Extracting ππ S -wave scattering lengths from cusp effect in heavy quarkonium dipion transitions
aa r X i v : . [ h e p - ph ] D ec Extracting ππ S -wave scattering lengths from cusp effect inheavy quarkonium dipion transitions
Xiao-Hai Liu ∗ , Feng-Kun Guo † , Evgeny Epelbaum ‡ Institut f¨ur Theoretische Physik II, Rhur-Universit¨at Bochum, D-44780 Bochum, Germany Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universit¨at Bonn, D–53115 Bonn, Germany
August 3, 2018
Abstract
Charge-exchange rescattering π + π − → π π leads to a cusp effect in the π π invariantmass spectrum of processes with π π in the final state which can be used to measure ππ S -wavescattering lengths. Employing a non-relativistic effective field theory, we discuss the possibility ofextracting the scattering lengths in heavy quarkonium π π transitions. The transition Υ(3 S ) → Υ(2 S ) π π is studied in details. We discuss the precision that can be reached in such an extractionfor a certain number of events. PACS : 13.25.Gv, 13.75.Lb ∗ E-mail address:
[email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected]
Introduction
Being much lighter than all the other hadrons, the pions play a unique role in the strong interacti-ons. They are the pseudo-Goldstone bosons of the spontaneous chiral symmetry breaking in quantumchromodynamics (QCD). Thus, to a large extent, the interaction between pions are governed by spon-taneous and explicit chiral symmetry breaking. The ππ scattering problem already has a long history,which began about half a century ago [1]. At low energies, the strength of the ππ S -wave interactionis described by the scattering lengths, which can shed light on the fundamental properties of QCD.The scattering lengths can be calculated in chiral perturbation theory (ChPT) [2, 3], the low-energyeffective theory of QCD, to a given order in the chiral expansion. Combining two-loop ChPT withRoy equations, the ππ scattering lengths were predicted with a high precision [4, 5]. For instance,the difference between the isospin I = 0 and I = 2 S -wave scattering lengths was predicted to be ( a − a ) M π + = 0 . ± . . A similar result of . ± . was obtained in Ref. [6] usingdispersion relations without input from ChPT.Experimentally, the ππ scattering lengths can be measured in several ways. The angular distribu-tions of K e decay is sensitive to the ππ phase shifts which are related with the scattering lengths. Thefirst experiment along these lines was carried out by the Geneva-Saclay Collaboration in the seventiesof the lest century [7]. A similar method was recently employed by the E865 and NA48/2 Collabora-tions [8, 9, 10, 11]. Pionium lifetime can also be related to the ππ scattering lengths, and the experi-mental result is well consistent with the prediction [12]. Another precise method relies on measuringthe cusp effect in the decay K + → π π π + , which results from the charge-exchange rescattering π + π − → π π [13, 14, 15, 16, 17, 18, 19, 20]. The cusp structure was also discussed in other pro-cesses including the K L → π , η → π and η ′ → ηππ [21, 22, 23]. Since the branching ratio of K + → π + π − π + , (5 . ± . , is much larger than that of K + → π π π + , (1 . ± . [24],the charge-exchange rescattering turns out to be important so that the cusp effect in the π π invariantmass spectrum appears to be enhanced. However, it is difficult to accurately measure the cusp effectin K L → π and η → π according to currently available experimental data. The process η ′ → ηππ is a promising candidate, in which the cusp effect is predicted to have an effect of more than inthe decay spectrum below π + π − threshold [23]. For