Signatures of the two K 1 (1270) poles in D + →ν e + VP decay
Guan-Ying Wang, Luis Roca, En Wang, Wei-Hong Liang, Eulogio Oset
aa r X i v : . [ h e p - ph ] F e b Signatures of the two K (1270) poles in D + → νe + V P decay
Guan-Ying Wang
School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan, 450001, China
Luis Roca
Departamento de F´ısica. Universidad de Murcia. E-30100 Murcia. Spain.
En Wang
School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China
Wei-Hong Liang
Department of Physics, Guangxi Normal University, Guilin 541004, China andGuangxi Key Laboratory of Nuclear Physics and Technology,Guangxi Normal University, Guilin 541004, China
Eulogio Oset
Departamento de F´ısica Te´orica e IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigacion de Paterna, Apdo 22085, 46071 Valencia, Spain
We analyze theoretically the D + → νe + ρ ¯ K and D + → νe + ¯ K ∗ π decays to see the feasibility tocheck the double pole nature of the axial-vector resonance K (1270) predicted by the unitary exten-sions of chiral perturbation theory (UChPT). Indeed, within UChPT the K (1270) is dynamicallygenerated from the interaction of a vector and a pseudoscalar meson, and two poles are obtained forthe quantum numbers of this resonance. The lower mass pole couples dominantly to K ∗ π and thehigher mass pole to ρK , therefore we can expect that different reactions weighing differently thesechannels in the production mechanisms enhance one or the other pole. We show that the differentfinal V P channels in D + → νe + V P weigh differently both poles, and this is reflected in the shape ofthe final vector-pseudoscalar invariant mass distributions. Therefore, we conclude that these decaysare suitable to distinguish experimentally the predicted double pole of the K (1270) resonance. PACS numbers:
I. INTRODUCTION
Semileptonic B and D meson decays have been for longconsidered as a good source to learn about non perturba-tive strong interactions, given the good knowledge of theweak vertex [1–3]. Refined methods have become avail-able more recently [4, 5] and the reactions are lookedupon with interest to even learn about physics beyondthe standard model [6, 7]. Explicit calculations correlat-ing a vast amount of data with the help of some selectedpieces of experimental information are also available [8].One of the relevant cases of these reactions consist on D meson decays leading to resonances in the final state,rather than the ordinary ground state of mesons, usuallystudied. In particular, semileptonic decays of hadronswhere the final hadron is a resonance are specially in-teresting. In this sense, the B and B s semileptonic de-cays leading to D ∗ (2400) and D ∗ s (2317) resonances werestudied in Ref. [9]. Similarly, the D decays into the scalarmesons f (500), K ∗ (800), f (980) and a (980) were ad-dressed in Ref. [10], with relevant results concerning thenature of these scalar mesons. A review of these and re-lated reactions can be seen in Ref. [11]. In this direction,the recent observation of the D + → νe + ¯ K (1270) reac-tion measured by the BESIII collaboration [12] offers anew opportunity to study the properties and nature of the K (1270) axial-vector resonance. Prior to this mea-surement the CLEO collaboration presented results onthe D + → νe + ¯ K (1270) [13], but the quality of data ismuch improved in the BESIII measurements. Interest-ingly there are theoretical results on these reactions inRefs. [1, 2] using quark models, in Ref. [14] using QCDsum rules and factorization approach, in Ref. [15] usinga covariant light front quark model and in Ref. [16] us-ing light cone sum rules. The branching ratios obtained,within 10 − − − , agree qualitatively with the one mea-sured by BESIII of about 2 . × − .Our interest in this reaction stems from the findingsof Refs. [17, 18] that there are two K (1270) resonancesinstead of one. The idea of the present work is to seewhich are the particular measurements in the BESIII re-action that could show evidence of these two states, forwhich we do theoretical calculations looking