SSearch for leptonic decays of D mesons M. Petriˇc, M. Stariˇc, I. Adachi, H. Aihara, K. Arinstein,
1, 30
T. Aushev,
17, 11
A. M. Bakich, V. Balagura, E. Barberio, K. Belous, V. Bhardwaj, M. Braˇcko,
19, 12
T. E. Browder, A. Chen, P. Chen, B. G. Cheon, I.-S. Cho, Y. Choi, J. Dalseno,
20, 38
M. Danilov, A. Das, Z. Doleˇzal, A. Drutskoy, W. Dungel, S. Eidelman,
1, 30
N. Gabyshev,
1, 30
P. Goldenzweig, B. Golob,
18, 12
H. Ha, J. Haba, H. Hayashii, Y. Horii, Y. Hoshi, W.-S. Hou, H. J. Hyun, T. Iijima, K. Inami, R. Itoh, M. Iwabuchi, Y. Iwasaki, N. J. Joshi, T. Julius, N. Katayama, T. Kawasaki, C. Kiesling, H. J. Kim, H. O. Kim, M. J. Kim, S. K. Kim, B. R. Ko, S. Korpar,
19, 12
P. Kriˇzan,
18, 12
P. Krokovny, T. Kuhr, A. Kuzmin,
1, 30
Y.-J. Kwon, S.-H. Kyeong, J. S. Lange, S.-H. Lee, J. Li, A. Limosani, C. Liu, D. Liventsev, R. Louvot, A. Matyja, S. McOnie, H. Miyata, Y. Miyazaki, R. Mizuk, G. B. Mohanty, T. Mori, M. Nakao, Z. Natkaniec, S. Nishida, O. Nitoh, T. Ohshima, S. Okuno, S. L. Olsen,
34, 6
G. Pakhlova, C. W. Park, H. Park, H. K. Park, R. Pestotnik, L. E. Piilonen, M. Prim, M. R¨ohrken, S. Ryu, Y. Sakai, O. Schneider, K. Senyo, M. E. Sevior, M. Shapkin, V. Shebalin,
1, 30
C. P. Shen, J.-G. Shiu, B. Shwartz,
1, 30
F. Simon,
20, 38
P. Smerkol, T. Sumiyoshi, M. Tanaka, N. Taniguchi, G. N. Taylor, Y. Teramoto, K. Trabelsi, T. Tsuboyama, S. Uehara, Y. Unno, S. Uno, G. Varner, K. Vervink, A. Vinokurova,
1, 30
C. H. Wang, M.-Z. Wang, P. Wang, M. Watanabe, Y. Watanabe, E. Won, B. D. Yabsley, Y. Yamashita, C. C. Zhang, Z. P. Zhang, V. Zhulanov,
1, 30
T. Zivko, and A. Zupanc (The Belle Collaboration) Budker Institute of Nuclear Physics, Novosibirsk Faculty of Mathematics and Physics, Charles University, Prague University of Cincinnati, Cincinnati, Ohio 45221 Justus-Liebig-Universit¨at Gießen, Gießen Hanyang University, Seoul University of Hawaii, Honolulu, Hawaii 96822 High Energy Accelerator Research Organization (KEK), Tsukuba Institute of High Energy Physics, Chinese Academy of Sciences, Beijing Institute of High Energy Physics, Vienna Institute of High Energy Physics, Protvino Institute for Theoretical and Experimental Physics, Moscow J. Stefan Institute, Ljubljana Kanagawa University, Yokohama Institut f¨ur Experimentelle Kernphysik, Karlsruhe Institut f¨ur Technologie, Karlsruhe Korea University, Seoul Kyungpook National University, Taegu ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana University of Maribor, Maribor Max-Planck-Institut f¨ur Physik, M¨unchen University of Melbourne, School of Physics, Victoria 3010 Nagoya University, Nagoya Nara Women’s University, Nara National Central University, Chung-li National United University, Miao Li Department of Physics, National Taiwan University, Taipei H. Niewodniczanski Institute of Nuclear Physics, Krakow Nippon Dental University, Niigata Niigata University, Niigata Novosibirsk State University, Novosibirsk Osaka City University, Osaka Panjab University, Chandigarh University of Science and Technology of China, Hefei Seoul National University, Seoul Sungkyunkwan University, Suwon School of Physics, University of Sydney, NSW 2006 Tata Institute of Fundamental Research, Mumbai Excellence Cluster Universe, Technische Universit¨at M¨unchen, Garching Tohoku Gakuin University, Tagajo Tohoku University, Sendai Department of Physics, University of Tokyo, Tokyo a r X i v : . [ h e p - e x ] J un Tokyo Metropolitan University, Tokyo Tokyo University of Agriculture and Technology, Tokyo IPNAS, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 Yonsei University, Seoul (Dated: October 27, 2018)We search for the flavor-changing neutral current decays D → µ + µ − and D → e + e − , and for thelepton-flavor violating decays D → e ± µ ∓ using 660 fb − of data collected with the Belle detectorat the KEKB asymmetric-energy e + e − collider. We find no evidence for any of these decays. Weobtain significantly improved upper limits on the branching fractions: B (cid:0) D → µ + µ − (cid:1) < . × − , B (cid:0) D → e + e − (cid:1) < . × − and B (cid:0) D → e + µ − (cid:1) + B (cid:0) D → µ + e − (cid:1) < . × − at 90%confidence level. PACS numbers: 13.20.Fc,11.30.Hv,12.15.Mm,12.60.-i
