Unveiling the K1 (1270) double-pole structure in the B ¯ →j /ψρ K ¯ and B ¯ →j /ψ K ¯ ∗π decays

Unveiling the K1ð1270Þdouble-pole structure in the ¯ B→J=ψρ ¯ Kand ¯ B→J=ψ¯ Kπdecays J. M. Dias,*G. Toledo ,1,†L. Roca,2,‡and E. Oset3,§ 1Instituto de Física, Universidad Nacional Autónoma de M´exico, AP 20-364, Ciudad de M´exico 01000, M´exico 2Departamento de Física, Universidad de Murcia, E-30100 Murcia, Spain 3Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, 22085, 46071 Valencia, Spain (Received 22 February 2021; accepted 24 May 2021; published 22 June 2021) By looking at the pseudoscalar-vector meson spectra in the ¯ B→J=ψρ¯ Kand ¯ B→J=ψ¯ Kπweak decays, we theoretically investigate the double-pole structure of the K1ð1270Þresonance by using the chiral unitary approach to account for the final-state interactions between the pseudoscalar (P) and vector (V) mesons. The K1ð1270Þresonance is dynamically generated through these interactions in coupled channels and influences the shape of the invariant mass distributions under consideration. We show how these shapes are affected by the K1ð1270Þdouble-pole structure to confront the results from our model with future experiments that might investigate the PV spectra in these decays. DOI: 10.1103/PhysRevD.103.116019 I. INTRODUCTION The observation of the axial-vector mesons K1ð1270Þ and K1ð1400Þ[1,2] were identified as the expected 1þ strange mesons from the quark model. Subsequent experiments have explored their properties and decay modes [3]. While the dominant K1ð1270Þdecay channel is ρK, the K1ð1400Þis observed to decay mostly through the Kð892Þπone. These states have usually been studied in terms of the mixing of the strange states of the JPC ¼1þþ and JPC ¼1þ−nonets (see, for example, Refs. [4–6]). Other exhaustive analyses of the 1þlow-lying mesons as dynamically generated resonances found that the S¼1and I¼1=2poles were not compatible with the above assignment, but rather they should be identified as a double-pole structure for the K1ð1270Þ[7]. The two-pole structure for the K1ð1270Þresonance is not unique: there are several cases where two poles were found for hadronic resonances, and a recent review can be seen in Ref. [8]. The discovery of the two-pole structure of the K1ð1270Þtriggered studies looking for scenarios where this prediction could be tested. The analysis of the K−p→K−πþπ−pdata at 63 GeV done in Ref. [2] provided additional support to the existence of two states: one with a mass of 1195 MeV coupling mostly to the Kð892Þπchannel, and one with a mass of 1284 MeV coupling to the ρKone. Several reactions aimed at observing these two states were proposed, such as D0→ πþVP [9], which is similar to B−→J=ψK− 1ð1270Þ with the hadronization involving three light mesons. More recently, another study considered the Dþ→ νlþK1ð1270Þdecay [10], identifying the signatures in the invariant-mass distributions of the decaying K1ð1270Þ. On the other hand, expected improvements in the experimental capabilities to study B-meson decays with higher statistics, like in the Belle II experiment, make these proposals an interesting scenario to look for. In this work we provide an additional reaction, considering decays of the form ¯ B0→J=ψVP, where VP are the vector and pseudoscalar meson pairs, ρ¯ Kand ¯ Kπ, using the chiral unitary approach, and we look for signatures of the two K1ð1270Þstates. A related study was done in Ref. [11], where the reaction B−→J=ψρ0K−was studied to look for signals of the Zcð4000Þ; however, simultaneously one K1ð1270Þshowed up in the ρ¯ Kmass distribution. Here we also look into the ¯ Kπchannel in order to see both K1ð1270Þstates. The work is organized as follows. In Sec. II we present the formalism for the elementary production at the quark level, where the different VP channels are related by SUð3Þ arguments. Then, we account for the final VP interaction by implementing meson-meson scattering based on the chiral unitary approach. In Sec. III we compute the *[email protected] †[email protected] ‡[email protected] §[email protected].es Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 103, 116019 (2021) 2470-0010=2021=103(11)=116019(13) 116019-1 Published by the American Physical Society invariant-mass distribution for the VP pair and its structure in terms of the individual poles, and identify the regions where the signature of such poles