into particu-lar final channels. The standard quark model picture formesons and baryons [19–23] has the great value to cor-relate a great amount of data on hadron spectroscopy,but the axial vector meson states are systematically notso well reproduced as the vector ones [19, 23]. With thisperspective it is not surprising that other pictures havebeen proposed to explain these states. The chiral uni-tary approach [24, 25] was applied to the study of thepseudoscalar-vector meson interaction, using the chiralLagrangian of Ref. [26] and it was shown that the inter-action, unitarized in coupled channels, gave rise to boundstates or resonances which could be identified with thelow lying axial-vector resonances [17, 27, 28]. An appeal-ing feature of these dynamically generated resonances isthat the reaction mechanisms producing them proceed ina different way than ordinary mechanisms that produceresonances. Indeed, one does not produce the resonancesdirectly, rather one produces the meson-meson compo-nents of the different coupled channels, which upon fi-nal state interaction among themselves generate the res-onances. This allows one to perform calculations andrelate many production channels, and often leads to par-ticular features in the invariant mass distributions [11].Concerning axial-vector meson production in different re-actions, work has been done recently in the study ofthe J/ψ → η ( η ′ ) h (1380) reaction [29], τ − → ν τ P A with P = π, K and A = b (1235), h (1170), h (1380), a (1260), f (1285) [30] and χ cJ decay to φh (1380) [31],among others quoted in those works.In Refs. [17, 18] it was shown that there were two K (1270) states, which coupled differently to the coupledchannels. One state appears at 1195 MeV and couplesmostly to K ∗ π . The other state appears at 1284 MeVcoupling mostly to ρK . In Ref. [18] some reactions dis-closing these final states were studied and it was shownthat they peaked at different energies, and the state ofhigher mass had a smaller width. The existence of thetwo K (1270) is directly linked to the chiral dynamicsof the problem and is similar to the appearance of thetwo Λ(1405) states in the baryon strange sector [32–34](see the review “ Pole structure of the
Λ(1405) region ”in the PDG [35]). With this picture in mind some reac-tions have been proposed to provide extra evidence of theexistence of these two K (1270) states. In Ref. [30] the τ − → ν τ K − K (1270) reaction is proposed looking at the ρK and K ∗ π final decay products of the K (1270) andtwo distinct peaks are seen in the results. In Ref. [36]the D → π + ρ ¯ K and D → π + π ¯ K ∗ reactions are alsosuggested in order to see the two peaks corresponding tothe two K (1270) resonances.In the present work, taking advantage of the re-cent BESIII measurement [12], we look at the D + → νe + ¯ K (1270) reaction, evaluating explicitly the decays¯ K (1270) → ρ ¯ K and ¯ K (1270) → π ¯ K ∗ showing thatthese final channels give different weights to the two K (1270) resonances and lead to invariant mass distri-butions that differ in the position and the shape. In viewof the results obtained here we can only encourage theBESIII collaboration to perform the analysis that we sug-gest here, which should shed valuable light on the issue ofthe two K (1270) states and the nature of the low lyingaxial-vector resonances. D + uu + dd + ss W d dsc s e + P, VV, P ν FIG. 1: Elementary D + → νe + V P process at the quark level.
II. FORMALISM
As explained in the introduction, within the chiralunitary approach (UChPT) of Refs. [17, 18], the axial-vector resonances are generated dynamically by the non-linear chiral dynamics involved in the unitarization pro-cedure of the elementary
V P scattering potential in s-wave, and there is no need to include them as explicitdegrees of freedom (by means of Breit-Wigner like am-plitudes or similar). (We refer to [17] for the semi-nal work on the UChPT approach for the axial-vectorresonances, and to [18, 36–38] for brief but illustrativesummaries). In particular, for the strangeness S = 1and isospin I = 1 / K (1270)resonances, looking at unphysical Riemann sheets of theunitarized V P scattering amplitudes. The poles are lo-cated at (1195 − i − i
