Roper-Like Resonances with Various Flavor Contents and Their Two-Pion Emission Decays

PHYSICAL REVIEW D 101, 111502(R) (2020) Rapid Communications

Roper-like resonances with various flavor contents and their two-pion emission decays

A. J. Arifi ,1 H. Nagahiro ,1,2 A. Hosaka ,1,3 and K. Tanida 3 1Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan 2Department of Physics, Nara Women’s University, Nara 630-8506, Japan 3Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan

(Received 17 April 2020; accepted 28 May 2020; published 16 June 2020) Λð6072Þ We study the three-body decay of the newly observed bottom baryon b by LHCb; Λð6072Þ → Λ ππ Λ b b . Its mass about 500 MeV above the ground state b and a broad width imply that the state could be an analogue of the Roper resonance of the nucleon Nð1440Þ. In terms of sequential Σ Σ processes going through b and b, we find that the observed invariant mass distribution is reproduced assuming its spin and parity JP ¼ 1=2þ. We discuss that the ratio of the two sequential processes and angular correlation of two pions are useful for the determination of spin and parity. We suggest further studies for the Roper resonance analogue in various flavor contents, raising an interesting and important question in baryon spectroscopy.

DOI: 10.1103/PhysRevD.101.111502

In this work, motivated by the recent observation of way, the mass ordering of positive and negative parity Λð6072Þ b by CMS and LHCb collaborations [1,2], with the baryons were well reproduced for nonstrange and strange mass and width M ¼ 6072.3 MeV and Γ ¼ 72 MeV mea- baryons. sured by LHCb [2], we discuss it as a state analogous to the On the other hand, similar states of JP ¼ 1=2þ with Roper resonance Nð1440Þ with spin and parity JP ¼ 1=2þ almost the same excitation energies have been known for in heavy flavor sectors. It is shown that the three-body some time for hyperons systematically [12,13]. The recent Λð6072Þ decay of two pion emission is particularly useful to observation of b could add another candidate in the determine its unknown spin and parity. list of the analogous states. In fact, there is also a candidate Λð2765Þ The Roper resonance is the first excited state of the in the charm sector, c . Note that in PDG it is not P þ Λ Σ nucleon of J ¼ 1=2 [3]. It has been a mysterious state yet determined whether it is c or c, but recent report Λ because its properties such as the mass and level ordering suggests that it is likely to be an isoscalar c [14]. However, with negative parity nucleons have not been easily the spin and parity of these charmed and bottom candidates explained by the conventional quark model (for a recent are not yet determined. Observing this situation, we show review see for instance [4]). Detailed studies by a dynami- possible candidate states in Fig. 1 with their excitation cal model for meson-baryon scatterings [5,6] and by the energies. They are similar not only in masses but also in measurement of the transition form factor in a wide range decays through two pion emission which is dominated by of momentum transfer Q2 [7] have convinced us that it is a the sequential processes going through intermediate baryon radial excitation of the nucleon. Furthermore, several lattice resonances as shown in Fig. 2, while a nonresonant simulations support this picture [8–10]. As demonstrated contribution is insignificant. These rather universal features in [5,6], one promising physical interpretation is that the in various flavor sectors may suggest important dynamics Roper resonance is a quark core coupled by meson clouds. of low energy QCD. The quark-meson interaction then has a significant effect in reducing the large mass of the quark core excitation. An alternative description was made in Ref. [11] where the pion (in general Nambu-Goldstone boson) exchange interaction reduced the mass of the quark core excitation. In this

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. þ Further distribution of this work must maintain attribution to FIG. 1. Excitation energies of the first excited JP ¼ 1=2 the author(s) and the published article’s title, journal citation, baryons and candidates with various flavor contents. Red bars and DOI. Funded by SCOAP3. are for those with undetermined spin and parity.