a brief review of the ππ scattering and a list ofexperimental measurements see Ref. [25].In this paper, we investigate the possibility of extracting the ππ scattering lengths using the cuspeffect in heavy quarkonium dipion transitions. These transitions are among the most important decaymodes of the heavy quarkonium states below open heavy-flavor thresholds. Taking the process ψ ′ → J/ψπ π as an example, the branching fraction is . ± . [24], and the BESIII and CLEO-cCollaborations have already accumulated huge data samples in this channel. In particular, the BESIIICollaboration has acquired a sample of 106 million ψ ′ events, and this number is still increasing[26]. Because of the Watson final-state theorem [27], it is possible to learn about the ππ interactionfrom the dipion transitions. There were suggestions of studying the ππ scattering phase shifts in the ψ ′ → J/ψπ + π − transitions [28, 29, 30]. There are also huge data samples for the bottomonium stateswhich were collected in the B -factories, and more are expected to come from the next-generation high-luminosity B -factories [31, 32]. In view of this situation, it is interesting to explore the cusp effect inheavy quarkonium transitions with two neutral pions in the final state. Because the isospin symmetryis well conserved here, one has B ( ψ ′ → J/ψπ + π − ) / B ( ψ ′ → J/ψπ π ) ≈ [24], and similarlyfor the other heavy quarkonium dipion transitions. This is similar to the η ′ → ηππ , which will makethe charge-exchange rescattering effect more important than those in the processes K L → π and η → π . In addition, it was found in Ref. [23] that two-loop rescattering is highly suppressed in the η ′ → ηππ process due to the approximate isospin symmetry, so that the cusp in the π π distribution2s completely dominated by one-loop contributions. The same conclusion should hold in our case.Therefore, it is safe to work up to only one-loop order. From the theoretical point of view, since theinteraction between a heavy quarkonium and pion is highly Okubo-Zweig-Iizuka (OZI) suppressed,we may further simplify the problem by neglecting this type of contributions.If we concentrate on the region near the ππ threshold, the three-momenta of all the final particlesare small in comparison with their masses. Thus, a nonrelativistic effective field theory (NREFT) canbe employed. The processes to be considered are similar to the η ′ → ηππ . We will follow here theNREFT framework developed and applied in Refs. [20, 23] and refer to these papers and referencestherein for more details. This method was firstly used in the study of cusp effect in K → π , and thenextended to other reactions such as e.g. η ′ → ηππ and η → π .Our paper is organized as follows. The framework of NREFT will be briefly introduced in Sec-tion 2, where the necessary terms in the effective Lagrangians are listed. The low-energy constantsentering the tree-level production amplitude are determined in Section 3 by matching to a relativisticdescription of the decay for the transition Υ(3 S ) → Υ(2 S ) π π in the framework of unitarized chiralperturbation theory. In the same section, we generate several sets of synthetic data using the MonteCarlo method, and investigate the accuracy of the extraction of the ππ scattering lengths from thesesynthetic data. A short summary is presented in Section 4, and an estimate of the J/ψπ scatteringlength is attempted in Appendix A.