The flavor-changing neutral current (FCNC) decays D → e + e − and D → µ + µ − [1] are highly suppressedin the standard model (SM) by the Glashow-Iliopoulos-Maiani mechanism [2]. With the inclusion of long dis-tance contributions the branching fractions can reach val-ues of around 10 − [3]. The SM short distance Feynmandiagrams for the D → µ + µ − decay are shown in Fig. 1.The lepton-flavor violating (LFV) decays D → e ± µ ∓ areforbidden in the SM, but are possible in extensions of theSM with nondegenerate neutrinos and nonzero neutrinomixings and are expected to be of the order of 10 − [3]in these scenarios. All these predictions are orders ofmagnitude below the current experimental sensitivity. FIG. 1. The SM short distance Feynman diagrams for the D → µ + µ − decay. In certain new physics (NP) scenarios, FCNC branch-ing fractions can be enhanced by many orders of mag-nitude. For example, R -parity violating supersymmetrycan increase the branching fractions of D → e + e − and D → µ + µ − up to 10 − and 10 − , respectively [4].The latter prediction is close to the current experimen-tal sensitivity. As another example, so far unobservedleptoquarks were suggested as a possible explanation ofthe small discrepancy between the measured value of the D s meson decay constant and the prediction of latticeQCD [5]. Leptoquarks could also enhance the D → (cid:96) + (cid:96) − branching fraction. In order to explain the mea-sured D + s → µ + ν width by a leptoquark contribution,and comply with other constraints arising from charmmeson decays, B (cid:0) D → µ + µ − (cid:1) should be enhanced to8 × − [6]. The above examples demonstrate the im-portance of FCNC and LFV decays searches in the explo-ration of possible NP contributions. It should be notedthat charm FCNC and LFV decays probe the couplings of the up-quark sector in contrast to B or K meson de-cays.In this paper, we report on a search for the decays D → µ + µ − , D → e + e − and D → e ± µ ∓ using660 fb − of data recorded in e + e − collisions at the center-of-mass (CM) energy of the Υ(4 S ) resonance and 60 MeVbelow by the Belle detector at the KEKB collider.The Belle detector, which is described in detail else-where [7], is a large-solid-angle magnetic spectrometerthat consists of a silicon vertex detector (SVD) [8], a 50-layer central drift chamber (CDC), an array of aerogelthreshold Cherenkov counters (ACC), a barrel-like ar-rangement of time-of-flight scintillation counters (TOF),and an electromagnetic calorimeter composed of CsI(Tl)crystals (ECL) located inside a superconducting solenoidcoil that provides a 1 . K L mesons and to identify muons (KLM). Two innerdetector configurations were used. A beam pipe witha radius of 2 . − , while a 1 . D mesons coming from c -quark production in the continuum e + e − → cc processare considered. The inclusion of D mesons from B de-cays would result in a higher combinatorial background.We normalize the sensitivity of our search to topologi-cally similar D → π + π − decays; this cancels varioussystematic uncertainties. The D → (cid:96) + (cid:96) − ( (cid:96) = e or µ )branching fraction is determined by B (cid:0) D → (cid:96) + (cid:96) − (cid:1) = N (cid:96)(cid:96) f (1)where N (cid:96)(cid:96) is the number of reconstructed D → (cid:96) + (cid:96) − decays and f is defined as f ≡ N ππ (cid:15) ππ (cid:15) (cid:96)(cid:96) B (cid:0) D → π + π − (cid:1) (2)Here B (cid:0) D → π + π − (cid:1) = (1 . ± . × − is the