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 ~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 
j12SAMSA341mi 32SB 12SC 12SA 11=2
hSCMSCSBMSB jSMSihSMSjSAMSA ; 1;mi
8>:1=2 1=2 SC
1=2 1=2 SB
SA 1 S
The flavor matrix element is
h32C 14B j12A 340 i XI;I3
hIC; I3C; IBI3BjIAI3Ai2IB 12IC 12IA 11=28>:I2 I3 IC
I1 I4 IB
IA 0 IA
where IiI1; I2; I3; I4 is the isospin of the quark qi.The imaginary part of the self-energy function in the
dispersion relation, Eq. (2), could be expressed as
2JA 1j ~Psjffiffiffi
XMJA ;MJB ;MJC
jMMJA ;MJB ;MJC sj2; (11)
where jPsj is the three momentum of B and C in theircenter of mass frame. So,
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis mB mC2s mB mC2p
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 for
MMJA ;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 . 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 . 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 be
P1s m2pot sXn
z s0z s ;
wherempot is the bare mass of a certain meson defined in the
HADRON LOOPS EFFECT ON MASS SHIFTS OF THE . . . PHYSICAL REVIEW D 84, 034023 (2011)
III. MIXING MECHANISM
In 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 and
1P1assignments. 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 .
The inverse of the propagator with two bare states mix-ing with each other reads
P1s M211s M212s
m2bare;1 s11s 12s
21s m2bare;2 s22s !
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-onalization
M2diags 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 ANALYSES
The bare masses of the related mesons are chosen at thevalues of the GIs work . As for the dimensionless
parameter, , 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 discussi...