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Hadron loops effect on mass shifts of the charmed and charmed-strange spectraZhi-Yong Zhou*Department of Physics, Southeast University, Nanjing 211189, Peoples Republic of ChinaZhiguang XiaoInterdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, China(Received 31 May 2011; published 11 August 2011)The hadron loop effect is conjectured to be important in understanding discrepancies between theobserved states in experiments and the theoretical expectations of the nonrelativistic potential model. Wepresent that in an easily operable procedure the hadron loop effect could shift the poles downwards toreduce the differences and provide better descriptions of both the masses and the total widths, at least, ofthe radial quantum number n 1 charmed and charmed-strange states. The 11P1 13P1 mixingphenomena could be naturally explained due to their couplings with common channels. The newlyobserved D states are also addressed, but there are still some problems that remain unclear.DOI: 10.1103/PhysRevD.84.034023 PACS numbers: 12.39.Jh, 13.20.Fc, 13.75.Lb, 11.55.FvI. INTRODUCTIONDiscoveries of more and more charmed or charmed-strange states in experiments attract great interest on thetheoretical side, because some members of them haveunexpected properties. In the Particle Data Group (PDG)table in Ref. [1], six lower charmed states, D0, D20070,D024000, D124200, D124300, D224600, and theirpartners have already been established. Recently, someevidence of three new charmed states, D2550, D2610,and D2760 have been reported by the BABARCollaboration [2], whose features lead to intense discus-sions and theoretical suggestions of the further ex-perimental investigations [38]. There are also ninecharmed-strange states quoted in the PDG table amongwhich some states quantum numbers are undetermined.The mass spectra of these charmed and charmed-strangestates are roughly depicted in the predictions of the non-relativistic potential model in the classic work by Godfreyand Isgur (referred to as GI in the following) [9]. However,the observed masses are generally lower than the predictedones. For example, the biggest discrepancies happening inboth spectra are the 13P0 states. The D02318 is about80 MeV lower than the expectation, while theDs02317 isabout 160 MeV lower. There are a body of theoreticalefforts at solving this problem usually by changing therepresentation of the potential (see Refs. [1014]and references therein). Lattice calculations have alsobeen made to explain the experiments [15,16]. However,the present systematic uncertainty of the Lattice calcula-tions does not allow determinations of the charmed mesonswith a precision less than several hundred MeV.Another expectation to shed light on this problem is totake the coupled channel effects (also called hadron loopeffects) into account, which play an important role in under-standing the enigmatic light scalar spectrum and theirdecays [17,18]. In the light scalar spectrum, the strongattraction of opened or virtual channels may dramaticallyshift the poles of the bare states to different Riemann sheetsattached to the physical region, and the poles on unphysicalRiemann sheets appear as peaks or just humps of themodulus of scattering amplitudes in the experimentaldata. The mass shifts induced by the intermediate hadronloops have also been shown to present a better descriptionof the charmonium states [1922]. The coupled channeleffects have already brought some insights into the natureof the charmed-strange DsJ2317 and some other states[2327]. However, although this effect could explain someof the observed charmed or charmed-strange states, there isstill a concern that this effect may also exist in those statespreviously consistent with the theoretical expectation [28].In this paper, we will address this point by considering themass shifts, induced by hadron loops, of all the firmlyestablished charmed and charmed-strange states. Here wepropose an easily operable way, in which we use theimaginary part of the self-energy function calculated fromthe quark pair creation (QPC) model [2931] in the disper-sion relation to obtain the analytically continued inversepropagator and extract the physical mass and width parame-ters, and then apply it to the charmed and charmed-strangespectra to interpret their masses and total decay widths in aconsistent way. It is found that the results of their massesand total widths are consistent with the experimental val-ues, at least for the nonradially excited states. This picturegives a natural explanation to the 11P1 13P1 mixing bythe coupling with the same channels instead of using aphenomenological mixing angle. This scheme has somesimilarities to the methods used by Heikkila et al. [19]and Pennington et al. [20] in their study of the charmoniumand bottomonium states, but there are significant differ-ences from them, as discussed in the text.