well-measured D → π + π − branching fraction [9], N ππ is thenumber of reconstructed D → π + π − decays, and (cid:15) (cid:96)(cid:96) and (cid:15) ππ are the reconstruction efficiencies for D → (cid:96) + (cid:96) − and D → π + π − decays, respectively.First, a general event selection is performed that is,apart from the particle identification criteria, the samefor all data samples. Later in the analysis, tighter opti-mized criteria specific for each decay mode are used.We use D mesons from the decay D ∗ + → D π + s witha characteristic low momentum pion, since this consider-ably improves the purity of the D → (cid:96) + (cid:96) − and π + π − samples. Each charged track is required to have at leasttwo associated vertex detector hits in each of the twomeasurement coordinates. To select pion and lepton can-didates, we impose standard particle identification crite-ria. Charged pions are identified using d E/ d x measure-ment from the CDC, Cherenkov light yields in the ACC,and timing information from the TOF [10]. Muon identi-fication is based on the matching quality and penetrationdepth of associated hits in the KLM [11]. Electron iden-tification is determined using the ratio of the energy de-posit in the ECL to the momentum measured in the SVDand CDC, the shower shape in the ECL, the matchingbetween the position of charged track trajectory and thecluster position in the ECL, the hit information from theACC and the d E/ d x information in the CDC [12]. Themuon and electron identification efficiencies are around90% with less than 1.5% and 0.3% pion misidentification,respectively, whereas the pion identification efficiency isaround 83%. D daughter tracks are refitted to a com-mon vertex, and the D production vertex is found byconstraining the D trajectory and the π s track to origi-nate from the e + e − interaction region; confidence levelsexceeding 10 − are required for both fits. A D ∗ + mo-mentum greater than 2 . /c in the CM frame of thecollisions is required to reject D -mesons produced in B -meson decays and to suppress combinatorial background.Candidate D mesons are selected using two kinematicobservables: the invariant mass of the D decay prod-ucts, M , and the energy released in the D ∗ + decay, q = ( M D ∗ + − M − m π ) c , where M D ∗ + is the invari-ant mass of the D π s combination and m π is the π + mass [9]. We require 1 .
81 GeV /c < M < .
91 GeV /c and q <
20 MeV.According to Monte Carlo (MC) simulation basedon
EvtGen [13] and
GEANT3 [14], the backgroundin D → (cid:96) + (cid:96) − decays originates predominantly fromsemileptonic B decays (80%) and from D decays (10%).The background events can be grouped into two cate-gories based on their M distribution: (1) a smooth com-binatorial background, and (2) a peaking backgroundfrom the misidentification of D → π + π − decays. Thedecay D → π + π − contributes to the background whenboth pions are misidentified as muons or as a muon-electron combination. MC studies showed that misidenti-fication of a K meson as a lepton does not produce a peakinside the mass region 1 .
81 GeV /c < M < .
91 GeV /c .The signal efficiencies (cid:15) (cid:96)(cid:96) and (cid:15) ππ are evaluated usingsignal MC simulation, which is also based on EvtGen and
GEANT3 , but includes final-state radiative effects(FSR) simulated by
PHOTOS [15]. Since we find dif-ferences between the widths of the D → π + π − signal in the data and MC simulation, we perform fits to the M and q distributions to obtain scaling factors for thesignal widths, and then tune the shapes in the MC sim-ulation by correcting M and q for each MC signal eventby M (cid:48) = m + ( M − m ) f m and q (cid:48) = q + ( q − q ) f q .Here, m and q denote the nominal D mass and thenominal energy released in the D ∗ + decay, respectively,and f m = 1 .
17 and f q = 1 .
