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Compositeness of dynamically generated states in a chiral unitary approach Tetsuo Hyodo∗ Department of Physics, Tokyo Institute of Technology, Meguro 152-8551, Japan Daisuke Jido Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan and J-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), 203-1, Shirakata, Tokai, Ibaraki, 319-1106, Japan Atsushi Hosaka Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan (Dated: January 5, 2012) 2 1 The structure of dynamically generated states in the chiral unitary approach is studied from a 0 viewpoint of their compositeness. We analyze the properties of bound states, virtual states, and 2 resonancesinasingle-channelchiralunitaryapproach,payingattentiontotheenergydependenceof thechiralinteraction. Wedefinethecompositenessofaboundstateusingthefieldrenormalization n constantwhichisgivenbytheoverlapofthebarestateandthephysicalstateinthenonrelativistic a J quantummechanics,orbytheresidueoftheboundstatepropagatorin therelativistic fieldtheory. Thefieldrenormalization constantenablesonetodefineanormalized quantitativemeasureofcom- 4 positenessoftheboundstate. Applyingthisschemetothechiralunitaryapproach,wefindthatthe boundstategeneratedbytheenergy-independentinteraction isalwaysapurelycompositeparticle, ] h while the energy-dependent chiral interaction introduces the elementary component, depending on t the value of the cutoff parameter. This feature agrees with the analysis of the effective interaction - l by changing the cutoff parameter. A purely composite bound state can be realized by the chiral c interaction only when the bound state lies at the threshold or when the strength of the two-body u attractive interaction is infinitely large. The natural renormalization scheme, introduced by the n property of the loop function and the matching with the chiral low-energy theorem, is shown to [ generate a bound state which is dominated by the composite structure when the binding energy is 2 small. v 4 PACSnumbers: 14.20.–c, 11.30.Rd,12.39.Fe 2 5 5 I. INTRODUCTION Theappearanceofhadronicmoleculesnearthethresh- . old regionis similar to the cluster phenomena in nuclear 8 structure, for instance, the appearance of the 0+ state 0 Recently, intensive attention has been paid to the dis- 2 1 cussion of hadronic molecular states, in which two (or of 12C close to the 3α threshold (the Hoyle state) [16]. 1 For hadrons, however, there are peculiar difficulties in more) hadrons are loosely bound by the interhadron : the discussion of the molecular states. For instance, be- v forces. In general, masses of such hadronic molecular i states lie close to some two-hadron threshold energies. causetheqq¯paircreationandannihilationcantakeplace X through the strong interaction, “the number of valence Classical examples are found in the light quark sector: r the Λ(1405) resonance below the K¯N threshold [1–3] quarks” is not a good classification scheme of the struc- a ture. In any case, the discrimination of the hadronic and the scalar mesons a (980) and f (980) close to the 0 0 K¯K threshold [4]. In relation with the recent progress molecules out of other structures is a subtle issue, as compared with the nuclear structure. To start with, we in the hadron spectroscopy at B factories, the charmo- nium X(3872) resonance is also discussed as the DD¯∗ need a theoretical framework with a clear definition of the hadroninc molecular states. molecule [5, 6]. It is important to clarify the property and the binding mechanism of the hadronic molecules Thestructureofaparticle,whetheritiselementaryor for the understanding of the nonperturbative aspects of composite, was intensively discussed in 1960s, using the the strong interaction. Various attempts to classify the field renormalization constant Z in nonrelativistic theo- internalstructure ofhadronshavebeen proposed,forin- ries[17–21]. A famousresultis foundin the study ofthe stance, by the N scaling method [7–12], by property c deuteron in Ref. [22], where a model-independent rela- changesalongwithchiralsymmetryrestoration[13],and tion of the compositeness with observables was derived by production rates in the heavy-ion collisions [14, 15]. in the small binding energy limit. The renormalization constantZ is first relatedto the p-n-dcoupling constant g and the binding energy of the deuteron B, which are then connected with the experimental observables such ∗ [email protected] as the scattering length and the effective range. In both 2 steps, small binding energy is assumed. The composite- II. CHIRAL UNITARY APPROACH ness through the field renormalization constant Z was also studied in a relativistic field theory [23], and there The meson-baryon scattering and baryon resonances have been severalattempts to extend the notion of com- havebeen welldescribed by the chiralunitary approach, positeness to resonances [24–27]. where the chiral low-energy theorem is combined with In this paper, we adopt the strategy to utilize the the unitarity condition of the scattering amplitude [29– field renormalization constant for the definition of the 32]. Adetailedformulationofthechiralunitaryapproach compositeness. First, following Ref. [22], we evaluate can be found in Ref. [33]. In this section we construct the field renormalization constant Z as the overlap of the single-channelversion of the chiral unitary approach the bare state in the free Hamiltonian with the physical along the same line with Ref. [34] and study the proper- bound state in the nonrelativistic quantum mechanics. ties of the generated states in the amplitude. It is our In addition to the universal relation in the weak bind- aim to establish a tractable model which will be used to inglimit[22],wealsoderivetheexpressioninexactform examine the argument of the compositeness in Sec. III. in terms of the scattering amplitude.1 Second, following Whileweadoptthemeson-baryonscatteringasanexam- Ref. [23], we define the field renormalization constant Z ple, the scattering of the Nambu-Goldstone (NG) boson as the residue of the full propagator of the bound state withaheavymesoncanbe describedinthe sameformu- in the Yukawa theory. This approach enables us to ana- lation (see Appendix C in Ref. [37]). The framework of lyzetheenergy-dependentinteraction. Itshouldbenoted the scattering of the NG bosons are the essentially the that we discuss the compositeness ofthe bound states in same, thanks to the chiral low energy theorem. the two-hadronscatterings,regardingthestable hadrons in the scattering states as elementary particles, in the sensethattheyformtheasymptoticstatesinthenonper- A. Construction of the scattering amplitude turbativeQCDvacuum. We donotconsiderthe internal structure of the hadrons in the asymptotic state. We consider a single-channel s-wave scattering of a We apply the notion of compositeness to the bound baryonwithmassM andaNGbosonwithmassm. The states in a chiral unitary approach [29–33]. In this chirallowenergytheorem[38,39]dictatestheleadingin- model, baryon resonances are described in the multi- teraction proportional to the boson energy ω W M ple scattering of the meson-baryon system whose inter- in the following form: ≃ − action is given by the chiral low-energy theorem. Al- though the generated resonance is expected to be a V(WT)(W)=C(W M), (1) − composite of a meson-baryon system, as pointed out in where C is a real-valued coupling constant with mass Ref.[34],the Castillejo-Dalitz-Dyson(CDD)pole contri- dimension minus two and W is the total center-of-mass butions [35,36] canbe hidden in the cutoff parameterin energy.2 Although the coupling constant is determined the renormalization procedure. To avoid this ambiguity, by the flavor content of the scattering system, here we the natural renormalization scheme has been proposed, simply regard it as a parameter of the model. The en- in which the subtraction constant is chosen to exclude ergydependenceoftheinteractionisaconsequenceofthe the CDD pole contribution from the loop function. To low energy theorem [40]. To appreciate the significance connect the argument of the compositeness to the natu- of the energy dependence, we also consider an energy- ral renormalization scheme, we discuss the properties of independent constant interaction whose strength is nor- the amplitude in the chiral unitary approachin detail. malized to the threshold value of the WT interaction: This paper is organized as follows. In Sec. II, we first analyzethepropertiesofthepolesingularitiesinthechi- V(const) =Cm=V(WT)(W =M +m). (2) ral unitary approach, using a single-channel model. We examine the energy-dependent chiral interaction and its ThisinteractiongeneratesthesameamplitudeastheWT energy-independent counterpart and discuss the proper- interaction only at the threshold, and the deviation be- tiesofthegeneratedstatesindetail. Next,inSec.III,we comes large in the energy region far from the threshold. define the compositeness of the bound state using field In relation with the pole structure of the Λ(1405) reso- renormalization constant Z in a general framework of nance[41,42], the energydependence ofthe chiralinter- field theories. We consider both nonrelativistic and rela- action has been recently studied [43, 44]. tivistic approaches,andexamine the limit of smallbind- The scattering amplitude can be obtained by solving ing. The numerical analysis of the compositeness of the the Bethe-Salpeter equation bound state in the chiral unitary approach is presented T(W)=V(W)+V(W)G(W)T(W), (3) in Sec. IV. 2 We may recover the usual WT interaction in the chiral unitary 1 A preliminary discussion on this issue has been given in a con- approachbyreplacingC→−Cg/(2f2)withthepiondecaycon- ferenceproceedings [28]. stantf andthegrouptheoretical factorCg. 