Here and in what follows, we consider ψ ′ dipion transitions as an example. The method can be easilyextended to bottomonium case. Let us describe the power counting scheme in the NREFT, whichis essentially a nonrelativistic velocity counting. The masses of all involved particles are counted as O (1) . Since we focus on the region close to the thresholds of two pions, even pions can be dealtwith nonrelativistically. The heavy quarkonium in the final state is also nonrelativistic because itsthree-momentum in the rest frame of the decaying particle does not exceed 500 MeV. We, therefore,count all these three-momenta as quantities of order O ( ǫ ) . The kinetic energy T i = p i − M i isthen counted as O ( ǫ ) . Another expansion parameter used in this scheme is the ππ scattering length,denoted by a ππ . Here one relies on the fact that low-energy interactions between two pions are weakdue to their Goldstone boson nature. In principle, J/ψπ scattering should also be taken into account.However, it should be suppressed according to the OZI rule because the
J/ψ and pion do not haveany common valence quark. A rough estimation of the
J/ψπ scattering length carried out in theappendix yields (cid:12)(cid:12) a J/ψπ (cid:12)(cid:12) . . fm which is consistent with the preliminary lattice result a J/ψπ =( − . ± . fm [33]. Thus, the J/ψπ scattering length is at least one order of magnitude smallerthan a − a . The bottomonium-pion scattering length would be even smaller. The situation here is,therefore, similar to the one in the process η ′ → ηππ , where ηπ interaction was found to play a minorrole in the ππ cusp structure, and its effect can be largely absorbed into the polynomial productionamplitude [23]. Thus, in the following, we will not take into account the heavy quarkonium-pionscattering.The relevant effective Lagrangians contain two parts L eff = L ψ + L ππ . (1)3ere, the first term describes the production mechanism and reads up to O ( ǫ ) L ψ = 12 X n =0 G n (cid:16) ψ ′† i ( W J/ψ − M J/ψ ) n J i Φ Φ + h.c. (cid:17) + X n =0 H n (cid:16) ψ ′† i ( W J/ψ − M J/ψ ) n J i Φ + Φ − + h.c. (cid:17) + · · · , (2)where W J/ψ = q M J/ψ − △ , with △ being the Laplacian. At this order, the production is purely S -wave, while the D -wave contribution starts from O ( ǫ ) . ππ interaction is described by [21]: L ππ = 2 X k =0 , ± Φ † k W k ( i∂ t − W k )Φ k + C x (Φ † Φ † Φ + Φ − + h.c. ) + 14 C (Φ † Φ † Φ Φ + h.c. )+ D x h (Φ † ) µ (Φ † ) µ Φ + Φ − + Φ † Φ † (Φ † + ) µ (Φ †− ) µ + h.c. i + 14 D h (Φ † ) µ (Φ † ) µ Φ Φ + Φ † Φ † (Φ † ) µ (Φ † ) µ + h.c. i + · · · , (3)where (Φ k ) µ = ( P k ) µ Φ k , ( P k ) µ = ( W k , − i ∇ )(Φ † k ) µ = ( P † k ) µ Φ † k , ( P † k ) µ = ( W k , i ∇ ) (4)and W k = q M k − △ . In Eq. (3), the couplings C x , C , D x and D can be obtained by matchingthe NREFT amplitude to the effective range expansion of ππ scattering amplitudes [10] T I ( s, t ) = 32 π ∞ X l =0 (2 l + 1) t Il ( s ) P l ( z ) , Re t Il ( s ) = q lab (cid:2) a Il + b Il q ab + O ( q ab ) (cid:3) , (5)where t Il is the partial wave amplitude with angular momentum l and isospin I , P l ( z ) are the Legendrepolynomials with z = cos θ , where θ is the scattering angle in the center-of-mass system, and q ab = (cid:2) λ ( s, M a , M b ) /s (cid:3) / / is the center-of-mass momentum with λ ( a, b, c ) = a + b + c − ab + bc + ac ) being the K¨all´en function. We thus have the following relations, C x = 16 π M π + ( a − a ) (cid:18) ξ (cid:19) , C = 16 π M π + ( a + 2 a )(1 − ξ ) ,D x = 4 π M π + ( b − b ) , D = 4 π M π + ( b + 2 b ) , (6)where ξ = (cid:0) M π + − M π (cid:1) /M π + , and the isospin breaking in the S -wave scattering lengths has beenconsidered at leading order in ChPT [34]. We have used the phase convention such that | π + i = −| , +1 i . Because we only consider S -wave scattering here, we have denoted the I = 0 and I = 2 