73) MeV, wherewe can identify the real part with the mass and the imagi-nary part with half the width. In Table IV of Ref. [18] thevalues of the different couplings to the different
V P chan-nels can be seen. The main observation is that the lowermass pole couples dominantly to K ∗ π and the highermass pole to ρK , but the couplings to the other V P channels are not negligible, and are actually considered.Following this philosophy, the way to produce a dynam-ically generated K (1270) resonance in a particular re-action is to create first all possible V P pairs and thenimplement their final state interaction. This later issuewill be addressed in the second part of this section butfirst we need to discuss the calculation of the elemen-tary production of the
V P states, and its depiction, atthe quark level, can be seen in Fig. 1. First the c quarkproduces an s quark through the Cabibbo favored ver-tex W cs and then hadronization into a final vector anda pseudoscalar meson is implemented by producing anextra ¯ qq with the P model [39–41].We are mostly interested in evaluating the relativeweight and momentum dependence of the different chan-nels modulo a global arbitrary normalization factor. Thedifferent weights among the allowed V P channels can beobtained from the following SU (3) reasoning.The flavor state of the final hadronic part after the ¯ qq is produced in the hadronization is | H i ≡ | s (¯ uu + ¯ dd + ¯ ss ) ¯ d i , which can be written as | H i = X i =1 | M i q i ¯ d i = X i =1 | M i M i i = | ( M ) i , (1)where we have defined q ≡ uds and M ≡ q ¯ q ⊺ = u ¯ u u ¯ d u ¯ sd ¯ u d ¯ d d ¯ ss ¯ u s ¯ d s ¯ s . (2)The hadronic states can be identified with the physicalmesons associating the M matrix with the usual SU (3)matrices containing the pseudoscalar and vector mesons: M ⇒ P ≡ π √ + η √ + η ′ √ π + K + π − − √ π + η √ + η ′ √ K K − ¯ K − η √ + η ′ √ ,M ⇒ V ≡ √ ρ + √ ω ρ + K ∗ + ρ − − √ ρ + √ ω K ∗ K ∗− ¯ K ∗ φ , (3)where the usual mixing between the singlet and octet togive η and η ′ [42] has been used in P . Also in the V ma-trix, ideal ω - ω mixing has been considered to produce ω and φ , to agree with the quark content of M in Eq. (2).Since the M in Eq. (1) can refer either to V P or P V ,we need to evaluate the contribution(
V P ) + ( P V ) = ρ + K − − √ ρ ¯ K + K ∗− π + − √ K ∗ π + 1 √ ω ¯ K + φ ¯ K (4)where we see that the ¯ K ∗ η channel has been cancelledmathematically and the η ′ is neglected because of itslarge mass as done in the original work of the V P in-teraction that generated the axial-vector K (1270) [17].The numerical coefficients in Eq. (4) in front of each V P channel provide the relative strength of the different
V P channels.The momentum structure of the amplitude corre-sponding to the mechanism in Fig. 1 can be evaluatedin a similar way to what was done in Refs. [9, 10]. In-deed, the amplitude, T , for the process of Fig. 1 can befactorized into the weak part and the hadronization part,and then it will be proportional to L µ Q ν V Had (5) where global constant factors are omitted since we willperform the calculations up to a global normalization. InEq. (5) L µ = ¯ u ν γ µ (1 − γ ) v l is the leptonic current and Q µ = ¯ u s γ µ (1 − γ ) u c the quark current. The hadroniza-tion part V Had will be discussed later on.When evaluating the D decay width of this process,we will need to square the amplitude and sum over thequark polarizations which gives (see Ref. [10] for explicitdetails and calculation)12 X pol | T | = 4 | V had | m l m ν m D M VP ( p l · p D )( p ν · p VP ) . (6)where p i are the four-momenta of the corresponding par-ticles, m i the masses, and the VP label refers to the final V P pair, which will eventually account for the K (1270)resonance.The final expression for the V P invariant mass, M VP ,distribution of the D + → νe + V P decay can be obtainedin the same way as in Ref.[10] (see the derivation leadingto Eq.