can be extracted. II. FORMALISM A. VP pseudoscalar and vector meson production The relevant contribution for the ¯ B→J=ψVP reaction is given at the quark level by the diagram shown in Fig. 1. The mechanism starts with a b¯ dquark pair, forming the initial ¯ B0meson, in which the bquark is converted into a cquark by emitting a Wboson, which then produces an anticharm ¯ c along with a strange quark s. In the end, we are left with a c¯ cpair making up a J=ψmeson, considered as a spectator, and a s¯ dpair. In order to produce a pseudoscalar as well as a vector meson, a ¯ qq pair with the quantum numbers of the vacuum is added to the already existing s¯ dpair, according to the 3P0model [12–14]. Therefore, the final mesonmeson hadronic state has the following quark flavor combination: jHi¼jsð¯ uu þ¯ dd þ¯ ssÞ¯ di:ð1Þ However, Eq. (1) above only refers to the quark content of the final mesonic states and it does not carry any information about the pseudoscalar or vector nature of those hadronic states. This is done by defining the q¯ qmatrix, denoted as M, written as M¼0 B B @ u¯ uu ¯ du ¯ s d¯ ud ¯ dd ¯ s s¯ us ¯ ds ¯ s 1 C C A ;ð2Þ in terms of which Eq. (1) reads jHi¼jsð¯ uu þ¯ dd þ¯ ssÞ¯ di ¼X ijs¯ qiqi¯ di¼jM3iMi2i¼jðMMÞ32i:ð3Þ The final meson-meson components are found by establishing the correspondence between Mand the pseudoscalar and vector meson SUð3Þmatrices, that is, M⇒P¼ 0 B B B B B B @ π0 ffiffi2 pþη ffiffi3 pþη0 ffiffi6 pπþKþ π−−π0 ffiffi2 pþη ffiffi3 pþη0 ffiffi6 pK0 K−¯ K0−1 ffiffi3 pηþffiffi2 3 qη0 1 C C C C C C A ; ð4Þ and M⇒V¼0 B B B @ ρ0 ffiffi2 pþω ffiffi2 pρþKþ ρ−−ρ0 ffiffi2 pþω ffiffi2 pK0 K−¯ K0ϕ 1 C C C A ;ð5Þ where the standard η−η0[15] and ω1−ω8mixings have been used for Pand V, respectively, in order to match the right flavor content of the matrix M. Since we aim at describing a reaction with a pseudoscalar along with a vector meson as final states, the matrix Min Eqs. (4) and (5) should be combined according to Eq. (3) in such a way that it gives the product PV or VP. There is nothing in our model that privileges one over the other, and thus we consider an equal-weighted combination between them in Eq. (3) so that it can be rewritten as jHi¼jðPVÞ32iþjðVPÞ32i:ð6Þ Therefore, the pseudoscalar and vector mesons produced in the reaction are jHi¼jρþ¯ K−i− 1 ffiffiffi 2 pjρ0¯ K0iþjK−πþi− 1 ffiffiffi 2 pj¯ K0π0i þ1 ffiffiffi 2 pjω¯ K0iþjϕ¯ K0i;ð7Þ where a term corresponding to the ¯ K0ηchannel has canceled out in the evaluation of Eq. (6) by using Eqs. (4) and (5). Note that the procedure adopted provides the final VP meson-meson components as well as their relative weights, which will play a significant role in the mass spectrum. The final jHistate can be written in the isospin basis by considering the following multiplets: ð−ρþ;ρ0;ρ−Þ, ð¯ K0;−K−Þ, and ð−πþ;π0;π−Þfor the ρ,¯ K, and πmesons, FIG. 1. Dominant diagram for the ¯ B→J=ψðPVÞreaction at the quark level. In the first step, a bquark decays into a cquark by emission of a gauge boson W, which then produces a strange quark salong with a ¯ cquark. Finally, we are left with a c¯ cpair, forming the J=ψ, and the s¯ dpair. This latter pair is hadronized in order to produce a pseudoscalar-vector meson pair emerging as the final state. J. M. DIAS, G. TOLEDO, L. ROCA, and E. OSET PHYS. REV. D 103, 116019 (2021) 116019-2 respectively. Then, the final VP states in isospin I¼1=2 are given by jρ¯ KiI¼1=2 I3¼1=2¼ffiffiffi 2 3 rjρþK−i−ffiffiffi 1 3 rjρ0¯ K0i; j¯ KπiI¼1=2 I3¼1=2¼−ffiffiffi 2 3 rjK−πþiþ ffiffiffi 1 3 rj¯ K0π0i:ð8Þ Using this, we can recast jHias jHi¼ ffiffiffi 3 2 rjρ¯ KiI¼1=2 I3¼1=2−ffiffiffi 3 2 rj¯ KπiI¼1=2 I3¼1=2 þ1 ffiffiffi 2 pjω¯ KiI¼1=2 I3¼1=2þjϕ¯ KiI¼1=2 I3¼1=2:ð9Þ This last equation also provides the relative weights, denoted as hi, in the isospin basis between the ith VP channels above. They are hρ¯ K¼ffiffiffi 3 2 r;h ¯ Kπ¼−ffiffiffi 3 2 r; hω¯ K¼1 ffiffiffi 2 p;h ϕ¯ K¼1:ð10Þ The differential decay width for the ¯ B→J=ψðPVÞ process, illustrated in Fig. 2,isgivenby dΓ dMinv ¼1 ð2πÞ3 1 4M2 B pJ=ψ ˜ pπð¯ KÞjTB→J=ψðPVÞj2;ð11Þ where MBis the ¯ B-meson mass while pJ=ψand ˜ pπð¯ KÞare the momentum associated with J=ψin the ¯ Brest frame and πð¯ KÞmesons in the PV rest frame, respectively. As a function of the PV invariant mass, Minv, they are pJ=ψ¼λ1=2ðM2 B;M 2 J=ψ;M 2 invÞ 2MB ;ð12Þ ˜ p¯ K¼λ1=2ðM2 inv;M2 ρ;m 2 ¯ KÞ 2Minv for the ρ¯ Kchannel;ð13Þ ˜ pπ¼λ1=2ðM2 inv;M 2 ¯ K;m 2 πÞ 2Minv for the ¯ Kπchannel;ð14Þ where λstands for the Käll´en function. For the evaluation of the full decay amplitude, which is needed in Eq. (11), we have to consider the diagrams in Fig. 3, where the final-state interaction mechanism is implemented to take into account the K1ð1270Þresonance contribution for the invariant-mass spectra we are interested in. Since we are interested in the distributions with ρ¯ Kand ¯ Kπas final VP states, we have T¯ B→J=ψρ ¯ K¼Cð ϵψ· ϵρÞhρ¯ KþX i hiGiðMinvÞtI¼1=2 i→ρ¯ KðMinvÞ ð15Þ and T¯ B→J=ψ¯ Kπ ¼Cðϵψ·ϵ¯ KÞh¯ KπþX i hiGiðMinvÞtI¼1=2 i→¯ KπðMinvÞ; ð16Þ where the index i, running from 1 to 4, stands for each possible VP channel involved in the loop, and Ccontains the information of the strength of the weak decay at the tree level. The channels are i¼1for ϕ¯ K,i¼2for ω¯ K,i¼3 for ρ¯ K, and i¼4for ¯ Kπ. These loops are represented by GiðMinvÞwhich is the G-loop function (given as a function of the invariant mass Minv) which we will discuss in Sec. II B. The hiare the relative weights in the isospin basis, defined in Eq. (10) above. Moreover, ϵψ,ϵρ, and ϵ¯ K are the polarization vectors for the J=ψ,ρ, and ¯ Kmesons, respectively. Furthermore, the amplitudes tI¼1=2 i→ρ¯ Kand tI¼1=2 i→¯ Kπin Eqs. (15) and (16) are the two-body scattering amplitudes for all possible transitions from the ith channel to ρ¯ K(and to ¯ Kπ), which in our approach encode the resonance K1ð1270Þas dynamically generated through these interactions [16], and are explained in the following subsection. B. Final-state interaction and the K1ð1270Þresonance Once the final meson-meson pair is produced at tree level in the ¯ B→J=ψVP reaction, they undergo final-state interaction from which the K1ð1270Þresonance emerges dynamically. In fact, in Ref. [16] this resonance was dynamically generated through the s-wave interaction between the pseudoscalar and vector mesons in the I¼1=2channel. The final-state interaction mechanism is introduced by adopting a unitarization procedure using the Bethe-Salpeter equation in coupled channels, from which some hadronic states show up as poles in the FIG. 2. Amplitude for the ¯ B→J=ψðPVÞdecay. The Cconstant is the parametrization of the weak vertex. UNVEILING THE K1ð1270ÞDOUBLE-POLE STRUCTURE …PHYS. REV. D 103, 116019 (2021) 116019-3 unphysical Riemann sheets of the scattering matrices. This approach is a unitary extension of chiral perturbation theory, called the chiral unitary approach [17–19], which has allowed to describe many hadronic resonances as composite states of mesons and/or baryons. In particular, in Ref. [16] the transition amplitudes tI¼1=2 i→jappearing in Eqs. (15) and (16) were unitarized by solving a coupledchannel scattering equation in an algebraic form, written in matrix form as t¼ð1−VGÞ−1V; ð17Þ where in Vij,tij the indices stand for the coupled channels: 1 for ϕ¯ K, 2 for ω¯ K, 3 for ρ¯ K, 4 for ¯ Kπ, and 5 for ¯ Kη.In addition, Vij is the interaction kernel, which corresponds to the tree-level amplitudes evaluated for all channels we are considering in this work by using the chiral Lagrangians from Ref. [20] given by LVVPP ¼− 1 4f2Trð½Vμ;∂νVμ½P; ∂νPÞ;ð18Þ where fis the pion decay constant (f¼93 MeV) and V,P are the matrices given in Eqs. (4) and (5). Furthermore, GkðsÞis the meson-meson loop function associated with the kth channel, which can be regularized either by dimensional or cutoff regularization schemes. In the present work, we follow Ref. [16] which employed the former scheme. In this case, the GkðsÞloop function is given by Gkðffiffiffi s pÞ¼ 1 16π2aðμÞþln M2 k μ2þm2 k−M2 kþs 2sln m2 k M2 kþqk ffiffiffi s p½lnðs−ðM2 k−m2 kÞþ2qkffiffiffi s pÞþlnðsþðM2 k−m2 kÞþ2qkffiffiffi s pÞ −lnð−sþðM2 k−m2 kÞþ2qkffiffiffi s pÞ−lnð−s−ðM2 k−m2 kÞþ2qkffiffiffi s pÞ;ð19Þ where Mkand mkstand for the vector and pseudoscalar meson masses in the kth channel, respectively. Moreover, aðμÞis the subtraction constant and in this work we take aðμÞ¼−1.85 for μ¼900 MeV, which is the scale of dimensional regularization, obtained in Ref. [16] by fitting the experimental K−p→K−πþπ−pdata. In addition, qk¼j qkjis the on-shell three-momentum of the meson in the loop, given in the center-of-mass frame by qk¼λ1=2ðs; M2 k;m 2 kÞ 2ffiffiffi s p:ð20Þ The ρand ¯ Kmesons have a relatively large width and hence a wide mass distribution. In order to take