 189 (1985).[29] T. Barnes, F. E. Close, P. R. Page and E. S. Swanson, Phys.Rev. D , 4157 (1997) [hep-ph / , 726 (2011).[31] M. K. Volkov, D. Ebert and M. Nagy, Int. J. Mod. Phys. A ,5443 (1998) [hep-ph / et al. [CRYSTAL BARREL Collaboration], Eur.Phys. J. C , 261 (2001).[33] F. E. Close and P. R. Page, Nucl. Phys. B , 233 (1995)[hep-ph / , 621(1993).[35] F. E. Close and P. R. Page, Phys. Rev. D , 1706 (1995) [hep-ph / , 1584 (1997) [hep-ph / ,114011 (1999) [hep-ph / ,129 (2011) [arXiv:1106.1010 [hep-ph]].[39] L. Y. Glozman, AIP Conf. Proc. , 64 (2011)[arXiv:1012.5046 [hep-ph]].[40] D. V. Bugg, Phys. Rev. D , 118501 (2013) [arXiv:1209.3481[hep-ph]].[41] K. Kittimanapun, K. Khosonthongkee, C. Kobdaj, P. Suebka,Y. Yan, Phys. Rev. C , 025201 (2009) [arXiv:0803.2028[hep-ph]].[42] A. Antonelli et al. [FENICE Collaboration], Phys. Lett. B ,427 (1996).[43] P. L. Frabetti et al. [E687 Collaboration], Phys. Lett. B ,240 (2001) [hep-ex / et al. , Phys. Lett. B , 290 (2004) [hep-ex / et al. [BABAR Collaboration], Phys. Rev. D ,052003 (2006) [hep-ex / , 145 (1981).[48] M. Atkinson et al. [Omega Photon Collaboration], Z. Phys. C , 333 (1985). [49] A. B. Clegg and A. Donnachie, Z. Phys. C , 677 (1990).[50] M. E. Biagini, S. Dubnicka, E. Etim and P. Kolar, Nuovo Cim.A , 363 (1991).[51] S. Dubnicka, Nuovo Cim. A , 1 (1988).[52] D. Alde et al. [IHEP-IISN-LANL-LAPP-KEK Collaboration],Z. Phys. C , 553 (1992).[53] D. Alde et al. [GAMS Collaboration], Nuovo Cim. A ,1867 (1994) [Z. Phys. C , 379 (1995)].[54] A. V. Anisovich, V. A. Nikonov, A. V. Sarantsev, V. V. Sarant-sev, C. A. Baker, C. J. Batty, D. V. Bugg and A. Hasan et al. ,Phys. Lett. B , 271 (1999).[55] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D , 092005 (2007) [Erratum-ibid. D , 119902 (2008)][arXiv:0708.2461 [hep-ex]].[56] A. V. Anisovich, V. V. Anisovich and A. V. Sarantsev, Phys.Rev. D , 051502 (2000) [hep-ph / , 094006 (2012) [arXiv:1203.4782 [hep-ph]].[58] P. Masjuan, E. R. Arriola and W. Broniowski,arXiv:1305.3493 [hep-ph].[59] D. V. Bugg, Phys. Rept. , 257 (2004) [hep-ex / , 354 (1965).[61] A. Forino and R. Gessaroli, Phys. Lett. , 65 (1965).[62] D. V. Amelin et al. [VES Collaboration], Nucl. Phys. A ,83 (2000).[63] L. Roca and E. Oset, Phys. Rev. D , 054013 (2010)[arXiv:1005.0283 [hep-ph]].[64] R. J. Abrams, R. L. Cool, G. Giacomelli, T. F. Kycia,B. A. Leontic, K. K. Li and D. N. Michael, Phys. Rev. D ,1917 (1970).[65] M. Coupland, E. Eisenhandler, W. R. Gibson, P. I. P. Kalmusand A. Astbury, Phys. Lett. B , 460 (1977).[66] D. Cutts, M. L. Good, P. D. Grannis, D. Green, Y. Y. Lee,R. Pittman, J. Storer and A. C. Benvenuti et al. , Phys. Rev. D , 16 (1978).[67] A. A. Carter, Nucl. Phys. B , 467 (1978).[68] A. A. Carter, M. Coupland, E. Eisenhandler, W. R. Gibson,P. I. P. Kalmus, D. P. Kimber, A. Astbury and D. P. Jones,Phys. Lett. B , 117 (1977).[69] A. D. Martin and M. R. Pennington, Nucl. Phys. B , 216(1980).[70] B. R. Martin and D. Morgan, Nucl. Phys. B , 355 (1980).[71] G. F. Chew and S. C. Frautschi, Phys. Rev. Lett. , 41 (1962).