2470-0010=2020=101(11)=111502(5) 111502-1 Published by the American Physical Society ARIFI, NAGAHIRO, HOSAKA, and TANIDA PHYS. REV. D 101, 111502 (2020)

T Σ−→Λ0π− T Λ0→Σ−πþ − T ½Σ−¼− b b b b ð Þ i b i i ; 3 m23 − mΣ− þ ΓΣ− b 2 b

− where m23 is the invariant mass of the subsystem π Λ0 → Λ0πþπ− Λ0 FIG. 2. Sequential decay processes of b b going (particle 2) and b (particle 3). The amplitudes with the ΣðÞ− ΣðÞþ Σþ Σ∓ through b and b . b intermediate state and those with b are computed similarly. The total amplitude is then the coherent sum of the four of them. In the present work, focusing on Λð6072Þ, we study in b The unknown coupling constants g ’s are constrained by detail its three-body decay Λð6072Þ → Λ ππ. In our i b b the heavy quark spin symmetry of QCD [16]; the ratios of previous preprint [15], we have shown that the Dalitz plot g1 to g2 and of g3 to g4 are computed when Σ and Σ are analysis is particularly useful for the determination of spin b b regarded as heavy quark spin partners. In the heavy quark and parity of Λð2765Þ when the decay is dominated by c limit, the spin J of a baryon is composed of a spin j of the sequential processes going through Σ and Σ. The relevant c c brown muck of light degrees of freedom [17] and spin 1=2 diagrams are shown in Fig. 2 (for charmed baryons, replace of the heavy quark, such that J ¼ j 1=2. the label b for bottom by c). Following this, we investigate For a spin 1=2 Λ , the brown muck spin can take two the sequential decays of Λð6072Þ with an intermediate b b values, j ¼ 0, 1. Assume further that the brown muck of state Σ or Σ. In principle there are other processes that b b ΣðÞ ¼ 1 may contribute. They are for instance, a direct process b has spin j as in the quark model, their total spin J 1 2 3 2 where two pions are emitted at once at a single vertex, and a can be = or = . Having this information, we can meson resonance process such as a scalar σ meson to which compute the ratio as [15,18] two pions couple. However, qualitative arguments are given pffiffiffi g2 1 for the reason that these processes may be suppressed for an ¼ 2 for j ¼ 0 and pffiffiffi for j ¼ 1 ð4Þ g1 2 analogous decay of Λcð2765Þ [15]. Therefore, in this paper we will consider exclusively the sequential process, which Λ P ¼ 1 2þ can indeed explain the so far observed data very well as we for b with J = . Combined with two-body phase will see later. space factors, when the same partial L-wave is possible for πΣ πΣ To compute these diagrams, let us first introduce the two final states b and b, we find the ratio amplitudes for various vertices. Because the bottom bary- ΓðΛð6072Þ → ΣπÞ g2 p2Lþ1 ons are sufficiently heavy, we employ a nonrelativistic R ¼ b b ¼ 2 2 ð5Þ Λ ΓðΛð6072Þ → Σ πÞ 2 2Lþ1 method. Assuming that the spin and parity of b equal b b g1 p1 1=2þ, the amplitudes at the first vertex of Fig. 2 are given by where the ratio of g2=g1 is constrained by the heavy quark spin symmetry as in Eq. (4). P ¼ 1 2þ ¼ 1 † For J = , the two decays are in p-wave, L . −iT Λ →Σ π ¼ g1χΣ ðσ · pÞχΛ ; ¼ 215 ¼ 192 b b b b Using p1 MeV and p2 MeV, we obtain † † −iT Λ →Σ π ¼ g2χΣ ðS · pÞχΛ ; ð1Þ b b b b R ¼ 1.43 for j ¼ 0 and 0.36 for j ¼ 1 ð6Þ and for the second vertex, As we will see shortly, this agrees well with the observed data from LHCb. On the other hand, if JP ¼ 1=2− the πΣ πΣ † partial waves of b and of b are s and d-waves, −iT Σ →Λ π ¼ g3χ ðσ · pÞχΣ ; b b Λb b respectively. For low momentum pion as in the present † case, the d-wave decay is suppressed as compared to the −iT Σ →Λ π ¼ g4χΛ ðS · pÞχΣ : ð2Þ b b b b s-wave decay. Therefore, we find R ≪ 1 which does not seem consistent with the experimental data. This demon- In Eqs. (1) and (2), gi’s are the coupling constants and χ’s strates the advantage in the use of the ratio R for spin and are the two-component spinors for the baryons as indicated parity. by lower indices. Moreover, we define the spin operator σ Now we discuss Dalitz plots and other related quantities. for spin-1=2 particle and the spin transition operator S for In what follows, we assume again JP ¼ 1=2þ for 3 2 1 2 Λð6072Þ transition from spin = to = . The pion momentum for b . For decays of unpolarized particles, Dalitz plots each vertex is denoted as p. are expressed as functions of two kinematical variables By using these vertex amplitudes, the amplitude of formed by the three decaying particles. In fact, they also Σ− the first diagram of Fig. 2 with the intermediate b is depend on the mass of the initial particle, which distributes ∼72 Λð6072Þ expressed as over a finite range of width MeV for b .