*zhouzhy@seu.edu.cnxiaozg@ustc.edu.cnPHYSICAL REVIEW D 84, 034023 (2011)1550-7998=2011=84(3)=034023(8) 034023-1 2011 American Physical Societyhttp://dx.doi.org/10.1103/PhysRevD.84.034023The paper is organized as follows: In Sec. II, the mainscheme and how to model the decay channels are brieflyintroduced. The mixing mechanism is introduced inSec. III. Numerical procedures and results are discussedin Sec. IV. Section V is devoted to our conclusions andfurther discussions.II. THE SCHEMEWe start by considering a simple model at the hadronlevel, in which the inverse meson propagator, P1s,could be represented as [19,20]P1s m20 ss m20 sXnns; (1)where m0 is the mass of the bare q q state and ns is theself-energy function for the n-th decay channel. Here, thesum is over all the opened channels or including nearbyvirtual channels (just virtual). ns is an analyticfunction with only a right-hand cut starting from the n-ththreshold sth;n, and so one can write its real part andimaginary part through a dispersion relationRens 1PZ 1sth;ndzImnzz s ; (2)where PRmeans the principal value integration. The poleof Ps on the unphysical Riemann sheet attached to thephysical region specifies its mass and total width of ameson by its position on the complex s plane, usuallydefined as spole Mp ip=22.One could recover a generalization of the familiar Breit-Wigner representation, usually used in experimental analy-ses, from Eq. (1), asP1s ms2 s imBWtots; (3)where ms2 m20 Res is the running squaredmass and tots Ims=mBW. mBW is determined atthe real axis where ms2 s 0 is fulfilled. The massand width parameters in these two definitions give similarresults when one encounters a narrow resonance, but theydiffer when the resonance is broad or when there areseveral poles interacting with each other.Based on the Cutkosky rule, the imaginary part of theself-energy function is expressed through the couplingsbetween the bare state and the coupled channels. Therelation could be pictorially expressed as Fig. 1.Thus, one key ingredient of this scheme is to model thecoupling vertices in the calculation of the imaginary part ofthe self-energy function. The QPC model [2931], alsoknown as the 3P0 model in the literature, turns out to beapplicable in explaining the Okubo-Zweig-Iizuka (OZI)allowed strong decays of a hadron into two other hadrons,which are expected to be the dominant decay modes of ameson if they are allowed. It is not only because this modelhas proved to be successful but also because it couldprovide analytical expressions for the vertex functions,which are convenient for extracting the shifted poles inour scheme. Furthermore, the vertex functions have expo-nential factors which give a natural ultraviolet suppressionto the dispersion relation and we need not choose one byhand as in Ref. [20].Here, we just make a brief review of the main results ofthe QPC model used in our calculation. (For a morecomplete review, see [3234].) In the QPC model, themeson (with a quark q1 and an antiquark q2) decay occursby producing a quark (q3) and antiquark (q4) pair from thevacuum. In the nonrelativistic limit, the transition operatorcan be represented asT3Xmh1m1mj00iZd3 ~p3d3 ~p43 ~p3 ~p4Ym1~p3 ~p42341m340 !340 by3 ~p3dy4 ~p4; (4)where is a dimensionless model parameter andYm1 ~p plYml p;p is a solid harmonic that gives themomentum-space distribution of the created pair. Herethe spins and relative orbital angular momentum of thecreated quark and antiquark (referred to by subscripts 3and 4, respectively) are combined to give the pair theoverall JPC 0 quantum numbers. 340 u u d dss= ffiffiffi3p and !340 ij, where i and j are the SU(3)-colorindices of the created quark and antiquark. 341m is a tripletof spin.Define the S matrix for the meson decay A ! BC ashBCjSjAi I 2iEf EihBCjTjAi; (5)and thenhBCjTjAi 3 ~Pf ~PiMMJAMJBMJC : (6)The amplitude turns out to beFIG. 