Pseudovector Meson with Strangeness and Closed Charm

RAPID COMMUNICATIONS

PHYSICAL REVIEW D 73, 111503(R) (2006) Pseudovector meson with strangeness and closed charm

Ting-Wai Chiu1 and Tung-Han Hsieh2

(TWQCD Collaboration)

1Department of Physics, National Taiwan University, Taipei, 10617, Taiwan 2Physics Section, Commission of General Education, National United University, Miao-Li, 36003, Taiwan (Received 12 April 2006; published 9 June 2006) We investigate the mass spectrum of 1 exotic mesons with quark content csc q=cqc s, using molecular and diquark-antidiquark operators, in quenched lattice QCD with exact chiral symmetry. For T the molecular operator fqicc5scisq5cg and the diquark-antidiquark operator fq Cic T T T T sC5c q C5csCi c g, both detect a 1 resonance with mass around 4010 50 MeV in the limit mq ! mu;d.

DOI: 10.1103/PhysRevD.73.111503 PACS numbers: 11.15.Ha, 11.30.Rd, 12.38.Gc, 14.40.Lb

I. INTRODUCTION cqc s and csc q. However, in general, whether any combination of two quarks and two antiquarks can emerge Since the discovery of D 2317 [1] by BABAR in April s as a hadronic state relies on the nonperturbative dynamics 2003, a series of new heavy mesons1 with open charm and between these four quarks. closed charm have been observed by Belle, CDF, CLEO, In this paper, we investigate the masses of the lowest- BABAR, and BES. Among these new heavy mesons, the lying states (with JP 1) of molecular and diquark- most intriguing ones are the charmoniumlike states, antidiquark operators with quark content csc q or X3872 [3], Y3940 [4], Y4260 [5], Z3930 [6], and cqc s. For two lattice volumes 243 48 and 203 40, X3940 [7]. Evidently, one can hardly interpret all of them each of 100 gauge conﬁgurations generated with single- as orbital and/or radial excitations in the charmonium plaquette action at 6:1, we compute point-to-point spectrum. Thus it is likely that some of them are exotic quark propagators for 30 quark masses in the range 0:03 (non-qq) mesons (e.g., molecule, diquark-antidiquark, and hybrid mesons). Theoretically, the central question is mqa 0:80, and measure the time-correlation functions of these exotic meson operators. The inverse lattice spacing whether the spectrum of QCD possesses these resonances, 1 with the correct quantum numbers, masses, and decay a is determined with the experimental input of f. The widths. strange quark bare mass msa 0:08 and the charm quark Recently, we have investigated the mass spectrum of bare mass mca 0:80 are ﬁxed such that the masses of the closed-charm exotic mesons with JPC 1 [8], and 1 corresponding vector mesons are in good agreement with [9], in lattice QCD with exact chiral symmetry [10–14]. By 1020 and J= 3097, respectively [15,16]. constructing molecular and diquark-antidiquark operators with quark content cqc q, we measured their time- II. THE MOLECULAR OPERATOR correlation functions, and extracted the mass spectrum of In general, one can construct many different molecular the hadronic states overlapping with these exotic meson operators with quark content cqc s or csc q such that operators. Our results suggest that Y4260 and X3872 the lowest-lying state of each operator has JP PC 1 . are in the spectrum of QCD, with J 1 and 1 , However, not every one of them has good overlap with respectively, and both with quark content cuc u . Note the lowest-lying hadronic state having the same quark that we have been working in the isospin limit (with mu content and quantum numbers. In the following we only md), thus our results [8,9] also imply the existence of present one of the good molecular operators.2 Explicitly, exotic mesons with quark content cdc d, even though we cannot determine their mass differences from those 1 M p fqicc5scisq5cg: (1) with cuc u . Moreover, we also observe heavier exotic 2 mesons with quark contents csc s and ccc c, for JPC We measure the time-correlation function, 1 [9], and 1 [8]. X Now, if the spectrum of QCD does possess exotic me- Ct hMx;~ tMy0~; 0i sons with quark content cqc q, then it is likely that there x~ are also exotic mesons with other quark contents, e.g.,

2 The operators we have investigated include qicc5s, 1 For recent reviews of these new heavy mesons, see, for cisq5c, and qicc5scisq5c, and they all example, Ref. [2], and references therein. yield compatible masses for the lowest-lying meson.

