Pentaquarks with anticharm or beauty revisited Physics Letters B 790 (2019) 248–250 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Pentaquarks with anticharm or beauty revisited Jean-Marc Richard a, Alfredo Valcarce b, Javier Vijande c,∗ a Université de Lyon, Institut de Physique Nucléaire de Lyon, IN2P3-CNRS-UCBL, 4 rue Enrico Fermi, 69622 Villeurbanne, France b Departamento de Física Fundamental and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain c Unidad Mixta de Investigación en Radiofísica e Instrumentación Nuclear en Medicina (IRIMED), Instituto de Investigación Sanitaria La Fe (IIS-La Fe), Universitat de Valencia (UV) and IFIC (UV-CSIC), Valencia, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 24 August 2018 Received in revised form 14 January 2019 Accepted 15 January 2019 Available online 21 January 2019 Editor: J.-P. Blaizot We use a constituent model to analyze the stability of pentaquark Q̄ qqqq configurations with a heavy antiquark c̄ or b̄, and four light quarks uuds, ddsu or ssud. The interplay between chromoelectric and chromomagnetic effects is not favorable, and, as a consequence, no bound state is found below the lowest dissociation threshold. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 . 1. Introduction There is a renewed interest in the spectroscopy of exotic hadrons containing one or two heavy constituents. For a review, see, e.g., [1–4]. New configurations are studied, and exotic states suggested in the 70s or 80s are revisited. Among the first multiquark candidates involving heavy flavors, there is the Q̄ qqqq pentaquark proposed independently and simul- taneously by the Grenoble group and by Harry Lipkin [5,6]. The word pentaquark was introduced in this context. For the chromomagnetic interaction, the P Q̄ = Q̄ qqqq is very similar to the H = uuddss of Jaffe [7], who realized that for a spin J = 0 and color-singlet state, the color-spin operator On = n∑ i< j λ̃i .λ̃ j σ i σ j , (1) applied to uuddss, reaches its largest eigenvalue O6 = 24, to be compared to O3 = 8 for each spin 1/2 baryon of the threshold. This means that in the limit of flavor symmetry SU(3)F , a chromo- magnetic operator H C M = −a O6 gives an additional downwards shift δM C M = 8 a = 1 2 (� − N) ∼ 150 MeV , (2) as compared to the threshold, provided the short-range correlation factor (in simple potential models, it is proportional to the expec- * Corresponding author. E-mail addresses: j-m.richard@ipnl.in2p3.fr (J.-M. Richard), valcarce@usal.es (A. Valcarce), javier.vijande@uv.es (J. Vijande). https://doi.org/10.1016/j.physletb.2019.01.031 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an open access artic SCOAP3 . tation value of δ(ri j )) is assumed to be the same for the H as for the ground-state baryons. Similarly, an eigenvalue O4 = 16 is found for a qqqq system in a state of color 3 and spin J q = 0 corresponding to a SU(3)F triplet of flavor. This means that in the limit where the mass of the heavy quark becomes infinite, i.e., the chromomagnetic en- ergy is restricted to the light sector, a downwards shift δM = 8 a ∼ 150 MeV is obtained for Q̄ qqqq, as compared to its lowest thresh- old Q̄ q + qqq. Again, the value δM ∼ 150 MeV is derived assuming that the qq short-range correlation is the same in Q̄ qqqq as in qqq. This mechanism of chromomagnetic binding was analyzed in several subsequent papers [8–11].1 When the SU(3)F symmetry is broken, the pentaquark is penalized (say for fixed mass m of u and d, and increased mass ms for the strange quark). Adopting a finite mass for the heavy quark also goes against the stability of the heavy pentaquark [9,10]. Note that the reasoning leading to δM = 8 a for a spin 1/2 Q̄ qqqq predicts a chromomagnetic binding δM = 16 a/3 ∼100 MeV for the spin 3/2 state. Hence both spin s = 1/2 and s = 3/2 states deserve some investigation. In this letter we adopt a generic constituent model, containing chromoelectric and chromomagnetic contributions, tuned to repro- duce the masses of the mesons and baryons entering the various thresholds and study the pentaquark configurations Q̄ uuds, Q̄ ddsu and Q̄ ssdu with Q = c or b, for both s = 1/2 and s = 3/2, using a powerful variational method. We switch on and off some of the contributions to understand why stability is hardly reached. 1 Sometimes, e.g., in [9], the ordering of the D̄� vs. D̄ s p threshold was not dis- cussed. le under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by https://doi.org/10.1016/j.physletb.2019.01.031 http://www.ScienceDirect.com/ http://www.elsevier.com/locate/physletb http://creativecommons.org/licenses/by/4.0/ mailto:j-m.richard@ipnl.in2p3.fr mailto:valcarce@usal.es mailto:javier.vijande@uv.es https://doi.org/10.1016/j.physletb.2019.01.031 http://creativecommons.org/licenses/by/4.0/ http://crossmark.crossref.org/dialog/?doi=10.1016/j.physletb.2019.01.031&domain=pdf J.-M. Richard et al. / Physics Letters B 790 (2019) 248–250 249 The paper is organized as follows. In Sec. 2, we present briefly the model and the variational method. The results are shown in Sec. 3. Some further comments are proposed in Sec. 4 2. Model We adopt the so-called AL1 model by Semay and Silvestre-Brac [12], already used in a number of exploratory studies of multiquark systems, for instance in our recent investigation of the hidden- charm sector c̄cqqq [13] or doubly-heavy tetraquarks Q Q q̄q̄ [14]. It includes a standard Coulomb-plus-linear central potential, sup- plemented by a smeared version of the chromomagnetic interac- tion, V (r) = − 3 16 λ̃i .