1. The imaginary part of the self-energy function.Rdmeans the integration over the phase space.ZHI-YONG ZHOU AND ZHIGUANG XIAO PHYSICAL REVIEW D 84, 034023 (2011)034023-2MMJAMJBMJC ~P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EAEBECp XMLA ;MSA ;MLB ;MSB ;MLC ;MSC ;mhLAMLASAMSA jJAMJAihLBMLBSBMSB jJBMJBihLCMLCSCMSC jJCMJCih1m1mj00ih32SCMSC14SBMSBj12SAMSA341mih32C 14B j12A 340 iIMLA ;mMLB ;MLC ~P: (7)The spatial integral IMLA ;mMLB ;MLC ~P is given byIMLA ;mMLB ;MLC ~P Zd3 ~kc nBLBMLB ~k 41 4~Pc nCLCMLC~k 32 3~Pc nALAMLA ~k ~PYm1 ~k; (8)where we have taken ~P ~PB ~PC andi is the mass of the i-th quark. c nALAMLA ~kA is the relative wave function of thequarks in meson A in the momentum space.The recoupling of the spin matrix element can be written, in terms of the Wigners 9-j symbol, as [32]h32SCMSC14SBMSBj12SAMSA341mi 32SB 12SC 12SA 11=2XS;MShSCMSCSBMSB jSMSihSMSjSAMSA ; 1;mi8>:1=2 1=2 SC1=2 1=2 SBSA 1 S9>=>;: (9)The flavor matrix element ish32C 14B j12A 340 i XI;I3hIC; I3C; IBI3BjIAI3Ai2IB 12IC 12IA 11=28>:I2 I3 ICI1 I4 IBIA 0 IA9>=>;; (10)where IiI1; I2; I3; I4 is the isospin of the quark qi.The imaginary part of the self-energy function in thedispersion relation, Eq. (2), could be expressed asImA!BCs 22JA 1j ~Psjffiffiffisp XMJA ;MJB ;MJCjMMJA ;MJB ;MJC sj2; (11)where jPsj is the three momentum of B and C in theircenter of mass frame. So,jPsjffiffiffisp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis mB mC2s mB mC2p2s: (12)Care must be taken when Eq. (11) is continued to thecomplex s plane. Since what is used in this model is onlythe tree level amplitude, there is no right-hand cut forMMJA ;MJB ;MJC s. Thus, the analytical continuation of theamplitude obeys Ms i Ms i Msi. The physical amplitude with loop contributions shouldhave right-hand cuts, and, in principle, the analytical con-tinuation turns to be Ms i Ms i Mns i by meeting the need of real analyticity.Ms i means the amplitude on the physicalRiemann sheet (the first sheet, in language of the analyticS matrix theory), andMns i means the amplitude onthe unphysical Riemann sheet (the n-th sheet) attachedwith the physical region.With the analytical expression of the imaginary part of thecoupled channel, one will be able to extract the poles or theBreit-Wigner parameters from the propagators by standardprocedures. In principle, all hadronic channels should con-tribute to themesonmass, as considered byHeikkila et al. instudying the charmonium states [19]. Even all the virtualchannels will contribute to the real parts ofs and renor-malize the bare mass. Pennington et al. proposed that aonce-subtracted dispersion relation will suppress contribu-tions of the faraway virtual channels and make the picturesimpler [20]. Since what we consider here is only the massshifts, we could make a once-subtracted dispersion relationat some suitable point s s0. It is reasonably expectedthat the lowest charmed state, D0, as a bound state, has themass defined by the potential model, uninfluenced by theeffect of the hadron loops. Its mass then essentially definesthe mass scale and thus fixes the subtracted point. So, we setthe subtracted point s0 m2D0 or s0 mc mu2 in apractical manner. The inverse of the meson propagator turnsout to beP1s m2pot sXns s0Z 1sth;ndzImnzz s0z s ;(13)wherempot is the bare mass of a certain meson defined in thepotential model.HADRON LOOPS EFFECT ON MASS SHIFTS OF THE . . . PHYSICAL REVIEW D 84, 034023 (2011)034023-3III. MIXING MECHANISMIn this scheme, all the states with the same spin-parityhave interference effects and could mix with each other.For example, the two J 1 states of the P wave areusually regarded as linear combinations of 3P1 and1P1assignments. Here in considering the coupled channeleffect, the mixing mechanism comes from the couplingvia common channels. It is also believed that the 23S1 and13D1 states mix with each other, similar to the interpreta-tion of the charmonium c 3770 state [35].The inverse of the propagator with two bare states mix-ing with each other readsP1s M211s M212sM212s M222s a;bs m2bare;1 s11s 12s21s m2bare;2 s22s !;(14)whereM2a;bs is the mass matrix and mbare;a represents themass parameter of the bare a state. The off-diagonal termsof the self-energy function is represented by the 1PI dia-gram for the two mixed states. The physical states shouldbe determined by the meson propagator matrix after diag-onalizationM2diags s1M2a;bss; (15)where the mixing matrix s satisfies sTs I, i.e.,s is a complex orthogonal matrix since M2a;bs issymmetric. The