scattering lengths by a and a for brevity in the above equations.4 ′ J/ψ π π (a) (b) (c) J/ψψ ′ π π π + π − J/ψψ ′ π π π π Figure 1: ψ ′ → J/ψπ π via tree diagram and ππ rescattering diagrams.In this paper, we work only up to one-loop order. Neglecting the J/ψπ interaction as explainedabove, the diagrams need to be considered are shown in Fig. 1. With the momenta defined as ψ ′ ( P ψ ′ ) → π ( p ) π ( p ) J/ψ ( p ) , (7)and s i = ( P ψ ′ − p i ) for i = 1 , , , the transition amplitudes at the tree and one-loop level are T tree = (cid:2) G + G ( p − M J ) (cid:3) ~ǫ ψ ′ · ~ǫ J , (8) T = 2 (cid:2) C x + D x ( s − M π + ) (cid:3) (cid:2) H + H ( p − M J ) (cid:3) J + − ( s ) ~ǫ ψ ′ · ~ǫ J + (cid:2) C + D ( s − M π ) (cid:3) (cid:2) G + G ( p − M J ) (cid:3) J ( s ) ~ǫ ψ ′ · ~ǫ J , (9)respectively, where J ab is a nonrelativistic loop integral defined as J ab ( P ) = Z d D li (2 π ) D w a ( ~l )( w a ( ~l ) − l ) 12 w b ( ~P − ~l )( w b ( ~P − ~l ) − P + l ) , (10)with w ( ~l ) = p M + ~l . Within the nonrelativistic power counting scheme, the loop integral measureis counted as O ( ǫ ) , and each of the two propagators is of order O ( ǫ − ) . Thus, the loop integrals J + − and J are of order O ( ǫ ) . Using dimensional regularization and taking D = 4 , we obtain J + − ( s ) = − π s M π + − s s , when s ≤ M π + , (11) J + − ( s ) = i π s s − M π + s , when s > M π + , (12) J ( s ) = i π s s − M π s . (13)One observes that J + − has a nonanalyticity at the π + π − threshold which gives rise to a cusp effect inthe π π invariant mass distribution. The expression for the decay amplitude up to O ( a ππ ǫ ) reads: T = (cid:2) G + G ( p − M J ) + 2 C x H J + − ( s ) + C G J ( s ) (cid:3) ~ǫ ψ ′ · ~ǫ J . (14)In the isospin limit, we have H = G in our phase convention, and it will be used in the following. In this section we explore the possibility to extract the ππ scattering lengths from the heavy quarko-nium dipion transitions. A known feature of the reaction ψ ′ → J/ψππ is that the kinematical region5 .28 0.29 0.30 0.31 0.32 0.33 0.3402468101214 M H ΠΠ L @
GeV D ∆ @ ° D Figure 2: Fit to the parametrization of ππ phase shifts introduced in Ref. [6] (solid line), where thedashed line represents the CHUA results.around the the ππ threshold we are interested in is strongly suppressed so that it only corresponds toa tiny fraction of the total events, see e.g. the BES and CLEO data for the ψ ′ → J/ψπ + π − [35, 36].A more promising reaction is the Υ(3 S ) → Υ(2 S ) π π , which we will concentrate on. The updateddata came from the CLEO Collaboration [37], and their analysis is based on a Υ(3 S ) yield about × . In order to show the cusp effect in the
Υ(3 S ) → Υ(2 S ) π π , it is necessary to determine the values of G and G . This will be achieved via matching to parameters entering the relativistic decay amplitudewhich can be fixed from fitting to the experimental data of the π π invariant mass spectrum. A precisedetermination is presently not possible due to the bad data quality. However, a rough estimate of theratio G /G can be obtained. We employ a simple parametrization of the tree-level relativistic decayamplitude [38] N ǫ · ǫ ′ ( s − C ) , (15)where N is an overall normalization constant, and ǫ ( ǫ ′ ) is the polarization vector of Υ(2 S ) ( Υ(3 S ) ).The cusp effect shows up only when the ππ FSI is considered. This may be taken into accountusing the chiral unitary approach (CHUA) [39, 40, 41, 42, 43], which has been used in studying thedipion transitions among heavy quarkonium states in Refs. [44, 45, 46]. In the CHUA, the ππ S -wavescattering amplitude after taking into account isospin symmetry is given by T ( s ) = V ( s ) (cid:2) − G ( s ) V ( s ) (cid:3) − , (16)where the × matrix V ( s ) contains the S -wave projected π π → π π , π + π − → π + π − and π π → π + π − amplitudes derived from the lowest order chiral perturbation theory with virtualphotons [47], V ( s ) = 1 F π M π / (cid:0) s − M π (cid:1) / √ (cid:0) s − M π (cid:1) / √ (cid:2) s + 4 (cid:0) M π + − M π (cid:1)(cid:3) / ! . (17)6 .27 0.28 0.29 0.30 0.31 0.32 0.33 0.34020406080100 M H Π Π L @