111502-2 ROPER-LIKE RESONANCES WITH VARIOUS FLAVOR … PHYS. REV. D 101, 111502 (2020) Z 1 ð ; 2 2 Þ ð 2 2 Þ¼ P m m12;m23 dm ð Þ P m12;m23 2 2 ; 8 N ðm − MΛ Þ þ ΓΛ =4 b b

where the factor N is for normalization and m for the Λbππ mass. The result is shown in the bottom panel of Fig. 3 as well as the invariant mass plot on the back face. As shown there, in the convoluted plot the peaks of kinematical reflections in the upper three panels are smeared out, while the real resonance peaks remain. Because the experimental data is shown for the sum of þ − signals of Λbπ and Λbπ , we have shown in Fig. 4 the corresponding one of our calculation (red solid curve) compared with the experimental data. We note that the blue − curve in Fig. 4 for the Λbπ invariant mass plot is the same as the red curve on the back face of the bottom panel in Fig. 3. Note that experimental resolution of around 1 MeV is not considered in the calculation. For theory side, we − þ have also shown the strengths of Λbπ and Λbπ separately by blue and green solid curves, respectively. We find that our theory calculation agrees remarkably well with experimental data not only in overall shape of the peaks and background. The relative strengths of the peaks is determined by the ratio R, while the background shape is reproduced by the convoluted kinematical reflection. It is FIG. 3. The Dalitz and invariant mass plots at three different initial masses of Λð6072Þ (upper three panels) and the con- interesting to see that even such a detailed structure, the b Σ Σ voluted one (bottom panel). relative height of the two resonance peaks of b and b are well reproduced. This is achieved by taking the sum of the − two charged states. If only Λbπ is included, the left peak is In experiments with low statistics, plots are often made by slightly higher than that of the right one as shown in the integrating the signals from such a finite mass range. bottom panel of Fig. 3 and by the blue curve in Fig. 4. This Therefore, to compare directly theory predictions with Λð6072Þ analysis so far supports the spin and parity of b is the actual data, this feature is properly treated. 1=2þ, and their decay is dominated by the two pion In the upper three panels of Fig. 3 are shown diagonal emission of the sequential processes. This is one of the views of the Dalitz plots for decay probabilities main conclusions of the present study. If there will be Z further high statistics data, it is interesting to compare the 1 1 Dalitz plots at each different energies. Pðm; m2 ;m2 Þ¼ jT j2dm2 dm2 ð7Þ 12 23 ð2πÞ3 32m3 12 23 Let us further look at the angular correlation (dependence) along the resonance band. The angle is the one Λð6072Þ at three different fixed masses of b around the peak value, mðΛbππÞ¼6.052, 6.072 and 6.092 GeV, as func- 2 2 tions of invariant mass squares m12 and m23. Note that the Λ π− ΣðÞ− invariant mass m23 is for b that couples to b , which will be relevant in the discussions below. In each layer, the solid red curve shown in the back face is the projected strength, namely the invariant mass plot. In all the three Σ− plots, we find four resonance bands corresponding to b , Σ− Σþ Σþ b , b , and b from the left to right. The first two are for the first diagram of Fig. 2 showing the genuine resonance band, while the latter two correspond to the second one that are the so called kinematical reflection. Now what we need is the sum of the plots like these three Λ0π− Λ0πþ FIG. 4. Invariant mass plots of b (blue), b (green), and ones. More precisely, the convolution is performed over the their sum (red). The data from LHCb is the sum of these two [2] finite mass range weighted by the Breit-Wigner function, and compared with the red curve.