TING-WAI CHIU AND TUNG-HAN HSIEH PHYSICAL REVIEW D 73, 111503(R) (2006) where the cc annihilation diagrams are neglected such that 1 : (q c)(c s)-(c s)(q c) Ct does not overlap with any conventional meson states. 5.37 i 5 i 5 Also, Ct is averaged over Ci (with i) for i 1,2,3, 3 where, in each case, the ‘‘forward-propagator’’ Cit and 24 x 48, 6.1 ‘‘backward-propagator’’ CiT t are averaged to in- 00 configs. crease the statistics. The same strategy is applied to all 4.92 time-correlation functions in this paper. Then the average of Ct over all gauge conﬁgurations is ﬁtted to the usual formula 4.47 (GeV)

Z m ema t emaTt 2 ma 4.03 to extract the mass ma of the lowest-lying state and its spectral weight 3.58 Z W : 2ma TWQCD/l24t48/g100/DVDS5A 0.02 0.04 0.06 0.08 0.10 Theoretically, if this state is a genuine resonance, then its m a mass ma and spectral weight W should be almost constant q for any lattices with the same lattice spacing. On the other FIG. 2 (color online). The mass of the lowest-lying state of M hand, if it is a 2-particle scattering state, then its mass ma is 3 versus the quark mass mqa, on the 24 48 lattice at 6:1. sensitive to the lattice volume, and its spectral weight is The solid line is the linear ﬁt. inversely proportional to the spatial volume for lattices with the same lattice spacing. In the following, we shall use the ratio of the spectral weights on two spatial volumes discriminate whether any hadronic state under investiga- 203 and 243 with the same lattice spacing ( 6:1)to tion is a resonance or not. In Fig. 1, the ratio (R W20=W24) of spectral weights of 3.0 the lowest-lying state extracted from the time-correlation _ _ _ _ function of M on the 203 40 and 243 48 lattices is 1 (q c)(c s)-(c s)(q c) i 5 i 5 plotted versus the quark mass mqa 20:03; 0:80. [Here q 2.5 (q) is always taken to be different from c (c) and s (s), even in the limit mq ! mc or ms]. Evidently, R ’ 1:0 for the entire range of quark masses, which implies that there exist 2.0 JP 1 resonances, with quark contents csc u and csc d, respectively. 24 In Fig. 2, the mass of the lowest-lying state extracted / W 1.5 from the molecular operator M is plotted versus m a.In 20 q 1 the limit mq ! mu;d ’ 0:002 65a (corresponding to m 135 MeV), it gives m 400734 MeV. R = W 1.0

III. THE DIQUARK-ANTIDIQUARK OPERATOR 0.5 In general, one can construct many different diquark- antiquark operators with quark content cqc s or csc q such that the lowest-lying state of each operator has JP 0.0 TWQCD/DVDS5A 1. However, not every one of them has good overlap with 0.01 0.1 1 the lowest-lying hadronic state having the same quark m a q content and quantum numbers. In the following we only present one of the good diquark-antidiquark operators. FIG. 1 (color online). The ratio of spectral weights of the Explicitly, lowest-lying state of the molecular operator M, for 203 40 3 and 24 48 lattices at 6:1. The upper-horizontal line, R 1 T T 3 D4xp fq CicxasC5c xa 24=20 1:728, is the signature of the 2-particle scattering 2 state, while the lower-horizontal line, R 1:0, is the signature of T T T a resonance. sCi c xaq C5cxag (2)

111503-2 RAPID COMMUNICATIONS

PSEUDOVECTOR MESON WITH STRANGENESS AND ... PHYSICAL REVIEW D 73, 111503(R) (2006) 3.0 where C is the charge conjugation operator satisfying T T T T T 1 T T CC and C5 C5. Here the diquark 1 [q C ic][sC 5c ][sCi c ][q C 5c] qT Q 2.5 operator xa for any Dirac matrix is deﬁned as T q Qxa abcqxbQxc Qxbqxc (3)

2.0 where x, fa; b; cg, and f; g denote the lattice site, color, and Dirac indices, respectively, and abc is the completely 24 antisymmetric tensor. Thus the diquark (3) transforms like / W 1.5 a color antitriplet. For C , it transforms like JP 20 5 0 , while for Ci (i 1; 2; 3), it transforms like 1 . In Fig. 3, the ratio (R W20=W24) of spectral weights of R = W 1.0 the lowest-lying state extracted from the time-correlation 3 3 function of D4 on the 20 40 and 24 48 lattices m a 2 : ; : 0.5 is plotted versus the quark mass q 0 03 0 8 . Evidently, R ’ 1:0 for the entire range of quark masses, which implies that there exist JP 1 resonances, with 0.0 TWQCD/DDS4v5A quark contents cdc s and cuc s, respectively. 0.01 0.1 1 In Fig. 4, the mass of the lowest-lying state of the diquark-antidiquark operator D4 is plotted versus mqa.In mqa the limit mq ! mu;d,itgivesm 401525 MeV. FIG. 3 (color online). The ratio of spectral weights of the lowest-lying state of the diquark-antidiquark operator D4, for IV. SUMMARY AND DISCUSSIONS 203 40 and 243 48 lattices at 6:1. The upper-horizontal line, R 24=203 1:728, is the signature of the 2-particle In this paper, we have investigated the mass spectra of scattering state, while the lower-horizontal line, R 1:0, is the the molecular operator M and the diquark-antidiquark signature of a resonance. operator D4, in quenched lattice QCD with exact chiral symmetry. Our results for M and D4 are summarized in Table I, where, in each case, the ﬁrst error is statistical, and the second one is our crude estimate of combined systematic uncertainty.