3 with V(W) being V(WT)(W) or V(const). The loop func- real axis in the first Riemann sheet below the threshold, tion G(W) is given by while a resonance is represented by a pole in the second Riemann sheet of the complex energy plane above the d4q 2M 1 threshold. A pole on the second Riemann sheet below G(W)=i , (2π)4(P q)2 M2+iǫq2 m2+iǫ the thresholdiscalledavirtualstate,whichisa peculiar Z − − − (4) phenomenon in the s-wave scattering. The amplitude in the second Riemann sheet is given by whereP =(W,0). Thisintegraldivergeslogarithmically. With the dimensional regularization,we obtain 1 T (W)= V(W), (8) II 1 V(W)G (W;a) II − 2M m2 M2+W2 m2 2Mq¯(W) G(W;a)= a+ − ln G (W;a)=G(W;a)+i . (9) (4π)2 2W2 M2 II 4πW q¯(W) + ln[W2 (M2 m2)+2Wq¯(W)] Note that below the threshold W < M + m, q¯(W) is W − − purely imaginary. Therefore, on the real axis below the +ln[W2n+(M2 m2)+2Wq¯(W)] threshold, the loop function is real in both the first and − ln[ W2+(M2 m2)+2Wq¯(W)] thesecondRiemannsheets. Forlaterreference,wepoint − − − outthatG(W;a)[GII(W;a)]ismonotonicallydecreasing (increasing) function of W below the threshold. ln[ W2 (M2 m2)+2Wq¯(W)] , − − − − Because the interactions in Eqs. (1) and (2) have no ! o singularityinthecomplexenergyplaneexceptfor W (5) | |→ ,wededucetheconditionforthepoleoftheamplitude ∞ where a is the subtraction constant at the subtraction as pointµ =M,whichplaysaroleofthe cutoffparameter s of the loop function. The three-momentum variable is 1 V(zR)G(zR;a)=0 (bound state), − defined as (1 V(zR)GII(zR;a)=0 (virtual state/resonance), − (10) [W2 (M m)2][W2 (M +m)2] q¯(W)= − − − . (6) 2W p where zR is the energy of the bound state, virtual state, or resonance. The pole position of z should be on the G(W;a) is an analytic function of W in the whole com- R real axis in the first Riemann sheet, while z can be plexplane,exceptforontherealaxisabovethethreshold R complex in the second Riemann sheet. Around the pole where it has a branch cut. In conventional approaches, atW =z we canformally expandthe amplitude in the the subtraction constant has been used to fit the exper- R Laurent series as imental data. In Ref. [34], on the other hand, the nat- ural value of the subtraction constant anatural was intro- g2 ∞ duced by the theoretical argument to exclude the CDD T(W)= + T(n)(W z )n, (11) R polecontributionfromthe loopfunction. We discussthe W zR − − n=0 X influence of the subtraction constants to the scattering amplitude. A solution of Eq. (3) is given by with constant coefficients T(n). The first term can be interpretedasthe propagatorofaparticlewithmassz , R 1 T(W)= V(W), (7) which couples to the scattering states through the cou- 1 V(W)G(W;a) pling constant g. Hence, we identify the residue of the − pole in the amplitude T(W) as the energy-independent with the help of the on-shell factorization [30, 45]. The coupling strength g2. same form of the amplitude can also be obtained by the N/D method with the N = 1 prescription [31]. For the energy-independent interaction (2), no factor- B. Bound state with constant interaction ization is needed so that both the N = 1 prescription and the N = V prescription in the N/D method pro- vide the equivalent amplitude (7). Note that Eq. (7) is Letusfirstconsiderthecasewhenthesystemdevelops a bound state with energy-independent interaction (2). obtainedbyneglectingtheunphysical(left-handandcir- The amplitude has a pole on the real axis in the first cular) cuts, although the unitarity cut is properly taken Riemann sheet: z M , Im M =0, and M M intoaccount. Thus,thisframeworkshouldnotbeapplied R B B B ≡ ≤ ≤ M +m where M =M is the lower limit of the mass of tothesystemwithadeeplyboundstate,whichmightbe B theboundstate;otherwise,thesystembecomesunstable. influenced by the unphysical cuts. The pole condition (10) leads to Iftheinteractionissufficientlyattractive,boundstates and resonances can be dynamically generated. A bound 1 CmG(M ;a)=0. stateappearsasapoleofthefullamplitudeT(W)onthe B − 4 With this relation,wecandeterminethe value ofC such Because the whole amplitude T(W) is invariant, the that a bound state is generated at W = M with the property of the bound state remains unchanged. B subtraction constant a as 1 C(MB;a)= . (12) C. Bound state with WT interaction mG(M ;a) B Here we have indicated that C can be also regardedas a Next we deal with the energy-dependent WT interac- functionofM anda,whentheboundstateisgenerated. B tion(1). Theconditionfortheboundstate(10)becomes The loop function G is real below the threshold, so it is consistent with the real-valued coupling strength C. If 1 C(MB M)G(MB;a)=0. (15) − − M were complex, then the loop function G(M ) would B B The coupling strength g can also be calculated from the be complexand no solutionwouldbe foundfor Eq.(12). residue of the pole as This means that there is no pole in the first Riemann sheet except for on the real axis, in accordance with the [g(MB;a)]2 = lim (W MB)T(W) causality. W→MB − 1 Note that Eq. (12) is the relation among C, MB, and = (16) a. In general, the parameters in this model are the − G′(MB)+ GM(MB−BM;a) strength of the interaction C and the subtraction con- In this case, the coupling constant depends on the sub- stant a. When there is a bound state, the system can traction constant a, in contrast to Eq. (13). This is alsobecharacterizedbythemassoftheboundstateM B closely related to the energy dependence of the interac- anda, withthe helpofthe relation(12). Using Eq.(11), tion,becauseEq.(16)isobtainedbydifferentiatingboth wecancalculatethecouplingconstantgfromtheresidue the numerator and the denominator, and the energy de- of the amplitude T as [46] pendence of the interaction leaves the G(M ;a) term in B [g(M )]2 = lim (W M )T(W) the final expression. Note also that if G(M ;a) = 0, B B B W→MB − the result coincides with that of the energy-independent 1 1 interaction (13), which, however, means C be- = . (13) → −∞ − ∂G(W;a)(cid:12) ≡−G′(MB) cause of the bound state condition (15). This limit will ∂W (cid:12)W=MB be important in the discussion of the compositeness. (cid:12) Below the threshold, G(W;(cid:12)(cid:12)a) is real and monotonically As before, the condition (15) relates C, MB, and a, decreasing function of W [34], so we have G′(MB) < 0. so we can characterize the system by (MB, a) instead of Thisindicatesthatthecouplingsquareisalwaysrealand (C,a), keeping the mass of the bound state unchanged. positive: In this case, the interaction strength C such that the bound state appears at M is calculated as [g(M )]2 0, B B ≥ 1 which leads to the real-valued coupling constant. The C(MB;a)= . (17) (M M)G(M ;a) equality holds for M =M +m where G′(M ) . B − B B B Note also that the dependence on the subtracti→on−co∞n- The state with zero binding energy corresponds to the stant a vanishes after taking the derivative in Eq. (13), upperlimitforthemassoftheboundstateMB =M+m, becausethesubtractionconstantdoesnotdependonthe which is called zero energy resonance [47]. The coupling energy.3 Namely, the coupling of the bound state to the strength for such state, together with the natural value scattering state g is unambiguously determined by the of the subtraction constant [34] (the natural subtraction massoftheboundstateMB andindependentofthesub- constantanatural is also explainedin Appendix B), is the tractionconstanta. Therefore,throughtherelation(12), critical coupling introduced in Refs. [37, 48]: we can freely change the subtraction constant a and the 1 couplingstrengthC,withoutalteringthepropertyofthe Ccrit =C(M +m;anatural)= , mG(M +m;a ) natural bound state. In fact, this is one of the consequences of the invariance of the amplitude T(W) under the trans- which is the smallest coupling strength to generate a formation bound state. As discussed in Ref. [34], the variation of the subtrac- −1 1 2M a a ∆a, C ∆am . (14) tion constant can be absorbed by the introduction of a → − → C − (4π)2 poletermintheinteractionkernel,keepingtheamplitude (cid:18) (cid:19) T(W) invariant: a a ∆a, 3 In the present framework, we perform the single subtraction → − to tame the divergence in Eq. (5). We can introduce further V(W) V˜(W)=C(W M) C (W −M)2 , (18) subtractions, whichresultsinthe energy-dependent subtraction → − − (W M ) eff terms. In this case, the coupling constant depends on the sub- − tractionconstants. Weconsider thecasewiththegeneral inter- (4π)2 M =M + . (19) actionandthegeneralsubtractiontermsinAppendixB. eff 2MC∆a 5 Equation(18) showsthat the existence ofa pole atW = sheet. If the real part of the pole is higher (lower) than M in the interaction kernel V˜(W). The residue of this the threshold, the state is a resonance (virtual state). eff pole is calculated as For the energy-independent interaction, the pole con- dition (10) becomes (4π)4 lim (W M )V˜(W)= . (20) W→Meff − eff − 4M2C(∆a)2 G (z ;a)= 1 . II R Cm Both the mass and the residue are determined by ∆a. This means that it is not possible to introduce an arbi- The right-handside is real,while the left-handside is,in trary pole term by the variation of the subtraction con- general, complex. Therefore, at the pole position W = stant; the mass and the coupling of the pole term are z theimaginarypartoftheloopfunctionshouldvanish: R related to each other through Eqs. (19) and (20). This pole term cannot be absorbed by simple shift Im [GII(zR;a)]=0. (22) of the coupling strength of the interaction kernel as in Eq.(14). Actually,Eq.(18)canbewritteninthesimilar Becausethesubtractionconstantaisrealandadditively form to Eq. (14) as includedintheloopfunctionGII(zR;a),thisconditionis independentofaandisthe necessaryconditionforz to R −1 1 2M be the pole position. We first consider the virtual state a a ∆a, C ∆a(W M) . → − → C − (4π)2 − onthe realaxis, zR =MV, Im MV =0,andM MV < (cid:18) (cid:19) ≤ (21) M +m. Below the threshold MV < M +m, there is no discontinuity, so the imaginary part of the G (M ) II V This is the transformation which leaves the amplitude functionvanishesandthecondition(22)issatisfied. This T(W)invariantfortheWTinteraction. Inthecaseofthe means that a virtual state can be produced on the real energy-independent interaction, the change of the sub- axis for a weakly attractive interaction. tractionconstantcanbecompensatedbythesimpleshift Resonances are associated with the pole z with a fi- R of the coupling strength C [Eq. (14)], and the coupling nite imaginary part. As shown in Appendix C, the pole constant of the bound state to the scattering state g is trajectory is continuous in the complex energy plane as independent of the subtraction constant [Eq. (13)]. In thecouplingstrengthC isvariedwithafixedthesubtrac- contrast, for the WT interaction, the change of the sub- tion constant a. Thus, the resonance solution should be traction constant causes the energy-dependent transfor- connectedtothevirtualstatesolutionontherealaxisat mation[Eq.(21)]andthecouplingconstantgdependson someenergy. Tohavea resonanceatz withIm z =0, R R thesubtractionconstant[Eq.(16)]. Thesearetheimpor- Eq. (22) should be satisfied in the vicinity of the6real tant consequences of the energy dependence of the WT axisintheregionM Re z <M+m. Thiscanbeim- R interaction. As we see below, this difference is crucial to plemented if the deri≤vative of the imaginary part in the the structure of the bound state. imaginarydirectioniszero. However,Cauchy-Riemann’s Letusconsiderthepropertiesofthecouplingsquarein theorem leads to Eq.