GeV D E v e n t s (cid:144) M e V H a L M H Π Π L @
GeV D È T @ a b it r a r yun it D H b L Figure 3: (a) Comparison of the best fit (histogram) to the π π invariant mass spectrum of the Υ(3 S ) → Υ(2 S ) π π data (points with error bars) measured in Ref. [37]. The fit is done by inte-grating the distribution bin-by-bin. The solid smooth curve is the invariant mass spectrum calculatedusing the best fit parameters and multiplied by an arbitrary normalization constant. (b) The phasespace subtracted spectrum around the π + π − threshold.Here F π is the pion decay constant and the difference between the charged and neutral pion masses istaken into account. G ( s ) = diag { G ( s ) , G + − ( s ) } is a diagonal matrix with G − ) ( s ) = − π " ˜ a ( µ ) + log M π µ + σ log (cid:18) σ + 1 σ − (cid:19) (18)denoting the usual scalar loop function. Here, σ = q − M π /s , and ˜ a ( µ ) is a subtractionconstant introduced to regularize the loop [42, 43]. In order for the ππ FSI to be consistent with ππ scattering, the value of ˜ a ( µ ) may be fixed by reproducing the S -wave ππ phase shifts in the isoscalarchannel, δ ( s ) . We fit to the parametrization of δ ( s ) introduced in Ref. [6], which is given byEq. (6) in that paper, and the central values of the parameters B i in the parametrization are used.Isospin breaking effects are neglected in the fit. The fit range is chosen to be from the ππ thresholdup to 340 MeV, which contains all the available phase space of the Υ(3 S ) → Υ(2 S ) π π . Our bestfit is shown in Fig. 2 and yields ˜ a (1 GeV ) = − . . With this value of ˜ a (1 GeV ) , we can fix thevalue of C in Eq. (15) by fitting to the π π invariant mass spectrum of the Υ(3 S ) → Υ(2 S ) π π data measured in Ref. [37]. The decay amplitude in the CHUA is given by N ǫ · ǫ ′ ( s − C ) h G ( s ) T ( s ) + √ G + − ( s ) T ( s ) i , (19)where T ( s ) refer to the unitarized amplitudes for the π π → π π ( π + π − → π π ) definedin Eq. (16). The best fit with χ /dof = 1 . is shown in Fig. 3 (a). A small cusp at the π + π − threshold shows up, which is more apparent in the phase-space-subtracted invariant mass spectrum inFig. 3 (b). From the fit, we obtain C = − . +0 . − . GeV = − . +0 . − . M π + . Matching G , to N and C in Eq. (15) leads to the relations7 .270 0.275 0.280 0.285 0.29001234 M H Π Π L @
GeV D d G (cid:144) d M Π Π @ a b it r a r yun it D Figure 4: The cusp effect at the π + π − threshold in the reaction Υ(3 S ) → Υ(2 S ) π π calculatedin the NREFT framework (solid line). The dashed line shows the result without charge-exchangerescattering. G = N (cid:2) ( M − m ) − C (cid:3) , G = − N M. (20)Thus, the ratio is determined to be G G = − . +0 . − . MeV . (21)It is small because G contains the mass difference of the two heavy quarkonia while G is propor-tional to the mass of the initial state. Using the central value and adopting the central values of thescattering lengths summarized in Ref. [25], a = 0 . and a = − . in units of M − π + , thecusp effect in the NREFT is plotted in Fig. 4. Certainly, if charge-exchange rescattering is switchedoff, the cusp would disappear as shown by the dashed line. When integrating the spectrum below thethreshold