(23) of Ref. [10]) and gives d Γ dM VP = 2(2 π ) m D M VP Z dM eν M eν | p VP | | ˜ p ν | | ˜ p V |× (cid:18) ˜ E D ˜ E VP − | ˜ p D | (cid:19) | V Had | (7)where M eν is the eν invariant mass and | p VP | = 12 m D λ / ( m D , M eν , M ) θ ( m D − M eν − M VP ) , | ˜ p V | = 12 M VP λ / ( M , m V , m P ) θ ( M VP − m V − m P ) , | ˜ p ν | = M eν , ˜ E D = m D + M eν − M M eν , ˜ E R = m D − M eν − M M eν , (8)with λ and θ standing for the K¨all´en and step functionsrespectively and we have neglected the positron mass.One of the main ingredients in the calculation of thehadronic part is the implementation of the final state in-teraction of the V P pairs produced in the mechanismof Fig. 1, which is depicted in Fig. 2. Note that, sincethe K (1270) resonance is generated dynamically withinour approach, it is not produced directly but, instead,the different V P pairs are produced and then rescatterinfinitely many times which is accounted for by the uni-tarized
V P scattering amplitude.Taking into account the six different possible interme-diate
V P pairs, ( K ∗− π + , ρ + K − , ¯ K ∗ π , ρ ¯ K , ω ¯ K and φ ¯ K ) the hadronic part of the amplitude for the decayinto the i − th final V P channel can be written as D + D + + + . . . PV = + PV + V VPPe + ν ν e + b)a) FIG. 2:
V P final state interaction. V Had ( D + → νe + V i P i ) = V p h i + X j =1 h j G j T I=1 / j,i (9)where V p is an arbitrary global normalization factor,which includes the weak coupling constant among otherfactors stemming from the quark matrix elements, h i arethe numerical coefficients in front of each V P state inEq. (4), G j is the vector-pseudoscalar loop function [18]and T I=1 / j,i are the unitarized ( V P ) j → ( V P ) i scatter-ing amplitude in isospin 1/2 from Ref. [18]. These arethe amplitudes that manifest the double pole structurein the complex energy plane associated to the K (1270).Note that in Ref. [18] the V P states are in isospin basisand here we are working with explicit charge basis, butwe can easily transform from one to the other basis usingthat | ρ ¯ K i I = ,I = = r | ρ + K − i − √ | ρ ¯ K i , | ¯ K ∗ π i I = ,I = = − r | K ∗− π + i + 1 √ | ¯ K ∗ π i . (10)Note that these unitarized V P scattering amplitudesdo not necessarilly have a Breit-Wigner shape in the realaxis (see explicit plots in Refs. [18, 36]). They actuallycontain the information of the whole
V P dynamics andnot only the resonant structure. However, in a actual ex-periment one would typically try to fit Breit-Wigner likeshapes and therefore we will also compare in the resultssection the results using for the scattering amplitudes T ij = g i g j s − s p , (11)where s p is the pole position which can be identified withthe mass and width of the generated resonances √ s p ≃ M R − i Γ R / g i are the couplings of the resonance tothe i − th V P channel which can be obtained from theresidues of the amplitudes at the pole positions and canbe found in Table IV of Ref. [18]. III. RESULTS
We first show in the left panels of Fig. 3 the differentcontributions to the
V P invariant mass distribution for the D + → νe + K ∗− π + and D + → νe + ρ + K − . The abso-lute normalization is arbitrary, but the relative strengthbetween the different curves and the different channelsare absolute (There is only a global normalization con-stant, the same for all the channels, see Eq. (9)). Thelabel “unitarized” stands for the results using for the V P → V P amplitudes, T I=1 / ij , the unitarized modelfrom Ref. [18], as explained above. These curves arecompared to the results using, instead, the explicit Breit-Wigner like shapes of Eq.