(16)whichmustberealandpositivetoobtainareal- valuedcouplingconstant. Becausethemassofthebound ∂ ∂GII(Re zR;a) Im G (z ;a) = >0, II R state should be larger than the mass of the constituent, ∂(Im z ) ∂(Re z ) R (cid:12)ImzR=0 R we have a condition (cid:12) (cid:12) for M Re zR < M +(cid:12)m. Thus, the existence of the M M 0. ≤ B branch of the pole trajectory to the resonance solution − ≥ is not allowed. This means that the pole with a finite It follows from Eq. (17) that G(M ;a) 0 for an at- B ≤ imaginarypartcannotappearfortheenergy-independent tractiveinteractionC <0 (equality holdsfor C ). →−∞ interaction. This is consistent with the interpretation in Therefore, G(M ;a)/(M M) 0 and we also have B B G′(M ) < 0. Thus, from E−q. (16≤), for the bound state the nonrelativistic potential model; simple attractive s- B wave potential has no centrifugal barrier and thus does generatedbytheattractiveWT interaction,wefindthat not produce a resonance state. the coupling square is always real and positive: Theresidueg2 canbeobtainedbythederivativeofthe [g(M ;a)]2 0. loop function B ≥ Equality holds for MB =M +m, where G′(MB)→−∞ [g(M )]2 = 1 . and a dependence of the coupling constant vanishes in V − G′ (M ) II V this limit. For the virtual state on the real axis Im M = 0, the V derivative of the loop function on the second Riemann D. Virtual states and resonances sheet is positive. Therefore, the coupling square is neg- ative. Because the pole singularities in the second Rie- Finally,weconsiderthecasewiththepoleonthecom- mann sheet are interpreted as Gamow states, the cou- plex energy plane at W = z in the second Riemann pling constant becomes pure imaginary. As in the case R 6 of the bound state with the constantinteraction,the de- III. FIELD RENORMALIZATION CONSTANT pendenceofthesubtractionconstantislostbytakingthe AND COMPOSITENESS derivative. For the WT interaction, the bound-state condition is In this section, we define the compositeness of the bound state in the scattering amplitude in the nonrel- 1 C(zR M)GII(zR;a)=0. (23) ativistic quantum mechanics [22] and in the relativistic − − field theory [23]. For this purpose, we describe the scat- Again,C isarealnumber,sotheequationtobesatisfied tering system with one bound state by the field theory at z is whose free spectrum contains one bare state and two- R body scattering states. We define the field renormaliza- Im [(z M)G (z ;a)]=0. (24) tionconstantZ astheoverlapofthephysicalboundstate R II R − and the bare state, which characterizes the elementarity Inthesamewayasintheenergy-independentinteraction, of the bound state. Expressing the constant Z in terms this condition is satisfied on the real axis z = M , so of physicalquantities, we obtain a master formula of the R V the virtual state can be formed on the real axis. As for compositeness. Withthismasterformula,wecandiscuss the resonance, in the present case, the compositeness ofbound states,either those observed in experiments or those obtained in a theory such as the ∂ chiral unitary approach. To be specific, we consider the Im [(z M)G (z ;a)] ∂(Im z ) R− II R s-wave scattering of a baryon ψ(JP = 1/2+) and a me- R (cid:12)(cid:12)ImzR=0 son φ(0−) in which a bound state B(1/2−) appears. It ∂ (cid:12) = [(Re zR M)GII(Re zR;(cid:12)a)] iseasytoderivesimilarexpressionsofthe compositeness ∂(Re z ) − R for other scattering systems with an s-wave bound state =GII(Re zR;a)+(Re zR−M)G′II(Re zR). (25) (see Appendix E and Ref. [23]). This can be zero for M Re z < M + m, R ≤ if the subtraction constant is properly chosen, be- A. Compositeness in nonrelativistic quantum cause G (Re z ;a) 0 from Eq. (23) and (Re z mechanics II R R M)G′ (Re z ) 0. ≤Namely, for the energy-dependen−t II R ≥ interaction, the pole can have the imaginary part and Let us first consider the two-body scattering system eventually an s-wave resonance can be produced. Sev- with one bound state in a nonrelativistic quantum me- eral examples of the s-wave single-channel resonance chanics. TheHamiltonianofthesystemH isdecomposed can be found in hadron physics, such as the σ meson into the free Hamiltonian H and the interaction V: 0 in the ππ scattering [13] and the lower energy pole of Λ(1405) [41, 42, 44]. H =H +V. 0 The condition (24) determines the relation between We assume that the eigenstates of the free Hamiltonian Re [z ] and Im [z ], which depends on the subtraction R R H are given by the continuum states labeled by the constant. Namely, for a given a, the pole trajectory is 0 relative momentum q and an elementary state4 B specified in the complex energy plane as a function of C 0 | i | i which is orthogonalto q . V stands for the interaction (for illustration, we show the trajectory of the pole po- | i among scattering states, as well as the coupling of the sition in Appendix C). The position of the pole is then scatteringstateto the elementarystate. B iselemen- determined by the coupling strength C and vice versa. 0 | i tary in the sense that it is an eigenstate of the Hamilto- The residue of the pole is nian H and the origin of B should be attributed to 0 0 | i 1 dynamics other than the interaction V.5 The eigenener- [g(zR;a)]2 =− G′ (z )+ GII(zR;a). (26) gies are given by II R zR−M H q =E(q) q H B = B B , 0 0 0 0 0 | i | i | i − | i For a virtual state (Im z = 0), the coupling square is R givenby the realnumber. Onthe realaxis, G′ (z ) and II R z M are positive, but the sign of the G (z ;a) de- R II R − pends on the value of the subtraction constant. Thus, 4 Ingeneral,wemayhaveseveralboundstatesaswellasmultiple the sign of the coupling square depends on the subtrac- scatteringchannels (seeAppendixD). tion constant and is determined by the balance of the 5 Onemayregard|B0iasaboundstateformedbyanotherHamil- first term and the second term in the denominator of tonianH′. UsingtheFeshbachprojectionformalismwithsingle Eq.(26). BecauseG′ (z )divergesatthe threshold,the bound state approximation [49, 50], we can set up this situa- coupling square is neIgIatRive for the virtual state close to tion. Thus, strictlyspeaking, |B0iisnotalways anelementary component; itshouldbeconsideredasaCDDpolecontribution thethreshold. Foraresonance(Im zR =0),thecoupling whichdoesnotoriginateinthepresentmodelspaceofthescat- constant is complex because of G , G6 ′ and (z M) tering. Inthispaper,however,weusetheword“elementary”for which are complex in the complexIIenergIIy plane.R − simplicity. 7 where E(q)=q2/(2µ) with µ being the reduced mass of Thisisanexactexpressionofthecompositeness1 Z . NR − the system. q and B0 are orthogonal and form the Nextweconsidertoexpresstheintegrandofthisequa- | i | i complete set: tion by the scattering amplitude. The formal solution of the Lippmann-Schwinger equation is written as q q′ =δ(q′ q), B B =1, 0 0 h | i − h | i B q =0, 1 h 0| i T(E)=V +V V. E H 1= B B + dq q q . (27) − 0 0 | ih | | ih | Z Inserting the complete set of the full Hamiltonian (29) and taking the matrix element by the initial and final We consider that the Hilbert space of the full Hamilto- scattering states with energy E, we obtain the Low’s nianalsocontainsoneboundstate B withthe binding | i equation [22], energy defined as B > 0. Namely, B is the eigenstate | i of the full Hamiltonian G (E)2 ∞ √E′ t(E′)2 t(E)=v+ | W | +4π 2µ3 dE′ | | , H|Bi=(H0+V)|Bi=−B|Bi, (28) E+B Z0 E−E′+iǫ p where B isnormalizedandorthogonaltothescattering where v (t) is the matrix element of the interaction | i states q,full as HamiltonianV (T operator)bythe scatteringstatesand | i we have used V q,full = T q . The second term of hq,full|q′,fulli=δ(q′−q), hB|Bi=1, thisequation,wh|ichcomiesfro|mtihe operator(E H)−1 B q,full =0, acting on the bound state discrete level, is a par−t of the h | i integrand of Eq. (30). Therefore, we can express the and they form the complete set as compositeness using the scattering amplitude as [28] 1= B B + dq q,full q,full . (29) ∞ √E | ih | | ih | 1 Z =4π 2µ3 dE t(E) v Z − NR Z0 E+B" − WenotethattheinteractionV shouldbeenergyindepen- p dent, to ensure the orthogonality and the completeness ∞ √E′ t(E′)2 4π 2µ3 dE′ | | . (31) of the Hilbert space of the full Hamiltonian. − Z0 E−E′+iǫ# WethendefinethefieldrenormalizationconstantZNR p astheprobabilityoffindingtheboundstateBinthebare Itisimportanttonotethatthescatteringamplitudet(E) state B : 0 is,ingeneral,complex,buttheimaginarypartofEq.(31) vanishes thanks to the optical theorem: Z B B 2. NR 0 ≡|h | i| This expresses the elementarity of the bound state B . Im t(E)= 12π 4π 2µ3√E t(E)2 | i − 2 · | | Itis clearfromthe completeness (27) that0 ZNR 1. = 4π2µq(Ep)t(E)2, (32) If B is a purely composite (elementary) ≤particle,≤we − | | | i obtain ZNR = 0 (ZNR = 1). Therefore, 1 ZNR serves where q(E) = √2µE is the momentum. Therefore, the − asthequantitativemeasureofthe“compositeness”ofthe imaginary part vanishes in the bracket in Eq. (31), and particle, which is given by the overlap Z is obtained as a real number. More NR explicitly, we can write 1 Z = dq q B 2. NR − |h | i| Z ∞ √E 1 Z =4π 2µ3 dE Re t(E) v Multiplying hq| to Eq. (28) from the left, we obtain − NR Z0 E+B" − p q V B 2 ∞ √E′ t(E′)2 1 Z = dq|h | | i| . 4π 2µ3 dE′ | | , − NR Z [E(q)+B]2 − PZ0 E−E′ # p Here the matrix element expresses the coupling strength where stands for the principal value integration. of the bound state to the scattering state with momen- P NowwetakethelimitB 0. ForsmallB,thebracket tum q. For an s-wave bound state, the coupling is in- → inEq.(31)isdominatedbytheboundstatepoletermin dependent of the angular variables and is a function of the T matrix, the energy. Then we can write q V B G (E), and W h | | i≡ obtain the energy integration form √E g2 Integrand in Eq. (31) W + (B0) , ∞ √E G (E)2 ≈E+B · E+B O 1 Z =4π 2µ3 dE | W | . (30) (cid:18) (cid:19) − NR Z0 (E+B)2 gW ≡GW(E =−B), p 8 because this is the only term with (B−1) for small E. of another baryon B (1/2−) as 0 O Then the E integration can be done analytically: =ψ¯(i∂/ M)ψ+ 1(∂ φ∂µφ m2φ2) L0 − 2 µ − 1 Z 4π 2µ3g2 ∞dE √E +B¯0(i∂/−MB0)B0. (34) − NR ≈ W (E+B)2 Z0 We consider the bound state of ψ and φ, which couples p g2 with the bare state B . Setting µ = Mm/(M +m), we =2π2 2µ3 W . (33) 0 √B obtainthesystemcorrespondingtothatdescribedbythe p Hamiltonian H0 in the previous section. The interaction This is the result of Ref. [22] which connects the com- Lagrangian,which corresponds to V in the previous sec- positeness ofthe bound state with the coupling constant tion, is introduced as a scalar-type Yukawa form, gliWmitanodf tthheebcoinudpilninggensterregnygBth.cSainncbeeZoNbRta≥ine0d,tahseupper Lint =g0ψ¯φB0+(H.c.), (35) whereg isthebarecouplingconstant. Wethenconsider 0 that the spectrum of the full theory has a bound state 1 B g2 . with mass M which is related to the binding energy as W ≤ 2π2s2µ3 B M =M +m B. The definition of the field renormal- B − izationconstant,whichwedenoteZ,istheresidueofthe TheequalitycorrespondstoZ =0,namely,thepurely NR full Green’s function at W =M , namely, B composite particle. Note that the upper limit decreases as we decrease the binding energy B and vanishes in the Z ∆(W)= . (36) weak binding limit B 0. W M B → − Itis importantto note thatinthe expression(33), the Forlaterconvenienceintheapplicationtothechiraluni- explicit dependence on the interaction V is lost when tary approach, we adopt the positive energy part of the B 0, and all the information of the interaction is rep- fermion propagator. → resentedbythecouplingconstantoftheboundstateg . W Let us calculate this Z constant with the Lagrangian Inthis sense, the resultis universalanddoes notdepend given in Eqs. (34) and (35). The free Green’s function ontheparticularchoiceoftheinteractionV. Incontrast, for B is given by 0 the exactformula(31)canbe appliedtothe boundstate with arbitrary binding energy. However, the result de- 1 ∆ (W)= . 