this feature into account in our formalism, we convolve the loop function GkðsÞwith the corresponding vector meson spectral function Im½DðsVÞ ¼ Im1 sV−M2 VþiMVΓV;ð21Þ where MVstands for the vector meson mass and ΓVis the vector meson width, considered here as energy independent. Choosing an energy-dependent form for ΓV,aswas done in Refs. [9,10], does not provide any significant change in our results compared with the usual uncertainties of our approach. The spectral function above is related to the exact propagator for the vector meson by using the Lehmann representation, which gives us DðsÞ¼− 1 πZ∞ sth dsV Im½DðsVÞ s−sVþiϵ;ð22Þ where sth is the corresponding vector meson threshold for the decay channels with the ρor ¯ Kmesons. Therefore, the convolution of the GkðsÞloop function defined in Eq. (19) FIG. 3. Relevant diagrams contributing to the amplitude T¯ B→J=ψðPVÞimplementing the final-state interaction. The first diagram on the right-hand side corresponds to the tree level. On the other hand, the second one with a loop encodes the final-state interaction mechanism, and actually it is a sum over all PiVipseudoscalar and vector mesons associated with the ichannel: i¼1for ϕ¯ K,i¼2for ω¯ K,i¼3for ρ¯ K, and i¼4for ¯ Kπ. J. M. DIAS, G. TOLEDO, L. ROCA, and E. OSET PHYS. REV. D 103, 116019 (2021) 116019-4 with the vector meson spectral function given by Eq. (22) provides Gðffiffiffi s p;M k;m kÞ ¼RðMVþ2ΓVÞ2 ðMV−2ΓVÞ2dsVGðffiffiffi s p;ffiffiffiffiffi sV p;m kÞ×Im½DðsVÞ RðMVþ2ΓVÞ2 ðMV−2ΓVÞ2dsVIm½DðsVÞ ;ð23Þ where the limits ðMV2ΓVÞ2are considered to be a reasonable cut in the integration above. These cuts cause a small deviation of the normalization of the Breit-Wigner distribution encoded in the spectral function in Eq. (21) and in order to reestablish it we divided by the normalization integral defined in the denominator in Eq. (23). By looking for poles of Eq. (17) in unphysical Riemann sheets of the complex ffiffiffi s pvariable, two poles are found in the I¼1=2channel, a broader one at ffiffiffiffiffi sp p¼ð1195 −i123ÞMeV, and a narrower one at ffiffiffiffiffi sp p¼ ð1284 −i73Þwhere, for poles not very far from the real axis, ffiffiffiffiffi sp pcan be approximated by ffiffiffiffiffi sp p¼ðMp−iΓ=2Þin which the real part stands for the pole mass, whereas the imaginary one is associated with half the width. For the sake of convenience, we shall refer to the former and latter as pole Aand B, respectively. For the sake of completeness, we show in Table Ithe parameters obtained in Ref. [16] for the numerical calculation of Eq. (17) described in this section, and the couplings to the ith-channel gAðBÞ iof each K1ð1270Þpole. This will be important in order to study the behavior of the distributions given in Eq. (11) considering each pole contribution individually. The amplitudes given in Eq. (17) contain the information about the whole dynamics for the VP interaction, including the resonance structure. Although both poles are intertwined in the highly nonlinear dynamics involved in the amplitude of Eq. (17),itis also interesting for illustrative purposes to differentiate the contribution from each individual pole. Since it is not possible to directly isolate each pole contribution from Eq. (17), this task can be achieved by adopting a Breit-Wigner approach for those amplitudes. Then, at the pole position, we have tIAðBÞ i;j ¼gAðBÞ igAðBÞ j s−sp ;ð24Þ where spis the pole position of the K1poles Aand B, whereas gAðBÞ iðjÞstands for the coupling of the iðjÞth channel to the pole AðBÞ. We know that the closer to the real axis these poles are, the better this approximation works. In addition, it is expected that experimentally these amplitudes are parametrized by a Breit-Wigner form so that, by adopting it in our formalism, a comparison between our results and those from future experiments is more reasonable. Furthermore, this parametrization can also be used to encode the double-pole K1structure if we assume that the amplitudes are given by a double-Breit-Wigner shape defined as Ti;j ¼tIA i;j þtIB i;j;ð25Þ where tIA i;j and tIB i;j are given by Eq. (24) for poles Aand B, respectively. It is interesting to mention that the chiral Lagrangian of Eq. (18) can be deduced from a more general framework— the local hidden gauge approach [21–24]—by exchanging vector mesons. This framework allows us to address a related source of