of π + π − , the cusp effect resulted from charge-exchange rescattering will reduce the num-ber of events in this region by about with respect to the tree-level contribution. The values of thisquantity in the processes K + → π π π + , η ′ → ηππ and η → π are about , and less than , respectively [23, 48]. An important question is to what precision the scattering lengths can be extracted from the consideredprocess. To explore this issue, we will first generate artificial data using the Monte Carlo (MC)method. The von Neumann rejection method is employed to select the random data points that followthe normalized distribution of the π π in range between 270 MeV and 290 MeV predicted in theNREFT. The MC data are generated using the central value of G /G given in Eq. (21), and a − a =0 . and a = − . [25] in units of M − π + as input. These data can then be divided into a numberof bins with the statistical errors given by the square root of the number of events in each bin. Varyingthe MC event numbers and the bin widths, one may investigate the impact of the event numbers aswell as the experimental energy resolution on the precision of the extraction of the scattering lengths.8 .270 0.275 0.280 0.285 0.2900.00.20.40.60.81.0 M H Π Π L @
GeV D e v e n t s (cid:144) . M e V M H Π Π L @
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GeV D e v e n t s (cid:144) M e V M H Π Π L @
GeV D e v e n t s (cid:144) . M e V M H Π Π L @
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GeV D e v e n t s (cid:144) M e V Figure 5: Various sets of MC events and their best fits. The event numbers in the range of [270 , MeV of the first, the second and the third rows are about × , × and × ,respectively. The bin widths of the first, the second and the third columns are 0.2, 0.5 and 1 MeV,respectively. The vertical dashed line indicates the π + π − threshold.We tried a number of different combinations of the event numbers and bin widths. Figure 5 showsthe ones with about × , × and × events in the range of [270 , MeV. We then fitthe π π invariant mass distribution calculated using Eq. (14) to the MC data. G /G is fixed to thesame value used in the data generation. The free parameters are an overall normalization constant, a − a and a . The results of the fits are collected in Table 1, where the uncertainties only reflect thestatistical errors in the fit. Because of the random fluctuation in the data generating process, the best fitvalues are not guaranteed to be the same as the input. An interesting observation is that the precisionof the extraction seems to be quite insensitive to the bin widths, at least up to 2 MeV. Comparing theextracted values with the input M π + ( a − a ) = 0 . and M π + a = − . , one sees that theprecision of the extracted value of a − a can reach − for × events in the range of [270 , MeV. For more events, the precision is better by a factor of around p N / N ′ , with N ′ and N the new and old event numbers, as it should be. From Table 1, one sees that the statistical precisionof Ref. [19], M π + ( a − a ) = 0 . ± . stat. ) , may be reached with × events. Thespectrum is rather insensitive to a such that the uncertainty is about 50% for × events. In fact,because a , independent of a − a , only enters through the π π → π π rescattering, its effect canbe largely absorbed into the polynomial production amplitudes. We have checked that if G /G was9in width Events × × × × χ /dof a − a . ± .
036 0 . ± .
012 0 . ± . . ± . χ /dof a − a . ± .
035 0 . ± .
014 0 . ± . . ± . χ /dof a − a . ± .
026 0 . ± .
012 0 . ± . . ± . χ /dof a − a . ± .
054 0 . ± .
010 0 . ± . . ± . χ /dof a − a . ± .
040 0 . ± .
