Relativistic corrections to decays of heavy baryons in the quark model
RRelativistic corrections to decays of heavy baryons in the quark model
Ahmad Jafar Arifi , ∗ Daiki Suenaga , † and Atsushi Hosaka
1, 2, ‡ Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan (Dated: February 9, 2021)We investigate relativistic corrections of order 1 /m , where m is the light quark mass, to heavybaryon decays by emitting one pseudoscalar meson in the quark model. This work is motivated byshortcomings in the previous studies in the nonrelativistic quark model for decays of the Roper-likestates such as Λ c (2765). We find that the relativistic corrections due to the internal motion ofquarks are essential ingredients in improving their decay properties such that the decay widths aresignificantly increased. In addition, such corrections can explain a phenomenological suppression ofthe quark axial-vector coupling constant g qA for the Σ c (2455) and Σ c (2520) decays. Introduction.—
In recent years, many states of hadronsthat contain heavy quarks are discovered in many ex-periments such as SLAC, KEK, and LHC experiments.Among them, the newly observed Λ b (6072) in LHC ex-periments [1, 2] is particularly interesting. From its massand decay properties, this state is suggested to be thefirst radial excitation of the Λ b baryon [3, 4], which is ananalogous state of the Roper resonance, N (1440) [5]. Infact, there are other candidates of the Roper-like stateswith heavy quark flavors such as Λ c (2765) and Ξ c (2970).Such analogy is further confirmed from the recent resultby Belle that Ξ c (2970) favors spin-parity 1 / + [6]. Inter-estingly, they appear to have not only a similar excitationenergy but also a large decay width [3].Historically, the Roper resonance has been a mysteri-ous state because the observed mass is much lower thanthat predicted by the quark model, and the inverse or-dering with the negative parity state is puzzling. Theseunexpected results have induced great amount of discus-sions to understand its nature both experimentally [7, 8]and theoretically [9–16]. One promising physical inter-pretation is that the Roper resonance is a quark corecoupled by meson clouds [17, 18]. Moreover, the rela-tivistic effects are found to be important [19–21].One of the difficulties of the Roper-like states is thattheir observed broad decay widths seem to contradictwith the prediction of the narrow width by the quarkmodel [22–27]. Due to a large discrepancy, one may ex-pect that the nonresonant process or f (500) contribu-tion may be essential. However, in our previous study, itis suggested that such a contribution is insignificant [3].The suppression is also supported by the chiral effectivemodel [28]. Therefore, the prediction of the narrow widthis indeed a serious problem of the previous studies in thequark model.In this letter, we investigate relativistic correctionsin the quark model primarily to solve the problem inthe decays of the Roper-like states. At the same time, ∗ arifi@rcnp.osaka-u.ac.jp † [email protected] ‡ [email protected] we also study the corrections to other low-lying states.For this purpose, we will use the Foldy-Wouthysen-Tani(FWT) transformation. This method was employed longago by Kubota and Ohta in analyzing the photoexcita-tion amplitudes of nucleon resonances [19]. They empha-sized that such corrections are crucial to give the correctsign of the photoexcitation amplitude of N (1440), lead-ing to better agreement with the data. The relativisticeffects in the photoexcitation amplitudes are also con-firmed by other computation [20]. Furthermore, the rel-ativistic treatments give better agreement for the massof N (1440) [21]. Motivated by these observations, we ex-pect that the relativistic corrections will play importantroles also for heavy baryons. Nonrelativistic quark model.—
The quark model com-putation of heavy baryon decays follows Refs. [22, 24].The harmonic oscillator wave functions of baryons areformed in the heavy quark basis. They are denoted as Y Q ( nl ξ , J ( j ) P ) where nl stand for the node and orbitalangular momentum quantum numbers, and ξ = λ or ρ indicate the two internal excitation modes of quarks. Itsspin ( J ) and parity ( P ) together with the brown muckspin ( j ) are denoted by J ( j ) P .In the quark model, the one-pion emission decay ofheavy baryon is depicted in Fig. 1. Here we employ theaxial-vector type coupling for the interaction between the π Y Q Y ′ Q p i p f q FIG. 1. Schematic picture of one-pion emission decay ofheavy baryon Y Q in the quark model where the pion is re-garded as a Nambu-Goldstone boson. a r X i v : . [ h e p - ph ] F e b TABLE I. Decay widths estimated by the nonrelativistic quark model with the relativistic corrections (NR+RC) and withoutthem (NR) for various charmed baryons in units of MeV. The minimum and maximum values of the decay widths are foundnumerically within the parameter range as indicated in the paragraph of
Model parameters .State Multiplet Channel Γ NR Γ NR+RC Γ Exp . RefΣ c (2455) ++ Σ c (1 S, / + ) Λ c π . ± .