0 pends on the matrix element v and, hence, onthe choice W M − B0 of the interaction Hamiltonian V. The full Green’s function ∆(W) is expressed in terms of From the expression of Eq. (30), we notice that the ∆ (W) by the Dyson equation: exact result of the compositeness can be calculated with 0 the knowledge of the energy dependence of the coupling ∆(W)=∆ (W)+∆ (W)g G(W)g ∆(W), 0 0 0 0 strength G (E) = q V B . In principle, G (E) rep- W W h | | i withtheunrenormalizedtwo-bodyloopfunctionofψand resents the coupling strength of the bound state to the φ, G(W). The solution of this equation is scattering state above the threshold, so it is an off-shell quantity. The coupling “constant” gW can be regarded 1 as the leading term in the expansion of G (E) around ∆(W)= W [∆ (W)]−1 g2G(W) E = B. In the present discussion, we assume that 0 − 0 − 1 the binding energy B and the coupling constant gW are = . W M g2G(W) known quantities. In this case, the approximation of − B0 − 0 1 Z with only known quantities should be in the form The divergence of the loop function G(W) is canceled − of Eq. (33) and any further contribution should depend by the infinite bare mass M . Using the renormalized on the choice of the interaction V because it is related B0 loopfunction G(W;a) with the parametera which char- withtheoff-shellquantityG (E). Toextendthemodel- W acterizesthefinitepart,thefullGreen’sfunctionisgiven independentformula(33),weneedtomeasurethehigher- by ordercoefficients ofthe expansionof G (E) experimen- W tally. 1 ∆(W)= . (37) W g2G(W;a) − 0 The renormalization condition can be obtained by re- B. Compositeness in relativistic field theory quiring the bound state pole at W = M .6 Equating B In this section, we define the compositeness following the method in Ref. [23]. To define the field renormaliza- 6 This procedure is in principle scale dependent. In the present tionconstant,weconsiderthe fieldtheorywith abaryon formulation, the renormalization scale is implicitly set at the fieldψ(JP =1/2+),amesonfieldφ(0−),andabarefield baryonmass,µs=M. 9 Eq. (37) with Eq. (36), we obtain the condition to interpret Eq. (41) as the compositeness, because the condition (40) is not always guaranteed and 1 Z may M =g2G(M ;a). − B 0 B become a complex number. Nevertheless, Eq. (41) may be applied to a narrowwidth resonance,because it coin- The field renormalization constant Z can be calculated cide with Eq. (39) in the zero width limit. as W M B Z = lim − W→MB W −g02G(W;a) IV. COMPOSITENESS OF DYNAMICALLY 1 GENERATED STATES = . (38) 1 g2G′(M ) − 0 B A. Application to chiral unitary approach The meson-baryon scattering amplitude is given by T(W) = g ∆(W)g , so the residue of the amplitude at 0 0 Here we analyze the compositeness of the bound state W =M is B ina specific model describedin sectionII.In the present lim (W M )T(W)=g2Z. section, we havederived three definitions of the compos- W→MB − B 0 iteness: The residue of the bound-state pole is the physical cou- (i) expression in nonrelativistic formalism, Eq. (31); pling constant squared, so we obtain the relation g2 =g2Z (ii) formula for small B, Eq. (33); 0 UsingthisrelationandEq.(38),wefinallywritethecom- (iii) expression in relativistic formalism, Eq. (39). positeness 1 Z as − For the application to the chiral unitary approach with 1 Z = g2G′(MB). (39) the energy-dependent interaction, it is fully legitimate − − to use Eq. (39). In the nonrelativistic formalism, we Note that the right-hand side is expressed only by the should choose the framework which can deal with the physical (renormalized) quantities, and no approxima- energy-dependent interaction. In this respect, the exact tion has been applied to the evaluation of the mass of result Eq.(31) cannot be used because the completeness the bound state. Thus, this is anexact expressionof the relation (29) is not always valid. However, the model- compositeness of the bound state in the relativistic field independent result for small B (33) may be appropriate theorywiththeinteractionLagrangian(35),whichisex- because it can be derived without using the complete- pressedintermsofthecouplingstrengthgofthephysical ness of the full Hamiltonian [22]. Thus, in the follow- bound state to the scattering state and the derivative of ing, we examine the expression of the compositeness in the loop function at the energy of the bound state. The the relativistic field theory in Eq. (39) and the model- expressionforthescalarmesonboundstateinthemeson- independent result for small binding energy, Eq. (33). meson scattering is derived in Appendix E. It should be Substituting Eqs. (13) and (16) into the exact result noted that the definition of the compositeness (39) de- in the field theory (39), we obtain pends onthe adopted interactionLagrangian . With int L a different interaction, the function form of Eq. (39) will 1 constant interaction, bemodifiedaccordingly,and,hence,theresultingexpres- sion for 1 Z will be changed. 1 Z = 1 WT interaction. MThe Mpos−+sibmle, wrehgeiorenGof′(tMhe )boisunredalstaantde nmeagsastiivseM, an≤d − 1+ (MBG−(MM)BG;a′()MB) B B (42) the c≤oupling square is always positive for the physical  bound state. Therefore, we find that the compositeness It is striking that the constant interaction always pro- is always positive, duces a purely composite bound state. This corresponds to the equivalence of the four-Fermi theory and Yukawa 1 Z 0. (40) − ≥ theory found in Ref. [23]. On the other hand, the com- In other words, the field renormalization constant is positenessoftheboundstategeneratedbytheWTinter- smaller than unity. action depends on the subtraction constant. This result The field renromalization constant is well defined for agrees with the fact of the appearance of the pole term resonances. Using the same argument, we find in the effective interaction for the WT interaction (18), which introduces the source of the elementarity. For a 1−Z =−g2G′II(zR), (41) general interaction V, using Eq. (D2), we obtain whereg2 isthe residueofthe pole inthe complex energy 1 plane of the scattering amplitude. This is a natural ex- 1−Z =1 V′(MB)G(MB), (43) tension of Eq. (39). However, it is not straightforward − V(MB)G′(MB) 10 where V′(M ) = ∂V(W)/∂W . In this expres- Notethatboththemomentumfactorsq( B)andq¯(M ) B |W=MB − B sion,GandV aredeterminedunderafixedrenormaliza- are pure imaginary. Substituting this expression into tion scheme. It is easy to see that Eq. (42) follows from Eq. (33), we obtain [28] thegeneralexpression(43)withtheinteractionkernel(1) and(2)inthephenomenologicalrenormalizationscheme. 1 Z = M|λ1/2(MB2,M2,m2)| g2, (46) Inaddition,theenergydependenceoftheinteractionker- − NR 16πM2(M +m M ) B − B nel is related to the deviation of the compositeness from forB 0. Thisform,incomparisonwiththerelativistic unity. → result (39), leads to the relation Let us recall the definition of the bound state that lMowBt−heMth≥res0haonldditnhdeicbaethesavGio′(rMofBt)h<e lo0.opRfeulnactitoionn(1b7e)- 1−Z ≈ −16πMMB2λ(1M/2(+Mm2,−MM2,Bm)G2)′(MB) (1−ZNR) implies G(M ;a) 0, so altogether we obtain (cid:18) | B | (cid:19) B ≤ A(M )(1 Z ) (47) B NR ≡ − G(M ;a) B 0, forthesmallbindingenergy. NotethatthefactorA(M ) (M M)G′(M ) ≥ B B − B dependsonlyonthemassoftheboundstate. Inthesmall binding limit, the definition of the field renormalization and, hence, with Eq. (40), for the WT interaction, we constantZ shouldbe consistentwith thatofZ ,so we obtain NR expect 0 1 Z 1. lim A(M )=1. ≤ − ≤ B MB→M+m This result indicates that the compositeness defined in Wewillcheckthevalidityofthisrelationinthenumerical Eq.(42)isnormalizedwithintherangefromzerotoone. calculation. This is a very useful aspect to discuss the structure of Finally we briefly mention the compositeness for vir- the bound state in a quantitative manner. tual states and resonances. The present formulation in For the WT interaction, a purely composite bound thenonrelativisticquantummechanicsisnotdirectlyap- state (Z =0) is achieved when plicabletoresonanceswhicharenotincludedinthecom- G′(M )= or G(M ;a)=0. (44) plete set in Eq. (27) and in Eq. (29).7 However, the B B −∞ residue of the resonance propagator can be evaluated without trouble in the relativistic field theory. This pro- Because the derivative of the loop function diverges at vides a natural extension of the compositeness of bound threshold, the first condition can be met if the bound states to virtual states and resonances. Equation (41) stateisproducedexactlyatthethreshold,irrespectiveof leads to thevalueofthesubtractionconstant. ForM =M+m, B 6 apurelycompositeboundstatewouldbeproducedonlyif 1 constant interaction, GTh(MusB, t;oa)reisalzizeeroZ, w=hi0c,hwinedsihcoautelds Cha→ve−∞ by Eq. (17). 1−Z =1+ G′IIG(zI1RI()z(RzR;a−)M) WT interaction. M =M +m or C . (45) (48) B →−∞  Notethatthecondition(45)isonlysatisfiedinthelimit- Foravirtualstate(Im zR =0),theexpressionprovidesa ingcasessoitimpliesthatthephysicalboundstate(finite real value for the compositeness. The virtual state gen- bindingenergyandfiniteinteractionstrength)cannotbe erated by the contact interaction always gives Z = 0, purely composite for the WT interaction. It is instruc- while the WT interaction can lead to Z < 0, depending tive to recall that the natural renormalization condition on the subtraction constant. This is because the virtual developed in Ref. [34] requires G(M;a ) = 0. In- statecannotbeincludedintheorthonormalbasis. Inthe natural terestingly, this condition was introduced to realize the case of a resonance,the field renormalizationconstant is hadronicmoleculestateintheamplitude,andagreeswith given by a complex number. Thus, the interpretation of Eq. (44) in the chiral limit (m 0). In the next section thefieldrenormalizationconstantasthecompositenessis we numerically study the com→positeness of the bound not as straightforwardas in the case of the bound state. state generated in the natural renormalization scheme. However,when the width is small, the imaginary partof Next we consider the small binding limit. Taking into therenormalizationconstantissmallandwecanreadoff account the convention of the scattering amplitude as the compositeness from its real part or from its absolute summarized in Appendix A, we write the coupling con- value. stantg inEq.(33)interms ofthe coupling strengthin W the chiral unitary approachas 7 The resonance state can be included in the extended complete- M q¯(M ) g2 = B g2. nessrelationusingthecomplexscalingmethod[51]. Thisisone W 16π3µMB q( B) possibilitytodefinethecompositeness ofresonances. −

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Compositeness of dynamically generated states in a chiral unitary approach PDF