111502-3 ARIFI, NAGAHIRO, HOSAKA, and TANIDA PHYS. REV. D 101, 111502 (2020)

of 0þ (initial brown muck) → 0þ (final brown muck) þ0− (kaon) does not occur. This selection rule applies to the other candidates and is a unique feature of the Roper like resonance. In recent analysis, LHCb reported several resonances in ΛcK [20]. One of them is Ξcð2965Þ with similar excitation energy, but it has significantly smaller decay width than Ξcð2970Þ. Therefore, Ξcð2965Þ could be a particle of different nature that can be identified as a negative parity 0 excitation of Ξc as pointed out in Ref. [21,22]. Likewise, in Ξð6227Þ the bottom sector, the b resonance observed in ΛbK and Ξbπ decays [23] would be this analogue of Λð6072Þ negative parity excitation of Ξ0 [24–28]. We expect a FIG. 5. The convoluted square Dalitz plot for b and its b Σ− Ξ P ¼ 1 2þ Ξ ππ corresponding angular correlation along b with a mass Roper-like b of J = decaying into b at cut MΣ− ΓΣ− . b b around 6.2 GeV. We may expect further candidates in the strangeness between the two pions θ12 in the intermediate resonance sector. For example, the resonance Ξ ð1820Þ listed in PDG rest frame. To see the correlation better, it is convenient to [12] may be identified with 3=2− resonance as observed in make a square plot as a function of m23 and cos θ12 as ΛK invariant mass [29]. However, a resonance with a shown in the lower plot of Fig. 5. As seen there the angular similar excitation energy is also observed in Ξππ invariant Σ correlation along the b band (near side) is flat due to its masses with a larger decay width [30]. This is a candidate spin 1=2. On the other hand, the angular correlation along of the Roper-like resonance. Σ the b band (far side) has a concave shape due to its Here we summarize several common features of the spin 3=2. Roper-like resonances in our observation: Σ In general the angular correlation along the b band (i) The excitation energy is around 500 MeV which is decaying from a particle of spin J contains the two terms rather flavor-independent. weighted by the helicity amplitudes Ah, where h is the (ii) The decay width is rather large with a significant conserved helicity in the decay, coupling to two-pion emission decay. 2 2 (iii) Their two-pion emission decays are dominated by ðθ Þ ∝ j ðΛ → Σ πÞj ð1 þ 3 θ Þ þ W 12 A1=2 b b × cos 12 the sequential processes going through 1=2 reso- þj ðΛ → ΣπÞj2 3 2 θ ð Þ nance, e.g., Σ and 3=2þ resonance, e.g., Σ, or their A3=2 b b × sin 12: 9 b b analogues, indicating that the nonsequential contri- Λ 1 2 When the initial b has spin = , the helicity amplitude butions including those from f0ð500Þ are insignifi- A3=2 vanishes. Thus, the angular correlation will have a cant as observed also for other baryons [31–33]. 2 1 þ 3 cos θ12 dependence (concave shape). In reality, (iv) The ratio R and angular correlation can be used to þ interference between the resonance of a finite width and determine their spin and parity 1=2 . background (kinematical reflection in the present case) Observations in the heavy flavor baryons and contaminates that angular correlation. Fitting the projected revisit to the light flavor baryons could provide new angular correlation as plotted in the upper panel of Fig. 3, insights into unique flavor independent dynamics of low our prediction for the angular correlation is energy QCD.