04 Belle [29]Σ c (2520) ++ Σ c (1 S, / + ) Λ c π . ± .
25 Belle [29]Λ c (2595) + Λ c (1 P λ , / − ) Σ c (2455) π . ± . c (2625) + Λ c (1 P λ , / − ) Σ c (2455) π c (2520) π < .
97 CDF [30]Λ c (2765) + Λ c (2 S λλ , / + ) Σ c (2455) π c (2520) π ± c (3136) + a Λ c (2 S ρρ , / + ) Σ c (2455) π c (2520) π (cid:48) + c Ξ (cid:48) c (1 S, / + ) Ξ c π . . . b . . . . . . . . .Ξ c (2645) + Ξ (cid:48) c (1 S, / + ) Ξ c π . ± .
13 Belle [32]Ξ c (2790) + Ξ c (1 P λ , / − ) Ξ (cid:48) c π . ± . c (2815) + Ξ c (1 P λ , / − ) Ξ (cid:48) c π c (2645) π . ± .
26 Belle [32]Ξ c (2970) + Ξ c (2 S λλ , / + ) Ξ (cid:48) c π c (2645) π c (2455) K . ± . c (3318) + a Ξ c (2 S ρρ , / + ) Ξ (cid:48) c π c (2645) π c (2455) K a Masses are estimated in the quark model for the ρ -mode Roper-like state. b The null results for the decay width are due to insufficient phase space. pion and a light quark inside a heavy baryon as L πqq = g qA f π ¯ qγ µ γ (cid:126)τ q · ∂ µ (cid:126)π. (1)This interaction is inspired by the low-energy theorem ofchiral symmetry. In many cases, the nonrelativistic cal-culations have been performed by considering the termsup to order 1 /m as given by H NR = g (cid:104) σ · q + ω π m ( σ · q − σ · p i ) (cid:105) , (2)where we define g = g qA / f π with g qA = 1 the quarkaxial-vector coupling constant and f π = 93 MeV thepion decay constant. Here we denote the energy andmomentum of the outgoing pion as ( ω π , q ). For kaonemission decays, the parameters such as the kaon decayconstant ( f K = 111 MeV), energy and momentum shouldbe changed accordingly. The initial and final momentaof the light quark are denoted by p i and p f . In the previous works [22–24], decay widths of heavybaryons were investigated by using the interaction inEq. (2). However, the resulting decay widths turned outto be too small for the Roper-like states, e.g. Λ c (2765)baryonas shown in the column denoted as Γ NR of Table I. Relativistic corrections of order /m .— To estimatethem properly, we perform the FWT transformation [33]for the Lagrangian in Eq (1). After some calculations,we obtain H RC = g m (cid:20) m π σ · q + 2 σ · ( q − p i ) × ( q × p i ) (cid:21) , (3)where m π is the pion mass. Note that m π should bereplaced by m K for the kaon emission decay. What wefound in this work is that the term proportional to p i in the second term of Eq. (3) plays an important rolenot only for the Roper-like state but also for Σ c ’s. Thisterm is due to the internal motion of the quarks inside aheavy baryon. In the electromagnetic interaction, such a TABLE II. Similar to Table. I, but for bottom baryons.State Multiplet Channel Γ NR Γ NR+RC Γ Exp . RefΣ b (5810) + Σ b (1 S, / + ) Λ b π . ± .
31 LHCb [35]Σ b (5830) + Σ b (1 S, / + ) Λ b π . ± .
47 LHCb [35]Λ b (5912) Λ b (1 P λ , / − ) Σ b π < .
25 LHCb [2]Λ b (5920) Λ b (1 P λ , / − ) Σ ∗ b π < .
19 LHCb [2]Λ b (6072) Λ b (2 S λλ , / + ) Σ b π ∗ b π ±
11 LHCb [2]Λ b (6469) Λ b (2 S ρρ , / + ) Σ b π ∗ b π b (5935) − Ξ (cid:48) b (1 S, / + ) Ξ b π < .
08 LHCb [36]Ξ b (5945) − Ξ (cid:48) b (1 S, / + ) Ξ b π . ± .
31 LHCb [36]Ξ b (6096) − b Ξ b (1 P λ , / − ) Ξ (cid:48) b π b (6102) − b Ξ b (1 P λ , / − ) Ξ (cid:48) b π ∗ b π b (6255) − b Ξ b (2 S λλ , / + ) Ξ (cid:48) b π ∗ b π b (6647) − a Ξ b (2 S ρρ , / + ) Ξ (cid:48) b π ∗ b π a Masses are estimated in the quark model for the ρ -mode Roper-like state. b Masses are taken from Ref [34]. term appears as the spin-orbit coupling in the relativisticcorrection [19].