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Preview Compositeness of dynamically generated states in a chiral unitary approach

Compositeness of dynamically generated states in a chiral unitary approach Tetsuo Hyodo∗ Department of Physics, Tokyo Institute of Technology, Meguro 152-8551, Japan Daisuke Jido Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502, Japan and J-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), 203-1, Shirakata, Tokai, Ibaraki, 319-1106, Japan Atsushi Hosaka Research Center for Nuclear Physics (RCNP), Ibaraki, Osaka 567-0047, Japan (Dated: January 5, 2012) 2 1 The structure of dynamically generated states in the chiral unitary approach is studied from a 0 viewpoint of their compositeness. We analyze the properties of bound states, virtual states, and 2 resonancesinasingle-channelchiralunitaryapproach,payingattentiontotheenergydependenceof thechiralinteraction. Wedefinethecompositenessofaboundstateusingthefieldrenormalization n constantwhichisgivenbytheoverlapofthebarestateandthephysicalstateinthenonrelativistic a J quantummechanics,orbytheresidueoftheboundstatepropagatorin therelativistic fieldtheory. Thefieldrenormalization constantenablesonetodefineanormalized quantitativemeasureofcom- 4 positenessoftheboundstate. Applyingthisschemetothechiralunitaryapproach,wefindthatthe boundstategeneratedbytheenergy-independentinteraction isalwaysapurelycompositeparticle, ] h while the energy-dependent chiral interaction introduces the elementary component, depending on t the value of the cutoff parameter. This feature agrees with the analysis of the effective interaction - l by changing the cutoff parameter. A purely composite bound state can be realized by the chiral c interaction only when the bound state lies at the threshold or when the strength of the two-body u attractive interaction is infinitely large. The natural renormalization scheme, introduced by the n property of the loop function and the matching with the chiral low-energy theorem, is shown to [ generate a bound state which is dominated by the composite structure when the binding energy is 2 small. v 4 PACSnumbers: 14.20.–c, 11.30.Rd,12.39.Fe 2 5 5 I. INTRODUCTION Theappearanceofhadronicmoleculesnearthethresh- . old regionis similar to the cluster phenomena in nuclear 8 structure, for instance, the appearance of the 0+ state 0 Recently, intensive attention has been paid to the dis- 2 1 cussion of hadronic molecular states, in which two (or of 12C close to the 3α threshold (the Hoyle state) [16]. 1 For hadrons, however, there are peculiar difficulties in more) hadrons are loosely bound by the interhadron : the discussion of the molecular states. For instance, be- v forces. In general, masses of such hadronic molecular i states lie close to some two-hadron threshold energies. causetheqq¯paircreationandannihilationcantakeplace X through the strong interaction, “the number of valence Classical examples are found in the light quark sector: r the Λ(1405) resonance below the K¯N threshold [1–3] quarks” is not a good classification scheme of the struc- a ture. In any case, the discrimination of the hadronic and the scalar mesons a (980) and f (980) close to the 0 0 K¯K threshold [4]. In relation with the recent progress molecules out of other structures is a subtle issue, as compared with the nuclear structure. To start with, we in the hadron spectroscopy at B factories, the charmo- nium X(3872) resonance is also discussed as the DD¯∗ need a theoretical framework with a clear definition of the hadroninc molecular states. molecule [5, 6]. It is important to clarify the property and the binding mechanism of the hadronic molecules Thestructureofaparticle,whetheritiselementaryor for the understanding of the nonperturbative aspects of composite, was intensively discussed in 1960s, using the the strong interaction. Various attempts to classify the field renormalization constant Z in nonrelativistic theo- internalstructure ofhadronshavebeen proposed,forin- ries[17–21]. A famousresultis foundin the study ofthe stance, by the N scaling method [7–12], by property c deuteron in Ref. [22], where a model-independent rela- changesalongwithchiralsymmetryrestoration[13],and tion of the compositeness with observables was derived by production rates in the heavy-ion collisions [14, 15]. in the small binding energy limit. The renormalization constantZ is first relatedto the p-n-dcoupling constant g and the binding energy of the deuteron B, which are then connected with the experimental observables such ∗ [email protected] as the scattering length and the effective range. In both 2 steps, small binding energy is assumed. The composite- II. CHIRAL UNITARY APPROACH ness through the field renormalization constant Z was also studied in a relativistic field theory [23], and there The meson-baryon scattering and baryon resonances have been severalattempts to extend the notion of com- havebeen welldescribed by the chiralunitary approach, positeness to resonances [24–27]. where the chiral low-energy theorem is combined with In this paper, we adopt the strategy to utilize the the unitarity condition of the scattering amplitude [29– field renormalization constant for the definition of the 32]. Adetailedformulationofthechiralunitaryapproach compositeness. First, following Ref. [22], we evaluate can be found in Ref. [33]. In this section we construct the field renormalization constant Z as the overlap of the single-channelversion of the chiral unitary approach the bare state in the free Hamiltonian with the physical along the same line with Ref. [34] and study the proper- bound state in the nonrelativistic quantum mechanics. ties of the generated states in the amplitude. It is our In addition to the universal relation in the weak bind- aim to establish a tractable model which will be used to inglimit[22],wealsoderivetheexpressioninexactform examine the argument of the compositeness in Sec. III. in terms of the scattering amplitude.1 Second, following Whileweadoptthemeson-baryonscatteringasanexam- Ref. [23], we define the field renormalization constant Z ple, the scattering of the Nambu-Goldstone (NG) boson as the residue of the full propagator of the bound state withaheavymesoncanbe describedinthe sameformu- in the Yukawa theory. This approach enables us to ana- lation (see Appendix C in Ref. [37]). The framework of lyzetheenergy-dependentinteraction. Itshouldbenoted the scattering of the NG bosons are the essentially the that we discuss the compositeness ofthe bound states in same, thanks to the chiral low energy theorem. the two-hadronscatterings,regardingthestable hadrons in the scattering states as elementary particles, in the sensethattheyformtheasymptoticstatesinthenonper- A. Construction of the scattering amplitude turbativeQCDvacuum. We donotconsiderthe internal structure of the hadrons in the asymptotic state. We consider a single-channel s-wave scattering of a We apply the notion of compositeness to the bound baryonwithmassM andaNGbosonwithmassm. The states in a chiral unitary approach [29–33]. In this chirallowenergytheorem[38,39]dictatestheleadingin- model, baryon resonances are described in the multi- teraction proportional to the boson energy ω W M ple scattering of the meson-baryon system whose inter- in the following form: ≃ − action is given by the chiral low-energy theorem. Al- though the generated resonance is expected to be a V(WT)(W)=C(W M), (1) − composite of a meson-baryon system, as pointed out in where C is a real-valued coupling constant with mass Ref.[34],the Castillejo-Dalitz-Dyson(CDD)pole contri- dimension minus two and W is the total center-of-mass butions [35,36] canbe hidden in the cutoff parameterin energy.2 Although the coupling constant is determined the renormalization procedure. To avoid this ambiguity, by the flavor content of the scattering system, here we the natural renormalization scheme has been proposed, simply regard it as a parameter of the model. The en- in which the subtraction constant is chosen to exclude ergydependenceoftheinteractionisaconsequenceofthe the CDD pole contribution from the loop function. To low energy theorem [40]. To appreciate the significance connect the argument of the compositeness to the natu- of the energy dependence, we also consider an energy- ral renormalization scheme, we discuss the properties of independent constant interaction whose strength is nor- the amplitude in the chiral unitary approachin detail. malized to the threshold value of the WT interaction: This paper is organized as follows. In Sec. II, we first analyzethepropertiesofthepolesingularitiesinthechi- V(const) =Cm=V(WT)(W =M +m). (2) ral unitary approach, using a single-channel model. We examine the energy-dependent chiral interaction and its ThisinteractiongeneratesthesameamplitudeastheWT energy-independent counterpart and discuss the proper- interaction only at the threshold, and the deviation be- tiesofthegeneratedstatesindetail. Next,inSec.III,we comes large in the energy region far from the threshold. define the compositeness of the bound state using field In relation with the pole structure of the Λ(1405) reso- renormalization constant Z in a general framework of nance[41,42], the energydependence ofthe chiralinter- field theories. We consider both nonrelativistic and rela- action has been recently studied [43, 44]. tivistic approaches,andexamine the limit of smallbind- The scattering amplitude can be obtained by solving ing. The numerical analysis of the compositeness of the the Bethe-Salpeter equation bound state in the chiral unitary approach is presented T(W)=V(W)+V(W)G(W)T(W), (3) in Sec. IV. 2 We may recover the usual WT interaction in the chiral unitary 1 A preliminary discussion on this issue has been given in a con- approachbyreplacingC→−Cg/(2f2)withthepiondecaycon- ferenceproceedings [28]. stantf andthegrouptheoretical factorCg. 