s matrix turns to be complex whenthe thresholds are open. The physical poles could be ex-tracted, in an equivalent way, by finding the zero points ofthe determinant of the inverse propagator, that is to solvedetP1s 0.IV. NUMERICAL ANALYSESThe bare masses of the related mesons are chosen at thevalues of the GIs work [9]. As for the dimensionlessparameter, , and the effective parameters in the QPCmodel to characterize the harmonic oscillator wave func-tions, we choose the same values as determined from thepotential in GIs work for self-consistency. The constituentquark masses are Mc 1:628 GeV, Ms 0:419 GeV,and Mu 0:22 GeV. 6:9 and the values of s arefrom Refs. [36,37]. The physical masses concerned in thefinal states are the average values in the PDG table. Therelative wave functions between the quarks in the mock-meson states are simple harmonic oscillator wave func-tions usually used in the QPC model calculation, whichbrings some uncertainties into the calculation, as discussedlater.There are some further explanations for the effective parameters of c u states. Godfrey and Isgur have onlypresented their results of n 1 S and P-wave charmedstates but not provided those of the D-wave and radialexcited states which are needed in our discussion of thenewly observed charmed states. We can only estimate thevalues by assuming that their ratios between the valuesof the c u states are similar to the ratios in the results fromthe other research groups. For example, we find the ratiosof the values between different charmed states inRef. [38] and Ref. [6] are almost same. Thus, the valuesof c u states used in our calculation, except those listed inRef. [37], are 13Dj 0:44 0:02 GeV, 21S0 0:47 0:02 GeV, and 23S1 0:44 0:02 GeV,respectively.The opened or nearby virtual two-body channels takeninto account in our calculations are all listed in Table I.Those channels with the meson are not considered,because meson is not regarded as a conventional q qstate in the potential model [9]. It will quickly decay intotwo pions and the three-body decays usually present aminor contribution.The masses and widths are simultaneously determined,as listed in Table II, for the charmed states, where wepresent the pole positions as well as the Breit-Wignerparameters for comparison. Remarkable improvements ofTABLE I. The channels of the charmed states considered in this paper.Mode Channel 113S1 013P0 111P1 113P1 213P2 021S0 123S1 113D1 313D30 0 D h h h h h hD h h hDsK h h h1 0 D h h h h h h hD h h hDsK h h0 1 D h hD! h h0 0 D0 h1T 0 D12420 h h h1S 0 D12430 h h h2 0 D22460 h h hZHI-YONG ZHOU AND ZHIGUANG XIAO PHYSICAL REVIEW D 84, 034023 (2011)034023-4the shifted masses of the already established charmedmesons could be found instantly. Furthermore, the totalwidths specified by twice of the imaginary part of the polepositions are also consistent in good quality with the valuesin the PDG table.The D11S0 state is a long-lived particle in the stronginteraction and there is no opened strong channel, so weregard it to be well described as a bound state in thepotential model and choose its squared mass as the sub-traction point of the dispersion relations.When we calculate the mass shift of the D13S1, theD00 and D threshold are both taken into account,because the D threshold is at about 2009 MeV, just2 MeV higher than the observed D20070. It is a typicaljust virtual channel and in principle it will contribute asignificant mass shift to the bare state. If the coupling to theD threshold is excluded, the pole mass will only beshifted to about 2031 MeV using this set of parameters.The pole of D13P0 is significantly shifted down to2275 MeV, which is 125 MeV down below the potentialmodel prediction. The pole width is about 250 MeV, whichis in accordance with the experimental value within errorbar.The shifted pole mass favors the BABAR and Belleresults [39,40] over the FOCUS result [41]. The D11P1and D13P1 states stay close to each other and they bothhave the same quantum numbers JP 1 and similardecay channels. The unmixed pole positions are atffiffiffiffiffiffiffiffiffiffiffiffis11P1p 2387 i28 MeV and ffiffiffiffiffiffiffiffiffiffiffiffis13P1p 2427i71 MeV, respectively. Both of the masses and the widthsof the unmixed 11P1 state have large differences from theexperimental values. It is the effect of their couplings withcommon channel D that significantly changes their polepositions to one narrower and the other broader. The