Model parameters.—
In the quark model, there arethree parameters: the light quark mass m , the heavyquark mass M , and the spring constant k . Following ourprevious study [24], for Λ c and Λ b baryons we will use theconstituent quark masses as m u ( d ) = 0 . ± .
05 GeV ,M c = 1 . ± . , and M b = 5 . ± . c and Ξ b baryons they consist of three different quarksso that we use the averaged mass m = 0 . ± .
05 GeVfor the u, d, and s quarks. In this work, the spring con-stant is k = 0 . ± .
01 GeV . From the above param-eters, we obtain the range parameter of the harmonicoscillator wavefunctions as a λ = 0 . ± .
04 GeV and a ρ = 0 . ± .
03 GeV for Λ c baryons. The values of therange parameters slightly vary for Ξ c , Λ b and Ξ b .In actual computations, we use the heavy baryonmasses from the experimental data when available. Oth-erwise, we use the theoretical input as given in the foot-notes of Tables I and II. In the following, we will lookat the decays of low-lying states one by one. Ground states.—
Let us start from the Σ c (2455) andΣ c (2520). These states are regarded as ground states be- cause the quarks are in the lowest S -wave orbit. However,they have an energy excess due to the spin-one (bad) di-quark that can decay into the spin-zero (good) diquarkby emitting one pion.The nonrelativistic quark model of order 1 /m overpre-dicts the decay widths of Σ c states and their siblings byfactor two as shown in the column denoted as Γ NR ofTable I and II. In our previous study [24], the discrep-ancy has led to the discussion of the suppression factorof about 3/4 for the quark axial-vector coupling constant g qA . In the literature, the universal suppression parame-ter is introduced to explain the experimental data [23].The necessity of the suppression factor for g qA has beenknown for long time for the nucleon g A ; in the nonrel-ativistic quark model g A = 5 /
3, about 30% larger thanthe observed value g A ∼ .
25 [37]. The situation is es-sentially the same for the decay of Σ c → Λ c π .Now, let us see the suppression mechanism by includ-ing the relativistic corrections in more detail. The matrixelement of the leading term of order 1 /m is the spin-isospin factor of σ i τ a times the overlap of the commonground state wave functions for Σ c and Λ c which is unityin the long-wavelength limit of the pion momentum. Forthe term of oder 1 /m , the matrix elements of σ · p i and σ · q cancel each other giving only a small contribution ofaround 0 .
1% of the total width by using the interactionin Eq. (2). The cancelation can be understood since theratio R p/q = (cid:104) σ · p i (cid:105) / (cid:104) σ · q (cid:105) is around 0.42 for this case.In the relativistic corrections of order 1 /m , the matrixelement of p i in Eq. (3) gives a factor proportional tothe square of the range parameter a . This term appearswith the opposite sign to the leading term of 1 /m . Thisexplains the reduction of the quark axial-vector couplingconstant g qA . As shown in Tables I and II, it is fare to saythat the agreement with the data is improved when ob-serving that the data marginally fall into the calculatedrange. Negative parity states.—
These are the first excitedstates of quark orbital motion in the P -wave (1 P state).We expect that they are dominated by the lower λ modes.We assume that this is the case not only for the 1 P states,but also for the 2 S Roper-like states in the following.The relativistic correction is found to be insignificantfor the negative parity states. For instance, the correc-tion to the decay of Λ c (2595) with J P = 1 / − is negli-gible and the interaction in Eq. (2) is sufficiently goodin explaining the experimental data. For this decay, theleading term of order 1 /m with σ · q is negligible becauseit results in a term proportional to q which is vanishingin the long-wavelength limit. Meanwhile, for the term oforder 1 /m the matrix element of σ · p i gives a finite termof order q . As a result, the σ · p i becomes the dom-inant term. This is in the line with the S -wave decayof Λ c (2595) → Σ c (2455) π . For the relativistic correctionterms of order 1 /m as in Eq. (3), the matrix elementis found to give only a term proportional to q resultingin a small contribution. For the case of Λ c (2625) with J P = 3 / − , the relativistic correction is found to be siz-able for Σ c (2455) π channel. However, because of the D -wave nature, the actual value is relatively small and theagreement with the data is still good. This behavior ap-plies to other siblings such as Ξ c (2790) and Ξ c (2815) asgiven in Tables I and II. Roper-like states.—
Now, let us come to the main re-sult of the present work. Here we found that the rela-tivistic correction is essential for the Roper-like states.As discussed earlier, the nonrelativistic quark model pre-dicts narrow widths around a few MeV which are smallerthan the experimental data by one order in magnitude.However, by taking into account the relativistic correc-tions in Eq. (3), the decay widths are significantly im-proved and have better agreement with the data as shownin the column denoted as Γ