interaction based on the exchange of pseudoscalar mesons. We address these two issues in Appendixes Aand B, respectively. Examples of reactions similar to ours, which look carefully into the final-state interactions of the mesons produced, are the D0→K0 Sπþπ−reaction studied in Ref. [25] and the Dþ→K−πþπþreaction studied in Refs. [26,27]. In Ref. [25] one of the mesons was kept as a spectator, while Refs. [26,27] dealt with a three-body interacting system. In Ref. [26] the transition amplitude was obtained as the sum of amplitudes classified in terms of isospin, based upon the dominant modes of weak decay, which are external and internal emission [28]. The finalstate interaction was taken into account by means of the TABLE I. Parameters obtained in Ref. [16]. In the first line, the values for the subtraction constant α, the scale of dimensional regularization μ, and the two K1poles are given. In the lower part, we list the channels and their corresponding couplings gAðBÞ ito each K1pole Aand B. αðμÞμscale Lower pole (A) Higher pole (B) −1.85 900 ð1195 −i123ÞMeV ð1284 −i73ÞMeV Channels Couplings gA igB i ϕ¯ K2096 −i1208 1166 −i774 ω¯ K−2046 þi821 −1051 þi620 ρ¯ K−1671 þi1599 4804 þi395 ¯ Kπ4747 −i2874 769–11171 UNVEILING THE K1ð1270ÞDOUBLE-POLE STRUCTURE …PHYS. REV. D 103, 116019 (2021) 116019-5 Omn`es representation in terms of experimental phase shifts. More detailed, and using models for the final-state interactions, is the work of Ref. [27] from which we can draw conclusions concerning our present work. The first consideration to be made is that while undoubtedly these works do a very good job concerning the final-state interaction of the meson components, they rely upon free parameters, some of which depend directly on the reaction studied. For instance, Ref. [26] required seven complex parameters that were adjusted to the data of the reaction, and in Ref. [27] the number of parameters adjusted to the data was of the order of 20, depending on the options. The use of these formalisms in a new reaction would contain unknown parameters. Here we benefit from the fact that we consider the J=ψinteraction with the light mesons to be weak, as found in the study of coupled channels in Ref. [29], and hence we only have to worry about the vector-pseudoscalar interaction of ρ¯ Ktogether with ¯ Kπ. On the other hand, we do not pretend to reproduce the whole phase space of the reaction, but rather a narrow region of the ρ¯ Kand ¯ Kπinvariant masses around the peaks of the K1ð1270Þresonances that we find. For this purpose, the work of Ref. [7] for the vector-pseudoscalar interaction, which predicted two K1ð1270Þresonances that were tested against data of the K−p→K−πþπ−preaction of Ref. [2] in Ref. [8], is sufficiently accurate. Also, by limiting ourselves to a narrow region we do not have to worry about possible contributions from scalar and tensor terms, which were considered in Refs. [26,27]. At this point, we have to address a problem concerning the VP interaction that was not considered in Ref. [7]. Indeed, in Refs. [7,16] the source of VP interaction was given by vector exchange extracted from the local hidden gauge approach [21–23]. We prove in Appendix Athat the vector exchange interaction leads to the contact chiral interaction of Ref. [20] used in Refs. [7,16]. However, there is another source of interaction based on the pseudoscalar exchange, as depicted in Fig. 7of Appendix B. This interaction was considered in Refs. [26,27,30]. The reason not to consider it is analogous to a similar source of interaction considered in the VV interaction in Refs [31,32]. Indeed, in these works, this new source of interaction was taken into account via the box diagram of Fig. 8in Appendix B. What was found there was that the real part of the new potential was negligible, and only the imaginary part, due to the large phase space for ρρ →ππ decay, was relevant. We take the opportunity to do the equivalent work here, and this is done in Appendix B.In Figs. 9,10,11, and 12 we show new mechanisms contributing to the VP interaction which involve pseudoscalar exchange. We find in all cases a very small contribution of a few percent relative to the large terms coming from vector exchange. It is interesting to see that this conclusion agrees with the observation made in Ref. [27], where the two interaction mechanisms—vector exchange and