(11), labeled as “BW poles” andalso considering the contribution of only the lower masspole ( A ) or the higher mass pole ( B ). The “tree level”curve represents the result removing the final V P stateinteraction, i.e. only the mechanism of Fig. 2a), whichis accounted for by considering only the first h i term inEq. (9). We have also implemented a convolution withthe final vector meson spectral function, in the same wayas in Ref. [36], in order to take into account the final vec-tor meson widths. This is specially relevant for the ρ ¯ K case due to the large width of the ρ meson and the factthat the ρK threshold lies around the K (1270) energyregion.We see that the invariant mass distributions in these D + decays are clearly dominated by the K (1270) res-onant contribution but the curves are clearly differentin shape and position of the peaks for the two finalchannels considered. Actually in the K ∗− π + channelthe peak of the distribution is located around 1160-1180 MeV, depending whether we use the unitarizedor the Breit-Wigner amplitudes for the V P scattering.However, for the ρ + K − distribution the curve peaksat around 1250-1270 MeV and is considerable narrower.This is a clear manifestation of the different weight thatthe two K (1270) poles have in both channels. In-deed, for the K ∗− π + final channel, the distribution isclearly dominated by the lower mass pole, the one at √ s p = (1195 − i K ∗ π . In the ρ + K − chan-nel the individual poles have a more comparable strengthamong themselves but the higher mass pole, the one at(1284 − i
73) MeV, shifts the final strength to higher en-ergies and narrows the distribution.It is also worth noting, however, that there is an im-portant interference effect between different mechanisms,particularly with the tree level contribution. This isclearly seen by comparing to the right panels, which havebeen evaluated removing the tree level terms, i.e. con-sidering only the mechanisms in Fig. 2b). This is whatone would obtain if the background, non-resonant termscould be ideally removed. In this later case the distribu-tions would more clearly manifest the effect of the indi-vidual poles. M K *- π + [MeV] d Γ / d M K * - π + [ a r b it r a r y un it s ] unitarizedBW polesBW pole ABW pole Btree level D + ν e + K *- π + (a) M K *- π + [MeV] d Γ / d M K * - π + [ a r b it r a r y un it s ] unitarizedBW polesBW pole ABW pole Btree level D + ν e + K *- π + (b) M ρ + Κ − [MeV] d Γ / d M ρ + Κ − [ a r b it r a r y un it s ] unitarizedBW polesBW pole ABW pole Btree level D + ν e + ρ + Κ − (c) M ρ + Κ − [MeV] d Γ / d M ρ + Κ − [ a r b it r a r y un it s ] unitarizedBW polesBW pole ABW pole Btree level D + ν e + ρ + Κ − (d) FIG. 3:
V P invariant mass distributions for D + → νe + K ∗− π + and D + → νe + ρ + K − . Left panels: including the interactionwith the tree level mechanism of Fig. 1. Right panels: without the interference with the tree level contribution. IV. SUMMARY
We show theoretically that the semileptonic decays ofthe D + meson into νe + K ∗− π + and νe + ρ + K − allowsto distinguish the two different poles associated to the K (1270) resonance as predicted by the chiral unitaryapproach [17, 18]. Using as the only input the lowest or-der chiral perturbation theory Lagrangian accounting forthe tree level interaction of a vector and a pseudoscalarmeson, the implementation of unitarity in coupled chan-nels allows to obtain the full V P scattering amplitudewhich dynamically develops two poles associated to the K (1270) resonance, without including them as explicitdegrees of freedom. The poles show up naturally from thehighly non-linear dynamics implied in the unitarization.Each pole has different features which could allow themto be distinguished in specifically devoted reactions, likethose considered in the present work. Indeed, each polecouples differently to different V P channels: the lowermass pole is wider and couples mostly to K ∗ π and thehigher mass pole is narrower and couples predominantlyto ρK .The semileptonic decays studied in the present workproceed first with the elementary V P production fromthe hadronization after the weak decay of the c quark viathe creation of a q ¯ q pair with the P model. The weightof the different channels are then related using SU (3) ar-guments. The K (1270) shows up in the decay after the implementation of the final state interaction of the V P pair, using the unitarized
V P amplitudes. In spite of thefact that in the full amplitudes there is always a mixtureof both poles, we obtain, by evaluating the