28 are the corresponding scal-ing factors. Another difference, a small shift of − . E miss ) between data and MC simulation is ob-served; we correct the MC distribution by subtracting,for every signal event, the above value from its E miss .We construct E miss from the difference between the beamenergy and the sum of the energies of all four vectors ofphotons and charged tracks, which are assumed to be pi-ons. The constants derived from D → π + π − are used tocorrect D → (cid:96) + (cid:96) − MC events. The uncertainties of thetuning procedure are included in the systematic error.In order to avoid biases, a blind analysis technique hasbeen adopted. All events inside the D signal region of | ∆ M | <
20 MeV /c and | ∆ q | < D → (cid:96) + (cid:96) − decays are not expected to be observed atthe current sensitivity, we optimize the selection criteriato obtain the best upper limits; we maximize the figureof merit, F = (cid:15) (cid:96)(cid:96) /N UL , where (cid:15) (cid:96)(cid:96) is the efficiency fordetecting D → (cid:96) + (cid:96) − decays obtained from the tunedsignal MC simulation and N UL is the Poisson average ofFeldman-Cousins 90% confidence level upper limits onthe number of observed signal events that would be ob-tained with the expected background and no signal [16].The average upper limits N UL are calculated from thenumber of generic MC background events, surviving theselection criteria and scaled to the data size. The samplecorresponds to 6-times the statistics of the data.For the optimization we select the following variables:signal region size (∆ M, ∆ q ), E miss , and minimal lep-ton identification probabilities. The quantities ∆ M and∆ q are measured relative to the nominal D mass andnominal energy released in the D ∗ + decay, respectively.The signal region in M is allowed to be asymmetricwith respect to the nominal D mass; for the µµ de-cay mode this provides some suppression of misidenti-fied D → π + π − decays, since their invariant mass dis-tribution peaks about 2 standard deviations below the D mass; for the ee and eµ modes an asymmetric re-quirement accounts for the low mass tail due to electronbremsstrahlung. The requirement on the maximal al-lowed missing energy in the event is chosen to suppressbackground from semileptonic B decays; these eventshave larger missing energy due to undetected neutrinos.We found a broad maximum in F for the lepton iden-tification probability, hence we repeated the procedureat fixed lepton identification criteria, optimizing only thesize of the signal region and the maximal allowed miss-ing energy in an event. The results are summarized inTable I. TABLE I. Optimal selection criteria. The requirements on M are asymmetric and are given as lower and upper boundson ∆ M .Mode ∆ M ∆ q E miss [ MeV /c ] [ MeV ] [ GeV ] µ + µ − ( − , ± e + e − ( − , ± e ± µ ∓ ( − , ± To estimate the number of combinatorial backgroundevents in the signal region, the sideband region | ∆ q | > D → π + π − decays. The com-parison of data and MC simulation shows good agree-ment in the combinatorial background distribution in thisregion. The distribution is parametrized as f ( M, q ) = A (1 − BM ) √ q , where the parameters A and B are deter-mined from a fit to the generic MC sample. The numberof combinatorial background events in the signal regionis calculated as N combbkg = p × N side , where N side is thenumber of events found in the sideband region and p isthe expected ratio of events in the signal and sidebandregion determined by integration of f ( M, q ).The peaking background in the signal region due tomisidentification of D → π + π − decays is estimated fromthe reconstructed D → π + π − decays found in data byreplacing the pion mass with the lepton mass and byweighting each event by w = u ( p , cos θ ) u ( p , cos θ ) v ( p , cos θ ) v ( p , cos θ ) (3)where p , and θ , are the momenta and polar an-gles of the outgoing pions and where u and v are thepion-lepton misidentification probability and pion iden-tification efficiency, respectively. The misidentificationprobabilities and efficiencies are measured in data using D ∗ + → D π + s , D → K − π + decays, binned in particlemomentum p and cosine of polar angle.The estimates for the number of background eventsin the signal region are summarized in Table II. Themisidentification of D → π + π − contributes significantlyonly to the D → µ + µ − decay channel (1 . D → µ + µ − ,zero candidates in the D → e + e − , and three candidatesin the D → e ± µ ∓ decay mode; the yields are consistentwith the estimated background.A binned maximum likelihood fit is used to deter-mine the yield of D → π + π − candidates for the nor- e v e n t s p er M e V / c a) µµ CombinatorialD !" + " - Signal at 90% C.L. b) eec) e µ M [ GeV/c ] FIG. 2. The dilepton invariant mass distributions for (a) D → µ + µ − , (b) D → e + e − , and (c) D → e ± µ ∓ . Thedashed vertical lines indicate the optimized signal window.Superimposed on the data (open histograms) are the esti-mated distribution for combinatorial background (filled his-togram), the misidentification of D → π + π − (cross-hatchedhistogram), and the signal if the branching fractions wereequal to the 90% confidence level upper limit (single hatchedhistogram). malization. We fit the invariant mass distribution usingthe same kinematic selections as for individual leptonicmodes, except for the criteria on M . The fit functionis the sum of two Gaussian distributions with the samemean and an FSR tail for the signal, and a first-orderpolynomial for the background. The shape of the FSRtail and its relative normalization are taken from the cor-responding signal MC simulation. The number of recon-structed D mesons in the π + π − mode is found to be51 . × , 44 . × , and 46 . × , using selectioncriteria for µµ , ee , and eµ modes, respectively. The in-variant mass distribution of D → π + π − using the µµ se-lection criteria with the fit curve superimposed is shownin Fig. 3. The relative uncertainties on N ππ are around0.5%.The signal efficiencies are determined from the tunedsignal MC simulation. In addition, event weighting is ap-plied to compensate for small differences in lepton andpion identification efficiencies between data and MC sim-ulation. The correction factors for lepton identificationwere obtained using γγ → (cid:96) + (cid:96) − and B → XJ/ψ ( → (cid:96) + (cid:96) − ) decays. The signal efficiencies are found to bebetween 5% and 7% for (cid:96) + (cid:96) − decays and about 11% for M [ GeV/c ] e v e n t s p er M e V / c FIG. 3. The invariant mass distribution of D → π + π − withthe fit superimposed using µµ selection criteria.TABLE II. Summary of the number of expected backgroundevents ( N bkg ), number of observed events ( N ) in the sig-nal region, the reconstruction efficiencies ( (cid:15) (cid:96)(cid:96) and (cid:15) ππ ) of the D → (cid:96) + (cid:96) − and D → π + π − decays, the factors f and thebranching fraction upper limits at the 90% confidence level. D → µ + µ − D → e + e − D → e ± µ ∓ N bkg . ± . . ± . . ± . N (cid:15) (cid:96)(cid:96) [%] 7 . ± .
34 5 . ± .
32 6 . ± . (cid:15) ππ [%] 12 . ± .
10 10 . ± .
09 11 . ± . f [10 − ] 4 . ± . . ± . . ± . − ] 1.4 0.79 2.6 π + π − decays. The uncertainties in (cid:15) (cid:96)(cid:96) are estimated tobe 0.3% and include contributions from MC statistics(0.2%), lepton identification efficiency corrections (0.2%),and MC tuning (0.1%). The uncertainty in (cid:15) ππ is smaller(0.1%), because of a larger MC sample, better knownpion efficiency corrections and a negligible contributionfrom MC tuning, since a wider range in M is used.From the number of reconstructed D → π + π − de-cays, from the efficiency ratio, and from the known D → π + π − branching fraction the factors f are calcu-lated with Eq. 2. The relative uncertainties are around5% (see Table II) and include the errors on N ππ , (cid:15) (cid:96)(cid:96) , (cid:15) ππ and the D → π + π − branching fraction, summed inquadrature.Finally, the branching fraction upper limits (UL) arecalculated using the program pole.f [17], which extendsthe Feldman-Cousins method [16] by the inclusion of sys-tematic uncertainties. We find that the inclusion of sys-tematic uncertainties produces nearly the same result asthe standard Feldman-Cousins