pseudoscalar exchange —were explicitly considered [see Figs. 4(a) and 2 of Ref. [27], respectively]. The authors of Ref. [27] stated that “we found that the effect of the diagram Fig. 4(a) connected to ðπþπ0ÞI¼1 P¯ K0is the most important among the three-body type diagrams that we consider”. The pseudoscalar exchange part for the VP interaction was considered as a Zgraph in Fig. 2 of Ref. [27]. If one selects the Z graphs related to the VP interaction, it is found that the vector exchange has a larger impact on the χ2than the pseudoscalar exchange [33]. One should add that, using Eq. (18), one finds that the strength of the vector exchange for the I¼1=2interaction that we consider here is twice as large as the one in the I¼3=2case (ρþ¯ K0) that is produced in the Dþdecay in Ref. [27], and it is attractive in I¼1=2, while it is repulsive for I¼3=2. This further magnifies the relevance of vector exchange in our case. (a) (b) FIG. 4. (a) dΓ=MρþK−invariant-mass distribution for the ¯ B0→J=ψρþK−reaction (black solid line), compared with the curves obtained by considering only pole A(red dotted line) and pole B(blue dot-dashed line). (b) dΓ=MK−πþdistribution for ¯ B0→ J=ψK−πþchannel also compared with distributions due to each pole contribution separately. BW is the Breit-Wigner parametrization, whereas UChPT stands for chiral unitary theory. J. M. DIAS, G. TOLEDO, L. ROCA, and E. OSET PHYS. REV. D 103, 116019 (2021) 116019-6 III. RESULTS In Figs. 4(a) and 4(b) we show the VP invariant-mass distributions for ¯ B0→J=ψρþK−and ¯ B0→J=ψK−πþ reactions, respectively. The dashed lines (labeled UChPT) represent the results obtained using the unitarized amplitudes for the two-body VP final-state interaction [Eq. (17)]. The solid lines (labeled BW) represent the curves obtained if we parametrize the two-body VP amplitudes in Eq. (17) by using a double Breit-Wignerlike shape [Eq. (25)]. In Fig. 4we also show the individual contributions of both poles A (dotted line) and B (dotdashed line) in the Breit-Wigner approach. Note that the phase spaces for both ¯ B0→J=ψρþK−and ¯ B0→J=ψK−πþdecays take nonzero values below the corresponding VP threshold as a consequence of the convolution with the vector meson spectral function in order to take into account the finite widths of the ρand ¯ K mesons. This effect is especially relevant for the ρ¯ K channel. On the other hand, the global normalization factor in Eqs. (15) and (16) is the same for both decay channels, and it does not play a relevant role in our results since what matters is the relative strengths and shapes between the mass distribution of the different channels and mechanisms considered. Actually, this global normalization, C,isthe only free parameter in our model. It is worth noting that the chiral unitary approach used in this case has a range of applicability up to about 1500–1600 MeV in the invariant mass. We plot the distributions in the whole range, but one should bear in mind that the predictions for the high invariant masses are less reliable. A first clear observation from Fig. 4is that the K1ð1270Þresonant shape dominates the distributions at low invariant masses. However, each distribution is mainly manifesting a different pole associated with the K1ð1270Þ. In fact, one would expect from the values of the couplings shown in Table Ithat the pole A would manifest more in the ¯ Kπdistribution and the pole B in ρ¯ K. Indeed, we see in Fig. 4(a) that the ρþK−mass distribution has a pronounced peak at 1284 MeV, which is just the energy region where the highest K1ð1270Þpole emerges (pole Bin Table I). On the other hand, in Fig. 4(b) the K−πþdistribution peaks at 1185 MeV, which is the energy region dominated by the lowest K1pole (pole Ain Table I). In addition, the former mass spectrum is narrower than the latter, manifesting the fact that pole B, which couples mostly to ρ¯ K, is the narrower one, with a width around 146 MeV. By contrast, pole A, with a width equal to 246 MeV, is broader than the pole Band couples mostly to ¯ Kπ, and then causes the peak in the K−πþdistribution to be wider. The previous discussion is also applicable if we look at the BW curves obtained by using the double Breit-Wignerlike amplitudes [Eq. (25)]. The reason for the difference between the UChPTand BW curves is that the unitarization amplitudes in Eq. (17) contain the full VP dynamics and not