3 with V(W) being V(WT)(W) or V(const). The loop func- real axis in the first Riemann sheet below the threshold, tion G(W) is given by while a resonance is represented by a pole in the second Riemann sheet of the complex energy plane above the d4q 2M 1 threshold. A pole on the second Riemann sheet below G(W)=i , (2π)4(P q)2 M2+iǫq2 m2+iǫ the thresholdiscalledavirtualstate,whichisa peculiar Z − − − (4) phenomenon in the s-wave scattering. The amplitude in the second Riemann sheet is given by whereP =(W,0). Thisintegraldivergeslogarithmically. With the dimensional regularization,we obtain 1 T (W)= V(W), (8) II 1 V(W)G (W;a) II − 2M m2 M2+W2 m2 2Mq¯(W) G(W;a)= a+ − ln G (W;a)=G(W;a)+i . (9) (4π)2 2W2 M2 II 4πW q¯(W) + ln[W2 (M2 m2)+2Wq¯(W)] Note that below the threshold W < M + m, q¯(W) is W − − purely imaginary. Therefore, on the real axis below the +ln[W2n+(M2 m2)+2Wq¯(W)] threshold, the loop function is real in both the first and − ln[ W2+(M2 m2)+2Wq¯(W)] thesecondRiemannsheets. Forlaterreference,wepoint − − − outthatG(W;a)[GII(W;a)]ismonotonicallydecreasing (increasing) function of W below the threshold. ln[ W2 (M2 m2)+2Wq¯(W)] , − − − − Because the interactions in Eqs. (1) and (2) have no ! o singularityinthecomplexenergyplaneexceptfor W (5) | |→ ,wededucetheconditionforthepoleoftheamplitude ∞ where a is the subtraction constant at the subtraction as pointµ =M,whichplaysaroleofthe cutoffparameter s of the loop function. The three-momentum variable is 1 V(zR)G(zR;a)=0 (bound state), − defined as (1 V(zR)GII(zR;a)=0 (virtual state/resonance), − (10) [W2 (M m)2][W2 (M +m)2] q¯(W)= − − − . (6) 2W p where zR is the energy of the bound state, virtual state, or resonance. The pole position of z should be on the G(W;a) is an analytic function of W in the whole com- R real axis in the first Riemann sheet, while z can be plexplane,exceptforontherealaxisabovethethreshold R complex in the second Riemann sheet. Around the pole where it has a branch cut. In conventional approaches, atW =z we canformally expandthe amplitude in the the subtraction constant has been used to fit the exper- R Laurent series as imental data. In Ref. [34], on the other hand, the nat- ural value of the subtraction constant anatural was intro- g2 ∞ duced by the theoretical argument to exclude the CDD T(W)= + T(n)(W z )n, (11) R polecontributionfromthe loopfunction. We discussthe W zR − − n=0 X influence of the subtraction constants to the scattering amplitude. A solution of Eq. (3) is given by with constant coefficients T(n). The first term can be interpretedasthe propagatorofaparticlewithmassz , R 1 T(W)= V(W), (7) which couples to the scattering states through the cou- 1 V(W)G(W;a) pling constant g. Hence, we identify the residue of the − pole in the amplitude T(W) as the energy-independent with the help of the on-shell factorization [30, 45]. The coupling strength g2. same form of the amplitude can also be obtained by the N/D method with the N = 1 prescription [31]. For the energy-independent interaction (2), no factor- B. Bound state with constant interaction ization is needed so that both the N = 1 prescription and the N = V prescription in the N/D method pro- vide the equivalent amplitude (7). Note that Eq. (7) is Letusfirstconsiderthecasewhenthesystemdevelops a bound state with energy-independent interaction (2). obtainedbyneglectingtheunphysical(left-handandcir- The amplitude has a pole on the real axis in the first cular) cuts, although the unitarity cut is properly taken Riemann sheet: z M , Im M =0, and M M intoaccount. Thus,thisframeworkshouldnotbeapplied R B B B ≡ ≤ ≤ M +m where M =M is the lower limit of the mass of tothesystemwithadeeplyboundstate,whichmightbe B theboundstate;otherwise,thesystembecomesunstable. influenced by the unphysical cuts. The pole condition (10) leads to Iftheinteractionissufficientlyattractive,boundstates and resonances can be dynamically generated. A bound 1 CmG(M ;a)=0. stateappearsasapoleofthefullamplitudeT(W)onthe B − 4 With this relation,wecandeterminethe value ofC such Because the whole amplitude T(W) is invariant, the that a bound state is generated at W = M with the property of the bound state remains unchanged. B subtraction constant a as 1 C(MB;a)= . (12) C. Bound state with WT interaction mG(M ;a) B Here we have indicated that C can be also regardedas a Next we deal with the energy-dependent WT interac- functionofM anda,whentheboundstateisgenerated. B tion(1). Theconditionfortheboundstate(10)becomes The loop function G is real below the threshold, so it is consistent with the real-valued coupling strength C. If 1 C(MB M)G(MB;a)=0. (15) − − M were complex, then the loop function G(M ) would B B The coupling strength g can also be calculated from the be complexand no solutionwouldbe foundfor Eq.(12). residue of the pole as This means that there is no pole in the first Riemann sheet except for on the real axis, in accordance with the [g(MB;a)]2 = lim (W MB)T(W) causality. W→MB − 1 Note that Eq. (12) is the relation among C, MB, and = (16) a. In general, the parameters in this model are the − G′(MB)+ GM(MB−BM;a) strength of the interaction C and the subtraction con- In this case, the coupling constant depends on the sub- stant a. When there is a bound state, the system can traction constant a, in contrast to Eq. (13). This is alsobecharacterizedbythemassoftheboundstateM B closely related to the energy dependence of the interac- anda, withthe helpofthe relation(12). Using Eq.(11), tion,becauseEq.(16)isobtainedbydifferentiatingboth wecancalculatethecouplingconstantgfromtheresidue the numerator and the denominator, and the energy de- of the amplitude T as [46] pendence of the interaction leaves the G(M ;a) term in B [g(M )]2 = lim (W M )T(W) the final expression. Note also that if G(M ;a) = 0, B B B W→MB − the result coincides with that of the energy-independent 1 1 interaction (13), which, however, means C be- = . (13) → −∞ − ∂G(W;a)(cid:12) ≡−G′(MB) cause of the bound state condition (15). This limit will ∂W (cid:12)W=MB be important in the discussion of the compositeness. (cid:12) Below the threshold, G(W;(cid:12)(cid:12)a) is real and monotonically As before, the condition (15) relates C, MB, and a, decreasing function of W [34], so we have G′(MB) < 0. so we can characterize the system by (MB, a) instead of Thisindicatesthatthecouplingsquareisalwaysrealand (C,a), keeping the mass of the bound state unchanged. positive: In this case, the interaction strength C such that the bound state appears at M is calculated as [g(M )]2 0, B B ≥ 1 which leads to the real-valued coupling constant. The C(MB;a)= . (17) (M M)G(M ;a) equality holds for M =M +m where G′(M ) . B − B B B Note also that the dependence on the subtracti→on−co∞n- The state with zero binding energy corresponds to the stant a vanishes after taking the derivative in Eq. (13), upperlimitforthemassoftheboundstateMB =M+m, becausethesubtractionconstantdoesnotdependonthe which is called zero energy resonance [47]. The coupling energy.3 Namely, the coupling of the bound state to the strength for such state, together with the natural value scattering state g is unambiguously determined by the of the subtraction constant [34] (the natural subtraction massoftheboundstateMB andindependentofthesub- constantanatural is also explainedin Appendix B), is the tractionconstanta. Therefore,throughtherelation(12), critical coupling introduced in Refs. [37, 48]: we can freely change the subtraction constant a and the 1 couplingstrengthC,withoutalteringthepropertyofthe Ccrit =C(M +m;anatural)= , mG(M +m;a ) natural bound state. In fact, this is one of the consequences of the invariance of the amplitude T(W) under the trans- which is the smallest coupling strength to generate a formation bound state. As discussed in Ref. [34], the variation of the subtrac- −1 1 2M a a ∆a, C ∆am . (14) tion constant can be absorbed by the introduction of a → − → C − (4π)2 poletermintheinteractionkernel,keepingtheamplitude (cid:18) (cid:19) T(W) invariant: a a ∆a, 3 In the present framework, we perform the single subtraction → − to tame the divergence in Eq. (5). We can introduce further V(W) V˜(W)=C(W M) C (W −M)2 , (18) subtractions, whichresultsinthe energy-dependent subtraction → − − (W M ) eff terms. In this case, the coupling constant depends on the sub- − tractionconstants. Weconsider thecasewiththegeneral inter- (4π)2 M =M + . (19) actionandthegeneralsubtractiontermsinAppendixB. eff 2MC∆a 5 Equation(18) showsthat the existence ofa pole atW = sheet. If the real part of the pole is higher (lower) than M in the interaction kernel V˜(W). The residue of this the threshold, the state is a resonance (virtual state). eff pole is calculated as For the energy-independent interaction, the pole con- dition (10) becomes (4π)4 lim (W M )V˜(W)= . (20) W→Meff − eff − 4M2C(∆a)2 G (z ;a)= 1 . II R Cm Both the mass and the residue are determined by ∆a. This means that it is not possible to introduce an arbi- The right-handside is real,while the left-handside is,in trary pole term by the variation of the subtraction con- general, complex. Therefore, at the pole position W = stant; the mass and the coupling of the pole term are z theimaginarypartoftheloopfunctionshouldvanish: R related to each other through Eqs. (19) and (20). This pole term cannot be absorbed by simple shift Im [GII(zR;a)]=0. (22) of the coupling strength of the interaction kernel as in Eq.(14). Actually,Eq.(18)canbewritteninthesimilar Becausethesubtractionconstantaisrealandadditively form to Eq. (14) as includedintheloopfunctionGII(zR;a),thisconditionis independentofaandisthe necessaryconditionforz to R −1 1 2M be the pole position. We first consider the virtual state a a ∆a, C ∆a(W M) . → − → C − (4π)2 − onthe realaxis, zR =MV, Im MV =0,andM MV < (cid:18) (cid:19) ≤ (21) M +m. Below the threshold MV < M +m, there is no discontinuity, so the imaginary part of the G (M ) II V This is the transformation which leaves the amplitude functionvanishesandthecondition(22)issatisfied. This T(W)invariantfortheWTinteraction. Inthecaseofthe means that a virtual state can be produced on the real energy-independent interaction, the change of the sub- axis for a weakly attractive interaction. tractionconstantcanbecompensatedbythesimpleshift Resonances are associated with the pole z with a fi- R of the coupling strength C [Eq. (14)], and the coupling nite imaginary part. As shown in Appendix C, the pole constant of the bound state to the scattering state g is trajectory is continuous in the complex energy plane as independent of the subtraction constant [Eq. (13)]. In thecouplingstrengthC isvariedwithafixedthesubtrac- contrast, for the WT interaction, the change of the sub- tion constant a. Thus, the resonance solution should be traction constant causes the energy-dependent transfor- connectedtothevirtualstatesolutionontherealaxisat mation[Eq.(21)]andthecouplingconstantgdependson someenergy. Tohavea resonanceatz withIm z =0, R R thesubtractionconstant[Eq.(16)]. Thesearetheimpor- Eq. (22) should be satisfied in the vicinity of the6real tant consequences of the energy dependence of the WT axisintheregionM Re z <M+m. Thiscanbeim- R interaction. As we see below, this difference is crucial to plemented if the deri≤vative of the imaginary part in the the structure of the bound state. imaginarydirectioniszero. However,Cauchy-Riemann’s Letusconsiderthepropertiesofthecouplingsquarein theorem leads to Eq.