NR+RC of Tables I and II.The shortcoming in the nonrelativistic quark modelcan be understood from the orthogonality of the wave-functions. The leading term of order 1 /m with σ · q ,which is the spin-flip transition process, contains a van-ishing overlap of the orthogonal orbital wave functions inthe long-wavelength limit. In contrast, the σ · p i termof order 1 /m in Eq. (2) provides a finite contribution. However, the odd power of the quark momentum oper-ator will translate into the pion momentum q and al-ways come with the pion energy ω π , which makes therole of σ · p i term not very important resulting in onlysmall decay widths up to order 1 /m [24]. On the otherhand, in the relativistic correction of 1 /m , the matrixelements consist of the higher terms of the quark mo-mentum of p i as given in Eq. (3). The even power of thequark momentum operator will translate into the squareof the range parameter a giving considerable contribu-tions. Together with the σ · p i term of order 1 /m withthe same sign, the corrections of order 1 /m lead to alarge increase of the total decay widths.As anticipated earlier, we have so far discussed the λ -mode excited states. To complete our discussions, wealso mention the results for ρ -mode ones. The excitationenergies of the ρ modes are expected to be larger; in theharmonic oscillator base, we expect that the mass of the ρ -mode Roper-like state is about 1 GeV above the groundstate. In more realistic calculations with a linear confine-ment potential, this energy is somewhat lowered [38]. Weexpect that the mass of the ρ -mode Roper-like state isabout 850 MeV above the ground state Λ c (2286). In Ta-ble I and II, results are shown by using this value. Theresulting widths are largely increased. We consider thatthis could be the reason that the ρ -mode Roper-like stateis not likely to be observed.For the Λ c (2765) and Λ b (6072) baryons, the computeddecay widths are found to be similar. This behavior fol-lows the heavy-quark flavor symmetry [39], i.e., the dy-namics of charmed and bottom baryons are similar. Also,the branching ratio R = Γ(Σ c (2520) π ) / Γ(Σ c (2455) π ) isnot significantly changed with the inclusion of the rela-tivistic corrections, and still consistent with the predic-tion from the heavy-quark spin symmetry [40]. For thecase of Roper-like Ξ c ( b ) baryons, the decay widths arefound to be smaller than Roper-like Λ c ( b ) baryons despitehaving similar phase space. This can be understood bythe fact that the Ξ c ( b ) baryons have only one light quarkthat couples to pion.For the case of Ξ c , the Σ c (2455) K channel is open. Inthis case, the relativistic correction for the kaon emis-sion decay is not large as compared to the pion emissiondecay because of the smaller phase space volume. As aresult, the ratio of Σ c (2455) ++ K − to Ξ c (2645) π + be-comes smaller around 10% when the relativistic correc-tion is included as compared to the case without it whichis around 40%. This prediction can be tested in the ex-periment to further clarify the role of relativistic effectsfor the Roper-like states. Summary.—
We have investigated relativistic correc-tions up to order 1 /m to the decays of low-lying heavybaryons through pseudoscalar meson emission in thequark model. As a result, we have found that the agree-ment with the data is significantly improved. In partic-ular, the decay widths of the Λ c (2765) and other Roper-like states are greatly increased by one order in magni-tude as compared to the previously calculated values upto order 1 /m .It is emphasized that we do not need a suppression ofthe quark axial-vector coupling constant g qA by hand [41],but rather it is naturally explained by the relativistic ef-fect. The fact that we can consistently use g qA = 1 sup-ports the discussion by Weinberg on the mended symme- try for the quark axial-vector coupling constant [42, 43]. Acknowledgements.—
A. J. A thanks Research Cen-ter for Nuclear Physics (RCNP) for the hospitality dur-ing his stay in completion of this work. We also thankKiyoshi Tanida for useful discussions. A. H. is supportedin part by Grants-in Aid for Scientific Research, GrantsNo. 17K05441(C) and by Grants-in Aid for ScientificResearch on Innovative Areas (No. 18H05407). [1] A. M. Sirunyan et al. [CMS], Study of excited Λ statesdecaying to Λ π + π − in proton-proton collisions at √ s =13 TeV, Phys. Lett. B , 135345 (2020).[2] R. Aaij et al. [LHCb], Observation of a new baryon statein the Λ π + π − mass spectrum, JHEP , 136 (2020).[3] A. J. Arifi, H. Nagahiro, A. Hosaka and K. Tanida,Roper-like resonances with various flavor contents andtheir two-pion emission decays, Phys. Rev. D ,111502 (2020).[4] K. Azizi, Y. Sarac and H. Sundu, New Λ b (6072) state asa 2 S bottom baryon, Phys. Rev. D , 034007 (2020).[5] L. D. Roper, Evidence for a P pion-nucleon resonanceat 556 MeV, Phys. Rev. Lett. , 340 (1964).[6] T. J. 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