just the resonant information. We see that this difference is more relevant for the ρ¯ Kdistribution. If we look at the individual contributions of the different poles, we clearly see the dominance of pole Bfor the ρþK−case and pole Afor the K−πþcase. Finally, we study the relative importance of the tree-level contribution in Fig. 3compared to the final-state interaction (implemented here by using UChPT or the Breit-Wigner approach). This is shown in Fig. 5where we confront the results obtained considering only the resonant part (dashed lines) with those for the whole mechanism: tree-level plus resonant parts (solid lines). We can see that for the ¯ Kπ channel the shapes of the UChPT curves with and without the tree-level contributions are similar in strength but shifted by about 50 MeV. For the double-pole Breit-Wigner parametrization the effect of turning off the (a) (b) FIG. 5. (a) dΓ=MρþK−invariant-mass distribution for the ¯ B0→J=ψρþK−reaction with and without the tree-level mechanism. (b) dΓ=MK−πþdistribution for the ¯ B0→J=ψK−πþchannel also compared with the distribution without interference between tree level and the resonant part of the amplitude. UNVEILING THE K1ð1270ÞDOUBLE-POLE STRUCTURE …PHYS. REV. D 103, 116019 (2021) 116019-7 tree-level contribution is more visible in the strength of the spectra than in the shift of the curves. It decreases the maximum strength of the peak by half its value. For the ρ¯ K channel the UChPT curves exhibit a noticeable difference in their shapes at high energies, but the strength is not altered much in the resonant region. IV. CONCLUSIONS We have theoretically investigated the double-pole structure of the K1ð1270Þresonance, which was shown in Ref. [16] to be dynamically generated through the pseudoscalar-vector meson interaction in coupled channels, by looking at the invariant-mass distributions for the ρ¯ K and ¯ Kπpairs, respectively, in the ¯ B→J=ψρ¯ Kand ¯ B→ J=ψ¯ Kπreactions. The final-state interaction mechanism was implemented employing the chiral unitary approach, in which the pseudoscalar-vector meson interaction gives rise to the two K1ð1270Þpoles that, in our model, affect both the ρ¯ Kand ¯ Kπdistributions differently. This feature allowed us to unveil the double-pole structure in these reactions. In particular, we have shown that the ρ¯ Kdistribution in the ¯ B→J=ψρ¯ Kreaction is dominated by the contribution from the K1highest mass pole, whereas the lowest mass pole contributes more for the ¯ Kπdistribution in the ¯ B→J=ψ¯ Kπdecay. As we have pointed out, this is due to the values of the coupling constants of those poles to the different channels considered in this work—more specifically, the ρ¯ Kand ¯ Kπchannels. On the other hand, it is important to stress that even though it is possible to see one pole dominate over the other in each distribution, both VP spectra still have the two K1ð1270Þpoles contributing to their shapes. In view of that, we have also modeled the two-body dynamics by using a double-pole BreitWigner parametrization such that the contributions of the two poles could be disentangled. In this case, one expects to observe the manifestation of each pole separately in the VP spectra to which each resonance couples most strongly. An experimental investigation of those reactions would be most welcome to shed light on the nature of K1ð1270Þ. ACKNOWLEDGMENTS We thank S. X. Nakamura for fruitful discussions. This work is partly supported by the Spanish Ministerio de Economia y Competitividad and by Generalitat Valenciana under contract PROMETEO/2020/023, and European FEDER funds under Contracts No. FIS2017-84038-C21-P B and No. FIS2017-84038-C2-2-P B. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824093 for the STRONG-2020 project. APPENDIX A: VP INTERACTION IN THE LOCAL HIDDEN GAUGE APPROACH We evaluate the VP interaction in the local hidden gauge (LHG) approach [21–24] through vector exchange, as depicted in Fig. 6. For this we borrow the VVV and VPP Lagrangians from the LHG, given by LVVV ¼ighðVμ∂νVμ−∂νVμVμÞVνi;ðA1Þ where g¼MV=2f(MV≈800 MeV, f¼93 MeV) and LVPP ¼−ighVμ½P; ∂μPi;ðA2Þ where h…istands for the trace in SUð3Þand V,Pare the matrices given in Eqs. (4) and (5). The chiral Lagrangian of Eq. (18) can be obtained from these Lagrangians in the following way. First, we make the approximation that the three-momenta of the vector mesons