012 0 . ± . . ± . Table 1: Results of fitting to various sets of MC data. The extracted scattering lengths are given inunits of M − π + .released as an additional free parameter, one would not get any useful information on a any more.Furthermore, the uncertainty of a − a would also increase by a factor of about 2 to 2.5. In fact, G /G also contributes to the Υ(3 S ) → Υ(2 S ) π + π − , and its value can be extracted by measuringthe π + π − spectrum in parallel, and thus not completely free. Notice that if we use a instead of a − a as a free parameter, the errors would be much larger. We should stress that to the precision atper cent level, radiative corrections, which is most important in the π + π − threshold region, should betaken into account [49, 23]. Nevertheless, we expect that the precision that can be achieved would notget worsened as the photon-exchange can be taken into account by a simple replacement of the π + π − loop function [23]. Since our aim is to explore the possibility of extracting a − a , they will not beconsidered here.From the CLEO data of the Υ(3 S ) → Υ(2 S ) π π , the events in the range of [270 , MeVcorrespond to around 15% of the total yield of the process. Thus, × , × , × and × events in [270 , MeV require the yields of the
Υ(3 S ) → Υ(2 S ) π π of about × , × , × and × , respectively. Given that the branching fraction of Υ(3 S ) → Υ(2 S ) π π is (1 . ± . [24], at least 2 billion Υ(3 S ) events have to be accumulated in order to obtain × events in the range of [270 , MeV. Other factors like the detecting efficiency will increasethe number even further.
In this paper, we investigate the possibility of extracting the ππ S -wave scattering lengths usingthe cusp effect in heavy quarkonium transitions emitting two neutral pions. These processes aredifferent from all the others for the cusp effect because all involved particles apart from the pionsare heavy. This has a theoretical advantage that the cross-channel rescattering of a pion off a heavyquarkonium is weak due to the OZI suppression. Due to the approximate isospin symmetry, B ( V ′ → V π + π − ) / B ( V ′ → V π π ) ∼ will lead to an enhanced cusp effect in π π invariant mass spectrum.Since we are dealing with the process where the relevant particles have low momenta, the frameworkof NREFT is adopted to calculate the decay amplitude which can be directly parameterized in terms of10igure 6: Schematic diagrams of the charmonium-pion scattering. Here the doubly-solid, dashed,solid and wiggly lines represent charmonia, pions, charmed mesons and gluons, respectively.the ππ threshold parameters. In the present analysis, we worked out the amplitude to order O ( a ππ ǫ ) .We then focus on the Υ(3 S ) → Υ(2 S ) π π , for which the parameters in the production amplitudeare determined by matching to a fit to the experimental data based on the chiral unitary approach.In order to have a feeling on the achievable accuracy of the extraction of the scattering length, wegenerated a number of sets of artificial data using the Monte Carlo method. We then fitted thesesynthetic data to using the values of a − a and a as free parameters. It is comforting to see that theresulting accuracy is insensitive to the bin width and energy resolution. A statistical precision of about2% and 1.5% of a − a can be reached with × and × events of the Υ(3 S ) → Υ(2 S ) π π ,which corresponds to at least × and × Υ(3 S ) events, respectively. The precision canbe worsened by a factor of about 2 in reality because G /G in the production amplitude cannotbe fixed completely. However, measuring the Υ(3 S ) → Υ(2 S ) π + π − in parallel is very helpful inconstraining G /G , and hence increasing the precision of the a − a extraction. The CLEO detectoralready recorded a sample of (5 . ± . × Υ(3 S ) decays [50], while this number is . × for the BaBar detector [51]. With future high-luminosity B -factories, the sample can be one or twoorder-of-magnitude larger. Acknowledgments
We would like to thank Bastian Kubis for very useful discussions and a careful reading, and Ulf-G.Meißner for comments on the manuscript. This work is supported in part by the EU HadronPhysics3project “Study of strongly interacting matter”, by the European Research Council (ERC-2010-StG259218 NuclearEFT), by the DFG and the NSFC through funds provided to the Sino-German CRC110 “Symmetries and the Emergence of Structure in QCD” and by the NSFC (Grant No. 11165005).