polesdetermined by the zero points of the inverse propagatormatrix are atffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis11P1p 2410 i7 MeV and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis13P1p 2377 i94 MeV respectively, which characterize thetwo observed states quite well. Their related Breit-Wigner parameters agree with the experimental valuesbetter. It is interesting to mention that in this scheme themixing matrix, as a function of s, are complex-valued andit is not easy to find its relation with the mixing anglecommonly used in phenomenological analyses. Here, theapproximate values of the mixing matrix at about2410 MeV isjs2:41 GeV2 0:57 0:26i 0:87 0:17i0:87 0:17i 0:57 0:26i ; (16)whose imaginary parts are small compared with the realparts. If one just neglects the imaginary parts and definesthe mixing matrix as usual, one obtains sin coscos sin 0:57 0:870:87 0:57 ; (17)which means the mixing angle 29 35. It is inagreement with the value 35:3 obtained by consider-ing the heavy quark symmetry [42].As for the other charmed states which are not quoted inthe PDG table, the evidences of several new charmedstates,D2550,D2610, andD2760, have been recentlyreported by the BABAR collaboration [2]. Several groupshave presented their tentative interpretations of the natureof these states [37], and we make a brief summary of theirconclusions here. D2550 is assigned to the 21S0, but itslarge decay width could not be explained by the QPCmodel, the chiral quark model, and the relativistic quarkmodel, so further experimental explorations were sug-gested. Although the potential model has predictedD23S1 to be located at about 2640 MeV, the QPC modeland the chiral quark model also favor D2610 to be amixed state of D23S1 and D13D1 to interpret its largewidth. There are conflicting opinions about the assignmentof D2760 as the heavier mixed state of D23S1 andD13D1, or as D13D3.In our calculation, the mass shift induced by the inter-mediate states also reduces the pole masses of D21S0,down to 2533 MeV, but its pole width is quite narrowcompared with the experimental value, as shown inTABLE II. Compilation of the experimental masses and the total widths (the PDG average values [1]) of the charmed states, theBreit-Wigner parameters, the shifted pole positions and the mass spectrum in the GIs model [9]. The experimental values of 21S0,23S1, and 13D1 are from Ref. [2]. Here we only list the neutral charmed states. The unit is MeV.JPn2s1LJ Expt. mass Expt. width mBW BW ffiffiffiffiffiffiffiffiffispolep M i=2 GI mass011S0 1867 1880113S1 2007 0:16 Table II. However, the mass of D23S1 is shifted toomuch down to about 2523 MeV due to many intermediatechannels opened. Actually, the pole is even shifted downabout 100 MeV below some thresholds, and its pole widthis fairly small as well. It seems to become a quasiboundstate due to its strong coupling with the D1 channel. Ofcourse, there is still some parameter space for the effec-tive parameter and the dimensionless coupling strengthparameter to be tuned to reduce the mass shift tofit the experiment signal, because these parameters havesignificant uncertainties. In our opinion, one possiblereason why the result seems to be inaccurate is theuncertainty of the simple harmonic oscillator functionwe used to estimate the coupling vertices, which mighthave a larger tail than the realistic one in the high sregion, which will contribute to the mass shift throughthe dispersion relation. A more realistic wave functionsolved from the linear potential model could be morereliable to describe the meson property. However, usually,this kind of wave function does not have an analyticalrepresentation and it can not be easily continued into thecomplex s plane in our scheme. On the other hand, it istoo early to get any firm conclusion, since these states stillneed further experimental confirmations. The pole ofD13D1 is shifted down to 2686 i66 MeV as well,whose Breit-Wigner mass is about 2730 MeV which iscloser to the mass of D2760. Unlike the 11P1 13P1case, with this set of parameters, the mixing mechanismdue to the coupling with their common channels does notchange their positions much. The unmixed pole ofD13D1 could also be estimated at about 2735 MeVbut its width is narrow.The discrepancies that happen in the charmed-strangespectrum could be well addressed qualitatively, owing totheir coupling with the opened thresholds and the nearbyvirtual