are very small with respect to the vector meson mass. This is so in our particular case, and hence one takes the limit of negligible three-momenta versus the vector meson mass. In this case, the vector field Vνin Eq. (A1) cannot correspond to an external vector of Fig. 6. This is so because if it were an external vector, then ν¼1, 2, 3 since ϵ0¼0when pV¼0. But then we have ∂νwhich gives rise to a vector threemomentum that is zero. Then,Vνis the V0vector exchanged in Fig. 6and one has a structure like in the VPP Lagrangian, only with the extra factor ϵμϵ0μ¼−ϵ0·ϵfor the external vectors. The amplitude for the diagram of Fig. 6is then given as −it¼−gðVμ∂νVμ−∂νVμVμÞijVν ji i q2−M2 V Vν0 lm½P;∂ν0Pml; ðA3Þ where the indices i,j,l,mare the matrix indices of Pand V in Eqs. (4) and (5) written explicitly to obtain the traces. FIG. 6. VP interaction through the exchange of vector mesons. J. M. DIAS, G. TOLEDO, L. ROCA, and E. OSET PHYS. REV. D 103, 116019 (2021) 116019-8 Since X pol ϵν jiϵν0 lm ¼−gνν0þqνqν0 M2 Vδjlδim;ðA4Þ we readily obtain, neglecting the term qνqν0=M2 Vconsistently with the approximations done, −it ¼−ig2 M2 VhðVμ∂νVμ−∂νVμVμÞ½P; ∂νPi;ðA5Þ and hence L¼− 1 4f2h½Vμ;∂νVμ½P; ∂νPi;ðA6Þ which is the chiral Lagrangian of Ref. [20], as shown in Eq. (18). This equivalence was already shown in a particular case for the ρπ interaction in Ref. [24]. Here we have made a general derivation. As shown in Ref. [7], the s-wave projected potential for the transition of channel ito jis given by Vij ¼Cij ϵ·ϵ0 8f23s−ðM2þm2þM02þm02Þ− 1 sðM2−m2ÞðM02−m02Þ;ðA7Þ where M,m,M0,m0are the initial and final vector and pseudoscalar masses, respectively, ϵand ϵ0are the polarization vectors of the initial and final vectors, and Cij are coefficients given in Table II of Ref. [7]. Of relevance here are the coefficients CρK;ρK¼CKπ;Kπ¼−2, CρK;Kπ¼1=2. APPENDIX B: PSEUDOSCALAR EXCHANGE IN THE VECTOR PSEUDOSCALAR INTERACTION In Ref. [30] it was pointed out that a source of the VP interaction is given by the diagram in Fig. 7. This interaction was also considered in Ref. [27] in the study of the VP interaction in the Dþ→K−πþπþreaction; however, this was done in addition to the vector exchange discussed here in Appendix A, and it was found that the vector exchange is far more important than the contribution of pseudoscalar exchange [27,33]. We address this issue here in connection with our VP channels ρ¯ K,¯ Kπthat we have in the problem under study. The effect of pseudoscalar exchange was already addressed in the study of the vector-vector interaction in Refs. [31,32]. The box diagram of Fig. 8was evaluated exactly with the full structure of the four intermediate propagators. Then, the VV →VV potential obtained there was added to the one obtained from VV →VV with a single vector exchanged discussed in Appendix Aand the whole potential was iterated with the Bethe-Salpeter equation. It was found that the real part of the box diagrams was negligible compared to the vector exchange, but the imaginary part provided a source of decay for the found VV molecular bound states. This was relevant because the bound VV states without this term had no width except for a small one when considering the width of the vector mesons. However, the PP intermediate states have a small mass and provide a large phase space for the decay. The box gave rise to a width of the VV states but no change in their mass. The equivalent diagram to Fig. 8for the VP interaction, based on Fig. 7, is given in Fig. 9. We evaluate its contribution here for the case of our interest ρ¯ K→ρ¯ K and the term is depicted in Fig. 10. However, unlike in the case of VV, where we need two steps in the reaction VV →PP ×PP →VV to obtain a VV →VV interaction term, here the single step VP →PV driven by pseudoscalar exchange already provides a VP → PV interaction term. In order to quantify the relevance of the pseudoscalar exchange versus vector exchange, we compare the contributions of Fig. 11(a) with Fig. 11(b) and Fig. 12(a) with Fig. 12(b). We find it sufficient to evaluate the diagrams close to the ρKthreshold to benefit from the approximations discussed FIG. 8. Box diagram in the VV interaction given by Pexchange and intermediate PP states. FIG. 7. Source for VP interaction through the exchange of pseudoscalars. UNVEILING THE K1ð1270ÞDOUBLE-POLE STRUCTURE …PHYS. REV. D 103, 116019 (2021) 116019-9