A An estimate of the
J/ψπ scattering length
Before presenting the formalism, let us first roughly estimate the
J/ψπ scattering length. Certainlythere are no scattering data available, but one may use the amplitude of ψ ′ π → J/ψπ as a reference forthe
J/ψπ elastic scattering amplitude. Considering a process of scattering a pion off a charmonium,two possible mechanisms are shown in Fig. 6: (a) corresponds to the situation in which the charmo-nium emits two soft gluons which hadronize into pions. This mechanism can be described by usingthe method of QCD multipole expansion. The charmonium-pion scattering can also occur throughintermediate charmed mesons, as depicted in Fig. 6(b), which represents a kind of non-multipoleeffect [52]. Noticing that the analytic structures of the amplitudes for these two mechanisms aredifferent, one concludes that there is no double counting.In the first mechanism, the difference between the transition ( ψ ′ π → J/ψπ ) and elastic (
J/ψπ → J/ψπ ) amplitudes is due to the charmonia-two-gluon vertex, which is proportional to a quantity called11 .30 0.35 0.40 0.45 0.50 0.55 0.60020406080100120140 M ΠΠ @ GeV D Figure 7: Phase space subtracted invariant mass spectrum of the ππ system for the decay ψ ′ → J/ψπ + π − (in arbitrary units). The original data are taken from Ref. [35].charmonium chromo-polarizability α c ¯ c , the definition of which can be found in Ref. [53]. Because α ψ ′ α J/ψ ≥ | α ψ ′ J/ψ | [53], one may expect that the elastic amplitude is somewhat larger than thetransition one, i.e., |A ( J/ψπ → J/ψπ ) ( a ) | & |A ( ψ ′ π → J/ψπ ) ( a ) | at the J/ψπ threshold. In thesecond mechanism, the elastic
J/ψπ scattering amplitude is proportional to g , and the transitionamplitude is proportional to g g ′ , where g ( g ′ ) are the J/ψ ( ψ ′ ) D ¯ D coupling constants. Neitherof these coupling constants can be measured directly. Based on a vector dominance model, it wasestimated in Ref. [54] that g = √ M J / ( M D f J/ψ ) with f J/ψ the
J/ψ decay constant. Thus, one mayestimate g ′ g ≈ f J/ψ f ψ ′ ≈ (cid:18) Γ( J/ψ → e + e − )Γ( ψ ′ → e + e − ) (cid:19) / ≈ . , which means |A ( ψ ′ π → J/ψπ ) ( b ) | & |A ( J/ψπ → J/ψπ ) ( b ) | . Combining with the estimate for thefirst mechanism, it is reasonable to assume |A ( J/ψπ → J/ψπ ) | ∼ |A ( ψ ′ π → J/ψπ ) | (22)at the J/ψπ threshold.Because of crossing symmetry, the ψ ′ π → J/ψπ scattering amplitude is related to the ψ ′ → J/ψππ . Assuming that the amplitudes are constant, denoted by e C , they are the same for scatteringand decay processes. This assumption is definitely not realistic, but it can be used to place an upperlimit of the J/ψπ scattering length. The
J/ψπ threshold occurs when the ππ invariant mass is √ s = M ππ = 415 MeV. In Fig. 7, we show the phase-space-subtracted invariant mass spectrum of the ππ system for the decay ψ ′ → J/ψπ + π − . That is, the experimental data [35] are divided by | ~p ∗ || ~p | ,where | ~p ∗ | = 12 √ s q λ (cid:0) s , M J , M π (cid:1) , | ~p | = 12 M ψ ′ r λ (cid:16) M ψ ′ , s , M J (cid:17) . (23)From Fig. 7, one can see that the physical decay amplitude at √ s = 415 MeV should be smaller thanthe assumed (nonrealistic) constant amplitude | e C | . From the decay width of ψ ′ → J/ψπ + π − , one12an extract the constant | e C | ≈ . . Using Eq. (22), we get an approximate upper limit for the J/ψπS -wave scattering length (cid:12)(cid:12) a J/ψπ (cid:12)(cid:12) . | e C | π ( M J + M π ) ≈ . fm . (24)Similarly, using the measured decay width of the Υ(3 S ) → Υ(2 S ) π + π − , we get (cid:12)(cid:12) a Υ(2 S ) π (cid:12)(cid:12) . . fm . (25) References [1] S. Weinberg, Phys. Rev. Lett. , 616 (1966).[2] S. 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