T T T T T Evidently, both the molecular operator M and the 1 [q C ic][sC 5c ][sCi c ][q C 5c] 5.37 diquark-antidiquark operator D4 detect a 1 resonance around 4010 50 MeV in the limit mq ! mu;d. Since its 243 x 48, 6.1 mass is just slightly above Y3940, high energy experi- 00 configs. ments should be able to see whether such a resonance, say, 4.92 Xs, exists in some decay channels, e.g., Xs ! KJ=,in the near future. Note that, in our calculations, we have omitted all inter- 4.47 nal quark loops generated by vacuum ﬂuctuations (i.e., the

(GeV) quenched approximation). However, we expect that the

m quenched approximation only results in a few percent systematic error in the mass spectrum, especially for 4.03 charmed mesons [16]. If it turns out that high energy experiments do not ﬁnd any charmoniumlike resonances

3.58 TABLE I. Masses of the lowest-lying states (JP 1) of the

TWQCD/l24t48/g100/DDS4V5A molecular operator M and the diquark-antidiquark operator D4. The last column R/S denotes the resonance (R) or scattering (S) 0.02 0.04 0.06 0.08 0.10 state. m a q Operator Mass (MeV) R/S FIG. 4 (color online). The mass of the lowest-lying state of p1 u cc sc su c 4007(34)(31) R 2 i 5 i 5 the diquark-antidiquark operator D4 versus the quark mass p1 fuTC csC c T 4015(25)(27) R 3 2 i 5 mqa, on the 24 48 lattice at 6:1. The solid line is the T T T u C5csC c g linear ﬁt. i

111503-3 RAPID COMMUNICATIONS

TING-WAI CHIU AND TUNG-HAN HSIEH PHYSICAL REVIEW D 73, 111503(R) (2006) around 3960–4060 MeV, in particular, in the KJ= ported in part by the National Science Council, Republic of channel, then it might imply that quenched QCD could China, under the Grant No. NSC94-2112-M002-016 be dramatically different from the unquenched QCD, even (T. W. C.), and Grant No. NSC94-2119-M239-001 for the mass spectrum. (T. H. H.), and by the National Center for High Performance Computation at Hsinchu, and the Computer ACKNOWLEDGMENTS Center at National Taiwan University. We thank Steve Olsen for his question regarding cuc s, which motivated our present study. This work was sup-

[1] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. [9] T. W. Chiu and T. H. Hsieh (TWQCD Collaboration), hep- 90, 242001 (2003). ph/0603207. [2] E. S. Swanson, hep-ph/0601110; T. Barnes, AIP Conf. [10] D. B. Kaplan, Phys. Lett. B 288, 342 (1992); Nucl. Phys. Proc. 814, 735 (2006); C. Quigg, hep-ph/0509332; J. L. B, Proc. Suppl. 30, 597 (1993). Rosner, AIP Conf. Proc. 815, 218 (2006). [11] R. Narayanan and H. Neuberger, Nucl. Phys. B443, 305 [3] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, (1995). 262001 (2003). [12] H. Neuberger, Phys. Lett. B 417, 141 (1998). [4] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 94, [13] P.H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 182002 (2005). (1982). [5] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. [14] T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003); Phys. Lett. 95, 142001 (2005). B 552, 97 (2003); hep-lat/0303008; Nucl. Phys. B, Proc. [6] S. Uehara et al. (Belle Collaboration), Phys. Rev. Lett. 96, Suppl. 129, 135 (2004). 082003 (2006). [15] T. W. Chiu and T. H. Hsieh, Nucl. Phys. A755, 471 [7] K. Abe et al. (Belle Collaboration), hep-ex/0507019. (2005). [8] T. W. Chiu and T. H. Hsieh (TWQCD Collaboration), [16] T. W. Chiu, T. H. Hsieh, J. Y. Lee, P.H. Liu, and H. J. Phys. Rev. D 73, 094510 (2006). Chang, Phys. Lett. B 624, 31 (2005).

111503-4