V P invariantmass distributions, that the D + → νe + K ∗− π + weighsmore the lower mass pole while in the D + → νe + ρ + K − decay the higher mass pole has a greater influence. Theshapes do not necessarilly reflect directly the pure res-onant shape of each pole since there are interferencesbetween the poles and non-resonant terms, but both theposition and shape of the invariant mass distributions areclearly different and reflect the dominance of either polein both channels considered and could be observed in ex-periments amenable to look at these mass distributions. V. ACKNOWLEDGMENTS
We would like to acknowlege the fruitful discussionswith Ju-Jun Xie and Li-Sheng Geng. This work is partlysupported by the National Natural Science Foundation ofChina under Grant Nos. 11505158, 11847217, 11975083and 11947413. It is also supported by the Academic Im-provement Project of Zhengzhou University. This workis partly supported by the Spanish Ministerio de Econo-mia y Competitividad and European FEDER funds un-der Contracts No. FIS2017-84038-C2-1-P B and No.FIS2017-84038-C2-2-P B. [1] N. Isgur, D. Scora, B. Grinstein and M. B. Wise, Phys.Rev. D (1989) 799.[2] D. Scora and N. Isgur, Phys. Rev. D (1995) 2783.[3] B. Bajc, S. Fajfer and R. J. Oakes, Phys. Rev. D (1996) 4957.[4] N. R. Soni, M. A. Ivanov, J. G. K¨orner, J. N. Pandya,P. Santorelli and C. T. Tran, Phys. Rev. D (2018)114031.[5] D. L. Yao, P. Fernandez-Soler, M. Albaladejo, F. K. Guoand J. Nieves, Eur. Phys. J. C (2018) 310.[6] S. Fajfer, J. F. Kamenik and I. Nisandzic, Phys. Rev. D (2012) 094025.[7] M. Tanaka and R. Watanabe, Phys. Rev. D (2013)034028.[8] L. R. Dai, X. Zhang and E. Oset, Phys. Rev. D (2018)036004.[9] F. S. Navarra, M. Nielsen, E. Oset and T. Sekihara, Phys.Rev. D (2015) 014031.[10] T. Sekihara and E. Oset, Phys. Rev. D (2015) 054038.[11] E. Oset et al. , Int. J. Mod. Phys. E , 1630001 (2016).[12] M. Ablikim et al. [BESIII Collaboration], Phys. Rev.Lett. (2019), 231801.[13] M. Artuso et al. [CLEO Collaboration], Phys. Rev. Lett. (2007) 191801.[14] R. Khosravi, K. Azizi and N. Ghahramany, Phys. Rev.D (2009) 036004.[15] H. Y. Cheng and X. W. Kang, Eur. Phys. J. C (2017)587. Erratum: [Eur. Phys. J. C (2017) no.12, 863]. [16] S. Momeni and R. Khosravi, J. Phys. G (2019) 105006.[17] L. Roca, E. Oset and J. Singh, Phys. Rev. D , 014002(2005).[18] L. S. Geng, E. Oset, L. Roca and J. A. Oller, Phys. Rev.D (2007) 014017.[19] S. Godfrey and N. Isgur, Phys. Rev. D , 189 (1985).[20] N. Isgur and G. Karl, Phys. Rev. D , 4187 (1978).[21] S. Capstick and N. Isgur, Phys. Rev. D , 2809 (1986)[AIP Conf. Proc. , 267 (1985)].[22] S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. ,S241 (2000).[23] J. Vijande, F. Fernandez and A. Valcarce, J. Phys. G ,481 (2005).[24] N. Kaiser, P. B. Siegel and W. Weise, Phys. Lett. B (1995) 23.[25] J. A. Oller and E. Oset, Nucl. Phys. A (1997) 438.Erratum: [Nucl. Phys. A (1999) 407].[26] M. C. Birse, Z. Phys. A , 231 (1996).[27] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A , 392 (2004).[28] Y. Zhou, X. L. Ren, H. X. Chen and L. S. Geng, Phys.Rev. D , 014020 (2014).[29] W. H. Liang, S. Sakai and E. Oset, Phys. Rev. D ,094020 (2019).[30] L. R. Dai, L. Roca and E. Oset, Phys. Rev. D , 096003(2019).[31] S. J. Jiang, S. Sakai, W. H. Liang and E. Oset, Phys.Lett. B (2019) 134831. [32] J. A. Oller and U. G. Meissner, Phys. Lett. B (2001)263.[33] D. Jido, J. A. Oller, E. Oset, A. Ramos and U. G. Meiss-ner, Nucl. Phys. A (2003) 181.[34] T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. (2012)55.[35] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D (2018) 030001.[36] G. Y. Wang, L. Roca and E. Oset, Phys. Rev. D (2019) 074018.[37] L. S. Geng, E. Oset, J. R. Pelaez and L. Roca, Eur. Phys. J. A (2009) 81.[38] L. Roca and E. Oset, Phys. Rev. D (2012) 054507.[39] L. Micu, Nucl. Phys. B , 521 (1969).[40] A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal,Phys. Rev. D (1973) 2223.[41] E. Santopinto and R. Bijker, Phys. Rev. C , 062202(2010).[42] A. Bramon, A. Grau and G. Pancheri, Phys. Lett. B283