(16)whichmustberealandpositivetoobtainareal- valuedcouplingconstant. Becausethemassofthebound ∂ ∂GII(Re zR;a) Im G (z ;a) = >0, II R state should be larger than the mass of the constituent, ∂(Im z ) ∂(Re z ) R (cid:12)ImzR=0 R we have a condition (cid:12) (cid:12) for M Re zR < M +(cid:12)m. Thus, the existence of the M M 0. ≤ B branch of the pole trajectory to the resonance solution − ≥ is not allowed. This means that the pole with a finite It follows from Eq. (17) that G(M ;a) 0 for an at- B ≤ imaginarypartcannotappearfortheenergy-independent tractiveinteractionC <0 (equality holdsfor C ). →−∞ interaction. This is consistent with the interpretation in Therefore, G(M ;a)/(M M) 0 and we also have B B G′(M ) < 0. Thus, from E−q. (16≤), for the bound state the nonrelativistic potential model; simple attractive s- B wave potential has no centrifugal barrier and thus does generatedbytheattractiveWT interaction,wefindthat not produce a resonance state. the coupling square is always real and positive: Theresidueg2 canbeobtainedbythederivativeofthe [g(M ;a)]2 0. loop function B ≥ Equality holds for MB =M +m, where G′(MB)→−∞ [g(M )]2 = 1 . and a dependence of the coupling constant vanishes in V − G′ (M ) II V this limit. For the virtual state on the real axis Im M = 0, the V derivative of the loop function on the second Riemann D. Virtual states and resonances sheet is positive. Therefore, the coupling square is neg- ative. Because the pole singularities in the second Rie- Finally,weconsiderthecasewiththepoleonthecom- mann sheet are interpreted as Gamow states, the cou- plex energy plane at W = z in the second Riemann pling constant becomes pure imaginary. As in the case R 6 of the bound state with the constantinteraction,the de- III. FIELD RENORMALIZATION CONSTANT pendenceofthesubtractionconstantislostbytakingthe AND COMPOSITENESS derivative. For the WT interaction, the bound-state condition is In this section, we define the compositeness of the bound state in the scattering amplitude in the nonrel- 1 C(zR M)GII(zR;a)=0. (23) ativistic quantum mechanics [22] and in the relativistic − − field theory [23]. For this purpose, we describe the scat- Again,C isarealnumber,sotheequationtobesatisfied tering system with one bound state by the field theory at z is whose free spectrum contains one bare state and two- R body scattering states. We define the field renormaliza- Im [(z M)G (z ;a)]=0. (24) tionconstantZ astheoverlapofthephysicalboundstate R II R − and the bare state, which characterizes the elementarity Inthesamewayasintheenergy-independentinteraction, of the bound state. Expressing the constant Z in terms this condition is satisfied on the real axis z = M , so of physicalquantities, we obtain a master formula of the R V the virtual state can be formed on the real axis. As for compositeness. Withthismasterformula,wecandiscuss the resonance, in the present case, the compositeness ofbound states,either those observed in experiments or those obtained in a theory such as the ∂ chiral unitary approach. To be specific, we consider the Im [(z M)G (z ;a)] ∂(Im z ) R− II R s-wave scattering of a baryon ψ(JP = 1/2+) and a me- R (cid:12)(cid:12)ImzR=0 son φ(0−) in which a bound state B(1/2−) appears. It ∂ (cid:12) = [(Re zR M)GII(Re zR;(cid:12)a)] iseasytoderivesimilarexpressionsofthe compositeness ∂(Re z ) − R for other scattering systems with an s-wave bound state =GII(Re zR;a)+(Re zR−M)G′II(Re zR). (25) (see Appendix E and Ref. [23]). This can be zero for M Re z < M + m, R ≤ if the subtraction constant is properly chosen, be- A. Compositeness in nonrelativistic quantum cause G (Re z ;a) 0 from Eq. (23) and (Re z mechanics II R R M)G′ (Re z ) 0. ≤Namely, for the energy-dependen−t II R ≥ interaction, the pole can have the imaginary part and Let us first consider the two-body scattering system eventually an s-wave resonance can be produced. Sev- with one bound state in a nonrelativistic quantum me- eral examples of the s-wave single-channel resonance chanics. TheHamiltonianofthesystemH isdecomposed can be found in hadron physics, such as the σ meson into the free Hamiltonian H and the interaction V: 0 in the ππ scattering [13] and the lower energy pole of Λ(1405) [41, 42, 44]. H =H +V. 0 The condition (24) determines the relation between We assume that the eigenstates of the free Hamiltonian Re [z ] and Im [z ], which depends on the subtraction R R H are given by the continuum states labeled by the constant. Namely, for a given a, the pole trajectory is 0 relative momentum q and an elementary state4 B specified in the complex energy plane as a function of C 0 | i | i which is orthogonalto q . V stands for the interaction (for illustration, we show the trajectory of the pole po- | i among scattering states, as well as the coupling of the sition in Appendix C). The position of the pole is then scatteringstateto the elementarystate. B iselemen- determined by the coupling strength C and vice versa. 0 | i tary in the sense that it is an eigenstate of the Hamilto- The residue of the pole is nian H and the origin of B should be attributed to 0 0 | i 1 dynamics other than the interaction V.5 The eigenener- [g(zR;a)]2 =− G′ (z )+ GII(zR;a). (26) gies are given by II R zR−M H q =E(q) q H B = B B , 0 0 0 0 0 | i | i | i − | i For a virtual state (Im z = 0), the coupling square is R givenby the realnumber. Onthe realaxis, G′ (z ) and II R z M are positive, but the sign of the G (z ;a) de- R II R − pends on the value of the subtraction constant. Thus, 4 Ingeneral,wemayhaveseveralboundstatesaswellasmultiple the sign of the coupling square depends on the subtrac- scatteringchannels (seeAppendixD). tion constant and is determined by the balance of the 5 Onemayregard|B0iasaboundstateformedbyanotherHamil- first term and the second term in the denominator of tonianH′. UsingtheFeshbachprojectionformalismwithsingle Eq.(26). BecauseG′ (z )divergesatthe threshold,the bound state approximation [49, 50], we can set up this situa- coupling square is neIgIatRive for the virtual state close to tion. Thus, strictlyspeaking, |B0iisnotalways anelementary component; itshouldbeconsideredasaCDDpolecontribution thethreshold. Foraresonance(Im zR =0),thecoupling whichdoesnotoriginateinthepresentmodelspaceofthescat- constant is complex because of G , G6 ′ and (z M) tering. Inthispaper,however,weusetheword“elementary”for which are complex in the complexIIenergIIy plane.R − simplicity. 7 where E(q)=q2/(2µ) with µ being the reduced mass of Thisisanexactexpressionofthecompositeness1 Z . NR − the system. q and B0 are orthogonal and form the Nextweconsidertoexpresstheintegrandofthisequa- | i | i complete set: tion by the scattering amplitude. The formal solution of the Lippmann-Schwinger equation is written as q q′ =δ(q′ q), B B =1, 0 0 h | i − h | i B q =0, 1 h 0| i T(E)=V +V V. E H 1= B B + dq q q . (27) − 0 0 | ih | | ih | Z Inserting the complete set of the full Hamiltonian (29) and taking the matrix element by the initial and final We consider that the Hilbert space of the full Hamilto- scattering states with energy E, we obtain the Low’s nianalsocontainsoneboundstate B withthe binding | i equation [22], energy defined as B > 0. Namely, B is the eigenstate | i of the full Hamiltonian G (E)2 ∞ √E′ t(E′)2 t(E)=v+ | W | +4π 2µ3 dE′ | | , H|Bi=(H0+V)|Bi=−B|Bi, (28) E+B Z0 E−E′+iǫ p where B isnormalizedandorthogonaltothescattering where v (t) is the matrix element of the interaction | i states q,full as HamiltonianV (T operator)bythe scatteringstatesand | i we have used V q,full = T q . The second term of hq,full|q′,fulli=δ(q′−q), hB|Bi=1, thisequation,wh|ichcomiesfro|mtihe operator(E H)−1 B q,full =0, acting on the bound state discrete level, is a par−t of the h | i integrand of Eq. (30). Therefore, we can express the and they form the complete set as compositeness using the scattering amplitude as [28] 1= B B + dq q,full q,full . (29) ∞ √E | ih | | ih | 1 Z =4π 2µ3 dE t(E) v Z − NR Z0 E+B" − WenotethattheinteractionV shouldbeenergyindepen- p dent, to ensure the orthogonality and the completeness ∞ √E′ t(E′)2 4π 2µ3 dE′ | | . (31) of the Hilbert space of the full Hamiltonian. − Z0 E−E′+iǫ# WethendefinethefieldrenormalizationconstantZNR p astheprobabilityoffindingtheboundstateBinthebare Itisimportanttonotethatthescatteringamplitudet(E) state B : 0 is,ingeneral,complex,buttheimaginarypartofEq.(31) vanishes thanks to the optical theorem: Z B B 2. NR 0 ≡|h | i| This expresses the elementarity of the bound state B . Im t(E)= 12π 4π 2µ3√E t(E)2 | i − 2 · | | Itis clearfromthe completeness (27) that0 ZNR 1. = 4π2µq(Ep)t(E)2, (32) If B is a purely composite (elementary) ≤particle,≤we − | | | i obtain ZNR = 0 (ZNR = 1). Therefore, 1 ZNR serves where q(E) = √2µE is the momentum. Therefore, the − asthequantitativemeasureofthe“compositeness”ofthe imaginary part vanishes in the bracket in Eq. (31), and particle, which is given by the overlap Z is obtained as a real number. More NR explicitly, we can write 1 Z = dq q B 2. NR − |h | i| Z ∞ √E 1 Z =4π 2µ3 dE Re t(E) v Multiplying hq| to Eq. (28) from the left, we obtain − NR Z0 E+B" − p q V B 2 ∞ √E′ t(E′)2 1 Z = dq|h | | i| . 4π 2µ3 dE′ | | , − NR Z [E(q)+B]2 − PZ0 E−E′ # p Here the matrix element expresses the coupling strength where stands for the principal value integration. of the bound state to the scattering state with momen- P NowwetakethelimitB 0. ForsmallB,thebracket tum q. For an s-wave bound state, the coupling is in- → inEq.(31)isdominatedbytheboundstatepoletermin dependent of the angular variables and is a function of the T matrix, the energy. Then we can write q V B G (E), and W h | | i≡ obtain the energy integration form √E g2 Integrand in Eq. (31) W + (B0) , ∞ √E G (E)2 ≈E+B · E+B O 1 Z =4π 2µ3 dE | W | . (30) (cid:18) (cid:19) − NR Z0 (E+B)2 gW ≡GW(E =−B), p 8 because this is the only term with (B−1) for small E. of another baryon B (1/2−) as 0 O Then the E integration can be done analytically: =ψ¯(i∂/ M)ψ+ 1(∂ φ∂µφ m2φ2) L0 − 2 µ − 1 Z 4π 2µ3g2 ∞dE √E +B¯0(i∂/−MB0)B0. (34) − NR ≈ W (E+B)2 Z0 We consider the bound state of ψ and φ, which couples p g2 with the bare state B . Setting µ = Mm/(M +m), we =2π2 2µ3 W . (33) 0 √B obtainthesystemcorrespondingtothatdescribedbythe p Hamiltonian H0 in the previous section. The interaction This is the result of Ref. [22] which connects the com- Lagrangian,which corresponds to V in the previous sec- positeness ofthe bound state with the coupling constant tion, is introduced as a scalar-type Yukawa form, gliWmitanodf tthheebcoinudpilninggensterregnygBth.cSainncbeeZoNbRta≥ine0d,tahseupper Lint =g0ψ¯φB0+(H.c.), (35) whereg isthebarecouplingconstant. Wethenconsider 0 that the spectrum of the full theory has a bound state 1 B g2 . with mass M which is related to the binding energy as W ≤ 2π2s2µ3 B M =M +m B. The definition of the field renormal- B − izationconstant,whichwedenoteZ,istheresidueofthe TheequalitycorrespondstoZ =0,namely,thepurely NR full Green’s function at W =M , namely, B composite particle. Note that the upper limit decreases as we decrease the binding energy B and vanishes in the Z ∆(W)= . (36) weak binding limit B 0. W M B → − Itis importantto note thatinthe expression(33), the Forlaterconvenienceintheapplicationtothechiraluni- explicit dependence on the interaction V is lost when tary approach, we adopt the positive energy part of the B 0, and all the information of the interaction is rep- fermion propagator. → resentedbythecouplingconstantoftheboundstateg . W Let us calculate this Z constant with the Lagrangian Inthis sense, the resultis universalanddoes notdepend given in Eqs. (34) and (35). The free Green’s function ontheparticularchoiceoftheinteractionV. Incontrast, for B is given by 0 the exactformula(31)canbe appliedtothe boundstate with arbitrary binding energy. However, the result de- 1 ∆ (W)= . 