2 Wðθ12Þ ∝ 1 þ 3.3 cos θ12: ð10Þ We are thankful for support from the Reimei Research Promotion project (Japan Atomic Energy Agency) in Λð6072Þ So far we have focused our discussions on b .As completion of this work. A. J. A. is supported by a anticipated in the beginning of this paper, there are other scholarship from the Ministry of Education, Culture, candidate baryons of similar nature, in particular Ξ bary- Science and Technology of Japan. H. N. is supported in ons. One is Ξcð2970Þ with excitation energy around part by Grants-in Aid for Scientific Research, Grants 500 MeV and decays into Ξcππ [19]. If it is regarded as No. 17K05443(C) and A. H. by Grants No. 17K05441(C) a Roper-like resonance, its decay into ΛcK is forbidden and by Grants-in Aid for Scientific Research on Innovative due to the brown muck selection rule, namely the transition Areas (No. 18H05407).

111502-4 ROPER-LIKE RESONANCES WITH VARIOUS FLAVOR … PHYS. REV. D 101, 111502 (2020)

[1] A. M. Sirunyan et al. (CMS Collaboration), Study of excited [18] H. Y. Cheng and C. K. Chua, Strong decays of charmed Λ0 Λ0πþπ− baryons in heavy hadron chiral perturbation theory, Phys. pbffiffiffistates decaying to b in proton-proton collisions at s ¼ 13 TeV, Phys. Lett. B 803, 135345 (2020). Rev. D 75, 014006 (2007). Ξ [2] R. Aaij et al. (LHCb Collaboration), Observation of [19] J. Yelton et al. (Belle Collaboration), Study of excited c Λ0πþπ− states decaying into Ξ0 and Ξþ baryons, Phys. Rev. D 94, a new baryon state in the b mass spectrum, arXiv: c c 2002.05112. 052011 (2016). Ξ0 [3] L. D. Roper, Evidence for a P-11 Pion-Nucleon Resonance [20] R. Aaij et al. (LHCb Collaboration), Observation of new c Λþ − at 556 MeV, Phys. Rev. Lett. 12, 340 (1964). baryons decaying to c K , arXiv:2003.13649. Ξ0 [4] V. D. Burkert and C. D. Roberts, Colloquium: Roper reso- [21] H. Yang, H. Chen, and Q. Mao, Identifying the c Ξ0 nance: Toward a solution to the fifty year puzzle, Rev. Mod. baryons observed by LHCb as P-wave c baryons, arXiv: Phys. 91, 011003 (2019). 2004.00531. [5] N. Suzuki, B. Julia-Diaz, H. Kamano, T. S. H. Lee, A. [22] K. Wang, L. Xiao, and X. Zhong, Understanding the newly Ξ0 Matsuyama, and T. Sato, Disentangling the Dynamical observed c states through their decays, arXiv:2004.03221. Origin of P-11 Nucleon Resonances, Phys. Rev. Lett. [23] R. Aaij et al. (LHCb Collaboration), Observation of a New Ξ− 121 104, 042302 (2010). b Resonance, Phys. Rev. Lett. , 072002 (2018). [6] H. Kamano, S. X. Nakamura, T. S. H. Lee, and T. Sato, [24] B. Chen, K. W. Wei, X. Liu, and A. Zhang, Role of newly Ξ ð6227Þ− Extraction of P11 resonances from πN data, Phys. Rev. C discovered b for constructing excited bottom 98 81, 065207 (2010). baryon family, Phys. Rev. D , 031502(R) (2018). [7] I. Aznauryan et al. (CLAS Collaboration), Electroexcitation [25] E. L. Cui, H. M. Yang, H. X. Chen, and A. Hosaka, Ξ ð6227Þ Σ ð6097Þ of nucleon