OZI-allowed strong thresholds in Table III, as thepicture proposed by van Beveren and Rupp for explainingthe DsJ2317 state [23]. Some the thresholds are openedas a result of the isospin breaking effects, e.g.,DsJ2317 ! Ds0,Ds0, whose contributions are highlysuppressed by a factor of about mu md=ms mu md=2 1=38, where the masses are the current quarkmasses. The coupling to such thresholds will contributetiny imaginary parts of the self-energy functions, whichhardly shift the mass of the state and only contribute to thedecay widths with an order of KeV. So we completelyneglect these thresholds with isospin breaking effects andthose OZI suppressed. When we choose the value of thedimensionless strength parameter 6:9 for thecharmed-strange spectrum, the mass shifts will be a littlelarger. We change to be around 5.5 and obtained theshifted masses of the charmed-strange S and P-wavestates, as listed in Table IV. One could regard this finetuning procedure as a fit, because we only want to give aqualitative description for the charmed-strange spectrum.Indeed the parameter in the QPC model, determined byfitting to experimental decay processes, usually has anuncertainty of about 30% [6,38]. Here we only list thepole positions, because they do not differ much with theBreit-Wigner parameters in this case as quasibound statesor narrow states.If the isospin breaking and other weak interaction chan-nels are virtual, Ds2112 and DsJ2317 are the boundstates when the bare 11S0 and 13S1 states are coupled to thejust virtualDK threshold. They show as the poles on thereal axis of the physical Riemann sheet. It is the couplingof the bare states to the lower isospin breaking Ds0thresholds and the other weak thresholds that shift thepoles to the unphysical Riemann sheets when they areopen in reality. When the mixing of the 11P1 and 13P1states is not considered, they are both the bound states at2478 MeV and 2493 MeV, respectively, below the DsKTABLE III. The opened and nearby closed channels of the charmed-strange states consideredin this paper.Mode Channel 113S1 013P0 111P1 113P1 213P20 0 DK h h h1 0 DK h h hTABLE IV. Compilation of the experimental masses and the total widths (the PDG averagevalues [1] of the charmed-strange states. The unit is MeV.JPn2s1LJ Expt. mass Expt. width ffiffiffiffiffiffiffiffiffispolep M i=2 GI mass011S0 1968 1980113S1 2112 0:5 thresholds. The mixing owing to coupling with the com-mon DsK thresholds shifts the 11P1 downwards along thereal s axis, and the 13P1 moves upwards and crosses DsKthresholds into the complex s plane of unphysical Riemannsheet.V. CONCLUSIONSIn this paper, we propose a simple procedure to extractthe pole positions or determine the Breit-Wigner parame-ters of the charmed states based on the parameters in thenonrelativistic potential model, by using the analyticalrepresentation of the QPC model to mimic the behaviorsof the imaginary part of the self-energy function of themeson propagator. Overall improvements could be foundbetween the pole positions or Breit-Wigner parametersand the well-established charmed and charmed-strangemesons. Several charmed-strange states could be regardedas the quasibound states, due to the coupling withnearby virtual OZI-allowed thresholds. In this model, the11P1 13P1 mixing is explained by the coupling withcommon channels and these resultant pole masses andwidths are consistent with the observed values. It is worthstressing that our calculation is the first one that system-atically addresses such a broad spectrum and the decays ofthe members by considering the coupled channel effects, asfar as we know. This calculation may help to improveour understanding the charmed and charmed-strangespectra.There are still some differences between the shifted polepositions and the parameters of the newly observed states.Since at the present stage the statistics of the data is still notenough to make a firm determination, further experimentalevidences are required for a confirmation of these mesons.ACKNOWLEDGMENTSWe are grateful to X. Liu and Z.-F. Sun for the valuablediscussions about the details of the QPC model. Z. X. issupported by the Fundamental Research Funds for theCentral Universities under Grant No. WK2030040020.Z. Z. is supported by the National Natural ScienceFoundation of China under Contract Nos. 10705009 and10875001.[1] K. Nakamura et al. (Particle Data Group), J. Phys. 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