Hadron loops effect on mass shifts of the charmed and charmed-strange spectra
Department of Physics, Southeast University, Nanjing 211189, Peoples Republic of China
Interdisciplinary 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 the
observed states in experiments and the theoretical expectations of the nonrelativistic potential model. We
present that in an easily operable procedure the hadron loop effect could shift the poles downwards to
reduce the differences and provide better descriptions of both the masses and the total widths, at least, of
the 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 newly
observed 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.Fv
Discoveries 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. , 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 , whose features lead to intense discus-sions and theoretical suggestions of the further ex-perimental investigations . 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) . However,the observed masses are generally lower than the predictedones. For example, the biggest discrepancies happening inboth spectra are the 13P0 states. The D
02318 is about
80 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. 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 loop
effects) 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 . The coupled channeleffects have already brought some insights into the natureof the charmed-strange DsJ2317 and some other states. 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 .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  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. and Pennington et al.  in their study of the charmoniumand bottomonium states, but there are significant differ-ences from them, as discussed in the text.
PHYSICAL REVIEW D 84, 034023 (2011)
1550-7998=2011=84(3)=034023(8) 034023-1 2011 American Physical Society
The 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 SCHEME
We 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 sXn
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 relation
Rens 1PZ 1sth;n
dzImnzz s ; (2)
where PRmeans the principal value integration. The pole
of 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), as
P1s 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 , 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 a
meson 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. .Here, we just make a brief review of the main results of
the QPC model used in our calculation. (For a morecomplete review, see .) 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 as
3 ~p43 ~p3 ~p4
y3 ~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 the
overall 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 as
hBCjSjAi I 2iEf EihBCjTjAi; (5)
hBCjTjAi 3 ~Pf ~PiMMJAMJBMJC : (6)
The amplitude turns out to be
FIG. 1. The imaginary part of the self-energy function.Rd
means the integration over the phase space.
ZHI-YONG ZHOU AND ZHIGUANG XIAO PHYSICAL REVIEW D 84, 034023 (2011)
MMJAMJBMJC ~P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EAEBECp XMLA ;MSA ;MLB ;MSB ;MLC ;MSC ;m
hLAMLASAMSA jJAMJAihLBMLBSBMSB jJBMJBi
j12SAMSA341mih32C 14B j12A 340 iI
MLB ;MLC ~P: (7)
The spatial integral IMLA ;m
MLB ;MLC ~P is given by
MLB ;MLC ~P
Zd3 ~kc nBLBMLB
c nALAMLA ~k ~PYm1 ~k; (8)
where we have taken ~