resonances from CLAS data on single pion Identifying the b and b as P-wave bottom P ¼ 3 2− 99 electroproduction, Phys. Rev. C 80, 055203 (2009). baryons of J = , Phys. Rev. D , 094021 (2019). [8] K. F. Liu, Baryons and chiral symmetry, Int. J. Mod. Phys. E [26] K. L. Wang, Q. F. Lu, and X. H. Zhong, Interpretation of Σ ð6097Þ Ξ ð6227Þ− 26, 1740016 (2017). the newly observed b and b states as the 99 [9] J. J. Wu, D. B. Leinweber, Z. W. Liu, and A. W. Thomas, P-wave bottom baryons, Phys. Rev. D , 014011 (2019). Structure of the Roper resonance from lattice QCD con- [27] H. Yang, H. Chen, E. Cui, A. Hosaka, and Q. Mao, Decay straints, Phys. Rev. D 97, 094509 (2018). properties of P-wave bottom baryons within light-cone sum 80 [10] C. Lang, L. Leskovec, M. Padmanath, and S. Prelovsek, rules, Eur. Phys. J. C , 80 (2020). Pion-nucleon scattering in the Roper channel from lattice [28] D. Jia, W. N. Liu, and A. Hosaka, Regge behaviors in 95 orbitally excited spectroscopy of charmed and bottom QCD, Phys. Rev. D , 014510 (2017). 101 [11] L. Glozman and D. Riska, The spectrum of the nucleons and baryons, Phys. Rev. D , 034016 (2020). [29] S. Biagi, M. Bourquin, R. Brown, H. Burckhart, P. the strange hyperons and chiral dynamics, Phys. Rep. 268, Extermann, M. Gailloud, C. P. Gee, W. Gibson, P. Jacot- 263 (1996). Guillarmod, J. Perrier, K. Ragan, P. Rosselet, P. Schirato, H. [12] M. Tanabashi et al. (Particle Data Group), Review of Siebert, V. Smith, K. Streit, J. Thresher, A. Wood, and C. particle physics, Phys. Rev. D 98, 030001 (2018) and 2019 Yanagisawa, Ξ resonances in Ξ− Be interactions. 2. update. Properties of Ξð1820Þ and Ξð1960Þ in the ΛK¯ 0 and Σ0K¯ 0 [13] M. Takayama, H. Toki, and A. Hosaka, Systematics of the channels, Z. Phys. C 34, 175 (1987). SU(3) baryon spectra and deformed oscillator quark model, [30] S. Apsell et al., Evidence for Ξ Resonances in the Ξð1530Þπ Prog. Theor. Phys. 101, 1271 (1999). System, Phys. Rev. Lett. 24, 777 (1970). [14] A. Abdesselam et al. (Belle Collaboration), Experimental þ þ [31] S. Prakhov et al. (Crystal Ball Collaboration), Measurement determination of the isospin of Λ ð2765Þ =Σ ð2765Þ , − 0 0 − c c of π p → π π n from threshold to pðπ Þ 750-MeV=c, arXiv:1908.06235. Phys. Rev. C 69, 045202 (2004). [15] A. J. Arifi, H. Nagahiro, A. Hosaka, and K. Tanida, [32] S. Prakhov et al. (Crystal Ball Collaboration), Reaction Three-body decay of Λcð2765Þ and determination of its − 0 0 K p → π π Λ from p − ¼ 514-MeV=c to 750-MeV=c, spin-parity, Phys. Rev. D 101, 094023 (2020). K Phys. Rev. C 69, 042202(R) (2004). [16] N. Isgur and M. B. Wise, Spectroscopy with Heavy Quark [33] R. Mizuk et al. (Belle Collaboration), Experimental Symmetry, Phys. Rev. Lett. 66, 1130 (1991). Constraints on the Possible JP Quantum Numbers of the [17] M. Neubert, Heavy quark symmetry, Phys. Rep. 245, 259 þ Λ ð2880Þ , Phys. Rev. Lett. 98, 262001 (2007). (1994). c

111502-5