0 pends on the matrix element v and, hence, onthe choice W M − B0 of the interaction Hamiltonian V. The full Green’s function ∆(W) is expressed in terms of From the expression of Eq. (30), we notice that the ∆ (W) by the Dyson equation: exact result of the compositeness can be calculated with 0 the knowledge of the energy dependence of the coupling ∆(W)=∆ (W)+∆ (W)g G(W)g ∆(W), 0 0 0 0 strength G (E) = q V B . In principle, G (E) rep- W W h | | i withtheunrenormalizedtwo-bodyloopfunctionofψand resents the coupling strength of the bound state to the φ, G(W). The solution of this equation is scattering state above the threshold, so it is an off-shell quantity. The coupling “constant” gW can be regarded 1 as the leading term in the expansion of G (E) around ∆(W)= W [∆ (W)]−1 g2G(W) E = B. In the present discussion, we assume that 0 − 0 − 1 the binding energy B and the coupling constant gW are = . W M g2G(W) known quantities. In this case, the approximation of − B0 − 0 1 Z with only known quantities should be in the form The divergence of the loop function G(W) is canceled − of Eq. (33) and any further contribution should depend by the infinite bare mass M . Using the renormalized on the choice of the interaction V because it is related B0 loopfunction G(W;a) with the parametera which char- withtheoff-shellquantityG (E). Toextendthemodel- W acterizesthefinitepart,thefullGreen’sfunctionisgiven independentformula(33),weneedtomeasurethehigher- by ordercoefficients ofthe expansionof G (E) experimen- W tally. 1 ∆(W)= . (37) W g2G(W;a) − 0 The renormalization condition can be obtained by re- B. Compositeness in relativistic field theory quiring the bound state pole at W = M .6 Equating B In this section, we define the compositeness following the method in Ref. [23]. To define the field renormaliza- 6 This procedure is in principle scale dependent. In the present tionconstant,weconsiderthe fieldtheorywith abaryon formulation, the renormalization scale is implicitly set at the fieldψ(JP =1/2+),amesonfieldφ(0−),andabarefield baryonmass,µs=M. 9 Eq. (37) with Eq. (36), we obtain the condition to interpret Eq. (41) as the compositeness, because the condition (40) is not always guaranteed and 1 Z may M =g2G(M ;a). − B 0 B become a complex number. Nevertheless, Eq. (41) may be applied to a narrowwidth resonance,because it coin- The field renormalization constant Z can be calculated cide with Eq. (39) in the zero width limit. as W M B Z = lim − W→MB W −g02G(W;a) IV. COMPOSITENESS OF DYNAMICALLY 1 GENERATED STATES = . (38) 1 g2G′(M ) − 0 B A. Application to chiral unitary approach The meson-baryon scattering amplitude is given by T(W) = g ∆(W)g , so the residue of the amplitude at 0 0 Here we analyze the compositeness of the bound state W =M is B ina specific model describedin sectionII.In the present lim (W M )T(W)=g2Z. section, we havederived three definitions of the compos- W→MB − B 0 iteness: The residue of the bound-state pole is the physical cou- (i) expression in nonrelativistic formalism, Eq. (31); pling constant squared, so we obtain the relation g2 =g2Z (ii) formula for small B, Eq. (33); 0 UsingthisrelationandEq.(38),wefinallywritethecom- (iii) expression in relativistic formalism, Eq. (39). positeness 1 Z as − For the application to the chiral unitary approach with 1 Z = g2G′(MB). (39) the energy-dependent interaction, it is fully legitimate − − to use Eq. (39). In the nonrelativistic formalism, we Note that the right-hand side is expressed only by the should choose the framework which can deal with the physical (renormalized) quantities, and no approxima- energy-dependent interaction. In this respect, the exact tion has been applied to the evaluation of the mass of result Eq.(31) cannot be used because the completeness the bound state. Thus, this is anexact expressionof the relation (29) is not always valid. However, the model- compositeness of the bound state in the relativistic field independent result for small B (33) may be appropriate theorywiththeinteractionLagrangian(35),whichisex- because it can be derived without using the complete- pressedintermsofthecouplingstrengthgofthephysical ness of the full Hamiltonian [22]. Thus, in the follow- bound state to the scattering state and the derivative of ing, we examine the expression of the compositeness in the loop function at the energy of the bound state. The the relativistic field theory in Eq. (39) and the model- expressionforthescalarmesonboundstateinthemeson- independent result for small binding energy, Eq. (33). meson scattering is derived in Appendix E. It should be Substituting Eqs. (13) and (16) into the exact result noted that the definition of the compositeness (39) de- in the field theory (39), we obtain pends onthe adopted interactionLagrangian . With int L a different interaction, the function form of Eq. (39) will 1 constant interaction, bemodifiedaccordingly,and,hence,theresultingexpres- sion for 1 Z will be changed. 1 Z = 1 WT interaction. MThe Mpos−+sibmle, wrehgeiorenGof′(tMhe )boisunredalstaantde nmeagsastiivseM, an≤d − 1+ (MBG−(MM)BG;a′()MB) B B (42) the c≤oupling square is always positive for the physical  bound state. Therefore, we find that the compositeness It is striking that the constant interaction always pro- is always positive, duces a purely composite bound state. This corresponds to the equivalence of the four-Fermi theory and Yukawa 1 Z 0. (40) − ≥ theory found in Ref. [23]. On the other hand, the com- In other words, the field renormalization constant is positenessoftheboundstategeneratedbytheWTinter- smaller than unity. action depends on the subtraction constant. This result The field renromalization constant is well defined for agrees with the fact of the appearance of the pole term resonances. Using the same argument, we find in the effective interaction for the WT interaction (18), which introduces the source of the elementarity. For a 1−Z =−g2G′II(zR), (41) general interaction V, using Eq. (D2), we obtain whereg2 isthe residueofthe pole inthe complex energy 1 plane of the scattering amplitude. This is a natural ex- 1−Z =1 V′(MB)G(MB), (43) tension of Eq. (39). However, it is not straightforward − V(MB)G′(MB) 10 where V′(M ) = ∂V(W)/∂W . In this expres- Notethatboththemomentumfactorsq( B)andq¯(M ) B |W=MB − B sion,GandV aredeterminedunderafixedrenormaliza- are pure imaginary. Substituting this expression into tion scheme. It is easy to see that Eq. (42) follows from Eq. (33), we obtain [28] thegeneralexpression(43)withtheinteractionkernel(1) and(2)inthephenomenologicalrenormalizationscheme. 1 Z = M|λ1/2(MB2,M2,m2)| g2, (46) Inaddition,theenergydependenceoftheinteractionker- − NR 16πM2(M +m M ) B − B nel is related to the deviation of the compositeness from forB 0. Thisform,incomparisonwiththerelativistic unity. → result (39), leads to the relation Let us recall the definition of the bound state that lMowBt−heMth≥res0haonldditnhdeicbaethesavGio′(rMofBt)h<e lo0.opRfeulnactitoionn(1b7e)- 1−Z ≈ −16πMMB2λ(1M/2(+Mm2,−MM2,Bm)G2)′(MB) (1−ZNR) implies G(M ;a) 0, so altogether we obtain (cid:18) | B | (cid:19) B ≤ A(M )(1 Z ) (47) B NR ≡ − G(M ;a) B 0, forthesmallbindingenergy. NotethatthefactorA(M ) (M M)G′(M ) ≥ B B − B dependsonlyonthemassoftheboundstate. Inthesmall binding limit, the definition of the field renormalization and, hence, with Eq. (40), for the WT interaction, we constantZ shouldbe consistentwith thatofZ ,so we obtain NR expect 0 1 Z 1. lim A(M )=1. ≤ − ≤ B MB→M+m This result indicates that the compositeness defined in Wewillcheckthevalidityofthisrelationinthenumerical Eq.(42)isnormalizedwithintherangefromzerotoone. calculation. This is a very useful aspect to discuss the structure of Finally we briefly mention the compositeness for vir- the bound state in a quantitative manner. tual states and resonances. The present formulation in For the WT interaction, a purely composite bound thenonrelativisticquantummechanicsisnotdirectlyap- state (Z =0) is achieved when plicabletoresonanceswhicharenotincludedinthecom- G′(M )= or G(M ;a)=0. (44) plete set in Eq. (27) and in Eq. (29).7 However, the B B −∞ residue of the resonance propagator can be evaluated without trouble in the relativistic field theory. This pro- Because the derivative of the loop function diverges at vides a natural extension of the compositeness of bound threshold, the first condition can be met if the bound states to virtual states and resonances. Equation (41) stateisproducedexactlyatthethreshold,irrespectiveof leads to thevalueofthesubtractionconstant. ForM =M+m, B 6 apurelycompositeboundstatewouldbeproducedonlyif 1 constant interaction, GTh(MusB, t;oa)reisalzizeeroZ, w=hi0c,hwinedsihcoautelds Cha→ve−∞ by Eq. (17). 1−Z =1+ G′IIG(zI1RI()z(RzR;a−)M) WT interaction. M =M +m or C . (45) (48) B →−∞  Notethatthecondition(45)isonlysatisfiedinthelimit- Foravirtualstate(Im zR =0),theexpressionprovidesa ingcasessoitimpliesthatthephysicalboundstate(finite real value for the compositeness. The virtual state gen- bindingenergyandfiniteinteractionstrength)cannotbe erated by the contact interaction always gives Z = 0, purely composite for the WT interaction. It is instruc- while the WT interaction can lead to Z < 0, depending tive to recall that the natural renormalization condition on the subtraction constant. This is because the virtual developed in Ref. [34] requires G(M;a ) = 0. In- statecannotbeincludedintheorthonormalbasis. Inthe natural terestingly, this condition was introduced to realize the case of a resonance,the field renormalizationconstant is hadronicmoleculestateintheamplitude,andagreeswith given by a complex number. Thus, the interpretation of Eq. (44) in the chiral limit (m 0). In the next section thefieldrenormalizationconstantasthecompositenessis we numerically study the com→positeness of the bound not as straightforwardas in the case of the bound state. state generated in the natural renormalization scheme. However,when the width is small, the imaginary partof Next we consider the small binding limit. Taking into therenormalizationconstantissmallandwecanreadoff account the convention of the scattering amplitude as the compositeness from its real part or from its absolute summarized in Appendix A, we write the coupling con- value. stantg inEq.(33)interms ofthe coupling strengthin W the chiral unitary approachas 7 The resonance state can be included in the extended complete- M q¯(M ) g2 = B g2. nessrelationusingthecomplexscalingmethod[51]. Thisisone W 16π3µMB q( B) possibilitytodefinethecompositeness ofresonances. −

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