[72] L. Micu, Nucl. Phys. B , 521 (1969).[73] A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Rev.D , 2223 (1973).[74] A. Le Yaouanc, L. Oliver, O. Pene and J. -C. Raynal, Phys.Rev. D , 1415 (1974).[75] A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys. Rev.D , 1272 (1975).[76] A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Phys.Lett. B , 57 (1977).[77] A. Le Yaouanc, L. Oliver, O. Pene and J. -C. Raynal, Phys.Lett. B , 397 (1977); “Hadron Transitions of the QuarkModel ”(Gordon and Breach, New York, 1988).[78] E. van Beveren, C. Dullemond and G. Rupp, Phys. Rev. D ,772 (1980) [Erratum-ibid. D , 787 (1980)].[79] E. van Beveren, G. Rupp, T. A. Rijken and C. Dullemond,Phys. Rev. D , 1527 (1983).[80] R. Bonnaz, B. Silvestre-Brac and C. Gignoux, Eur. Phys. J. A , 363 (2002) [hep-ph / , 171(1992).[82] J. Lu, W. -Z. Deng, X. -L. Chen and S. -L. Zhu, Phys. Rev. D , 054012 (2006) [hep-ph / , 074020(2009) [arXiv:0901.0505 [hep-ph]].[84] H. G. Blundell and S. Godfrey, Phys. Rev. D , 3700 (1996)[hep-ph / , 189 (1995) [hep-ph / , 2809 (1986).[87] S. Capstick and W. Roberts, Phys. Rev. D , 4570 (1994)[nucl-th / ,6811 (1996) [hep-ph / , 094004(2005) [hep-ph / , 123(2005) [hep-ph / , 509 (2007) [hep-ph / , 617 (2007) [hep-ph / , 094017 (2007) [arXiv:0704.0075 [hep-ph]];X. Liu, C. Chen, W.Z. Deng, and X.L. Chen, Chinese Phys. C , 424 (2008).[94] D. -M. Li and B. Ma, Phys. Rev. D , 074004 (2008)[arXiv:0801.4821 [hep-ph]]; D. -M. Li and B. Ma, Phys.Rev. D , 094021 (2008) [arXiv:0803.0106 [hep-ph]]; D. -M. Li and S. Zhou, Phys. Rev. D , 054013 (2008)[arXiv:0805.3404 [hep-ph]]; D. -M. Li and S. Zhou, Phys.Rev. D , 014014 (2009) [arXiv:0811.0918 [hep-ph]].[95] Z. -F. Sun and X. Liu, Phys. Rev. D , 074037 (2009)[arXiv:0909.1658 [hep-ph]]. [96] X. Liu, Z. -G. Luo and Z. -F. Sun, Phys. Rev. Lett. , 122001(2010) [arXiv:0911.3694 [hep-ph]].[97] Z. -F. Sun, J. -S. Yu, X. Liu and T. Matsuki, Phys. Rev. D ,111501 (2010) [arXiv:1008.3120 [hep-ph]].[98] J. -S. Yu, Z. -F. Sun, X. Liu and Q. Zhao, Phys. Rev. D ,114007 (2011) [arXiv:1104.3064 [hep-ph]].[99] X. Wang, Z. -F. Sun, D. -Y. Chen, X. Liu and T. Matsuki, Phys.Rev. D , 074024 (2012) [arXiv:1202.4139 [hep-ph]].[100] Z. -C. Ye, X. Wang, X. Liu and Q. Zhao, Phys. Rev. D ,054025 (2012) [arXiv:1206.0097 [hep-ph]].[101] M. Jacob and G. C. Wick, Annals Phys. , 404 (1959) [AnnalsPhys. , 774 (2000)].[102] R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B , 217 (2001) [hep-ex / et al. [Particle Data Group Collaboration], Eur. Phys.J. C , 1 (1998).[104] B. Ketzer, PoS QNP , 025 (2012) [arXiv:1208.5125 [hep-ex]].[105] M. N. Oakden and M. R. Pennington, Nucl. Phys. A , 731(1994).[106] J. Alspector, K. J. Cohen, W. C. Harrison, B. Maglich,F. Sannes, D. Van Harlingen, G. Cvijanovich and M. Matin et al. , Phys. Rev. Lett. , 511 (1973).[107] R. Kokoski and N. Isgur, Phys. Rev. D , 907 (1987).[108] D. V. Bugg, B. S. Zou and A. V. Sarantsev, Nucl. Phys. B ,59 (1996).[109] M. J. Corden, J. D. Dowell, J. Garvey, M. Jobes, I. R. Kenyon,J. Mawson, T. McMahon and I. F. Corbett et al. , Nucl. Phys.B157