method. The results aresummarized in Table II. Note that B (cid:0) D → e ± µ ∓ (cid:1) de- notes the sum B (cid:0) D → e + µ − (cid:1) + B (cid:0) D → µ + e − (cid:1) .In summary, we have searched for the FCNC decays D → µ + µ − and D → e + e − , and the LFV decays D → e ± µ ∓ using the Belle detector and have foundno evidence of these decays. The upper limits on thebranching fractions at the 90% confidence level are B (cid:0) D → µ + µ − (cid:1) < . × − , B (cid:0) D → e + e − (cid:1) < . × − , B (cid:0) D → e ± µ ∓ (cid:1) < . × − . Previously, the best upper limits on these decays werepublished by the BaBar Collaboration [18] using 122 fb − of data. Our results improve these limits by a factor of 9for D → µ + µ − decay, by a factor of 15 for D → e + e − decay, and by a factor of 3 for D → e ± µ ∓ decay. Re-cently, the CDF collaboration reported a preliminaryresult on the UL for the D → µ + µ − branching frac-tion [19]; our result is lower by a factor of 3 and canfurther constrain the size of certain R -parity violatingcouplings. It also strongly disfavors a leptoquark con-tribution [6] as the explanation for the anomaly in themeasured D + s → µ + ν width [20]. ACKNOWLEDGMENTS
We thank the KEKB group for the excellent operationof the accelerator, the KEK cryogenics group for the ef-ficient operation of the solenoid, and the KEK computergroup and the National Institute of Informatics for valu-able computing and SINET3 network support. We ac-knowledge support from the Ministry of Education, Cul-ture, Sports, Science, and Technology (MEXT) of Japan,the Japan Society for the Promotion of Science (JSPS),and the Tau-Lepton Physics Research Center of NagoyaUniversity; the Australian Research Council and theAustralian Department of Industry, Innovation, Scienceand Research; the National Natural Science Foundationof China under Contracts No. 10575109, No. 10775142,No. 10875115, and No. 10825524; the Ministry of Ed-ucation, Youth and Sports of the Czech Republic un-der Contracts No. LA10033, and No. MSM0021620859;the Department of Science and Technology of India; theBK21 and WCU program of the Ministry of Education,Science and Technology, National Research Foundationof Korea, and NSDC of the Korea Institute of Scienceand Technology Information; the Polish Ministry of Sci-ence and Higher Education; the Ministry of Educationand Science of the Russian Federation and the RussianFederal Agency for Atomic Energy; the Slovenian Re-search Agency; the Swiss National Science Foundation;the National Science Council and the Ministry of Edu-cation of Taiwan; and the U.S. Department of Energy.This work is supported by a Grant-in-Aid from MEXTfor Science Research in a Priority Area (“New Develop-ment of Flavor Physics”), and from JSPS for CreativeScientific Research (“Evolution of Tau-lepton Physics”). [1] Throughout this paper charge-conjugate modes are in-cluded.[2] S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev.D, , 1285 (1970).[3] G. Burdman, E. Golowich, J. Hewett, and S. Pakvasa,Phys. Rev. D, , 014009 (2002).[4] E. Golowich, J. Hewett, S. Pakvasa, and A. A. Petrov,Phys. Rev. D, , 114030 (2009).[5] B. A. Dobrescu and A. S. Kronfeld, Phys. Rev. Lett., , 241802 (2008).[6] I. Dorsner, S. Fajfer, J. F. Kamenik, and N. Kosnik,Phys. Lett., B682 , 67 (2009).[7] A. Abashian et al. , Nucl. Instrum. Methods A, , 117(2002).[8] Z. Natkaniec et al. , Nucl. Instrum. Methods A, , 1(2006).[9] C. Amsler et al. (Particle Data Group), Phys. Lett. B, , 1 (2008). [10] E. Nakano, Nucl. Instrum. Methods A, , 402 (2002).[11] A. Abashian et al. , Nucl. Instrum. Methods A, , 69(2002).[12] K. Hanagaki, H. Kakuno, H. Ikeda, T. Iijima, andT. Tsukamoto, Nucl. Instrum. Methods A A, , 490(2002).[13] D. J. Lange, Nucl. Instrum. Methods A, , 152 (2001).[14] R. Brun, F. Bruyant, M. Maire, A. C. McPherson, andP. Zanarini, (1987), C ERN-DD/EE/84-1.[15] E. Barberio and Z. Was, Comput. Phys. Commun., ,291 (1994).[16] G. J. Feldman and R. D. Cousins, Phys. Rev. D, , 3873(1998).[17] J. Conrad, O. Botner, A. Hallgren, and C. P´erez de losHeros, Phys. Rev. D, , 012002 (2003).[18] B. Aubert et al. , Phys. Rev. Lett.,93