λ̃ j [ λ r − κ r − � + V S S (r) mi m j σ i .σ j ] , (3) V S S = 2 π κ′ 3 π 3/2 r30 exp ( − r 2 r20 ) , r0 = A ( 2mi m j mi + m j )−B , where λ = 0.1653 GeV2 , � = 0.8321 GeV, κ = 0.5069, κ′ = 1.8609, A = 1.6553 GeVB−1 , B = 0.2204, mu = md = 0.315 GeV, ms = 0.577 GeV, mc = 1.836 GeV and mb = 5.227 GeV. Here, λ̃i .λ̃ j is a color factor, suitably modified for the quark–antiquark pairs. We disregard the small three-body term of this model used in [12] to fine-tune the baryon masses vs. the meson masses. Note that the smearing parameter of the spin–spin term is adapted to the masses involved in the quark–quark or quark–antiquark pairs. It is worth to emphasize that the parameters of the AL1 potential are con- strained in a simultaneous fit of 36 well-established meson states and 53 baryons, with a remarkable agreement with data, as could be seen in Table 2 of Ref. [12]. Before implementing any constraint of symmetry, a system q̄1q2q3q4q5 has three possible color components for an overall color singlet, five spin components for a spin s = 1/2, and four for s = 3/2. The configurations with s = 5/2 do not support any bound state in the simple chromomagnetic model and thus are not further studied in the present paper. As for color, a singlet q̄1q2 is associated with a q3q4q5 singlet, and a q̄1q2 octet can be neutral- ized by any of the two q3q4q5 octets. Three alternative bases can be obtained by replacing q2 by either q3 , q4 or q5 . In the limit of large m Q , attention was focused in the configu- ration q2q3q4q5 with optimal chromomagnetic attraction. It corre- sponds to a color 3 and spin s2345 = 0, which is a combination of the state with s23 = s45 = 0 and the one with s23 = s45 = 1. For fi- nite m Q , the three spin states with s2345 = 1 also contribute, that can match s = 1/2 when coupled to s1 = 1/2. These three latter states with s2345 = 1 allow one to build an overall s = 3/2, as well as the quark state s2345 = 2. We calculate the binding energy of mesons, baryons and pen- taquarks by means of an expansion on a set of correlated Gaus- sians, schematically �α(x1, . . .) = N∑ i=1 γi [ exp(− X̃ . Ai . X /2) ± · · · ] , (4) where the ellipses stand for terms deduced by permutations dic- tated by the symmetries of the system. The subscript α refers to the spin–isospin–color components which are coupled by the interaction (3). The vector X stands for the set of Jacobi coordi- nates describing the relative motion, namely X̃ = {x1, . . . , xn−1} for a n-body system. The matrices Ai are symmetric and definite pos- itive. The weight factors γi and the range matrices Ai are tuned by standard techniques to minimize the energy, for an increasing number of terms N , until a reasonable convergence is reached. We Table 1 Threshold masses for c̄uuds and b̄uuds, and pentaquark theoretical estimate, 5q. All masses are in GeV. D̄ � 3.016 B � 6.447 J = 1/2 D̄ s p 2.958 B s p 6.357 5q 2.966 5q 6.361 D̄∗ � 3.170 B∗ � 6.504 J = 3/2 D̄∗s p 3.098 B∗s p 6.413 5q 3.101 5q 6.418 push our calculation until the difference of introducing a new term is smaller than 2 MeV. In principle, the results are independent of the choice of any particular set of the Jacobi coordinates for the five-quark problem shown in Fig. 1 of Ref. [13]. However, some sets lead to matrices Ai which are closer to a diagonal form and thus leads to faster convergence to the lowest eigenvalue. Thus, changing the set of Jacobi coordinates and the initial values of the parameters entering the matrices Ai is a routine consistency check of such variational methods that has been carried in the present study as well as in Ref. [13]. The first concern is whether or not a state is bound below the lowest threshold, say M B , where M is a meson, and B a baryon. An immediate strategy is to detect the ground state energy lower than the threshold energy M + B . If the state is unbound, one observes a slow decrease toward M + B as N increases. It turns out useful to look also at the content of the variational wave function, which comes very close to 100% in the singlet–singlet channel of color in the M B basis. On the other hand, if a variational state converges to a bound state as N increases, then it includes sizable hidden-color components even for low N . 3. Results 3.1. Results for Q̄ uuds A calculation of the masses of mesons D(cū), . . . , B s(sb̄) and baryons p(uud), . . . �b(bud) leads to the threshold masses shown in Table 1, which also displays the best 5-body energy with the re- quired convergence, N = 5 or N = 6 in Eq. (4). No binding is found, as seen from the variational energy remaining above the threshold and from the color-content of the variational wave function, 100% in the lowest M B channel of the threshold. This negative result survives a number of changes in the model, by modifying some parameters. One of these checks consists in recalculating the threshold and pentaquark energies with the strength of the hyperfine interaction, the parameter κ′ in Eq. (1), artificially increased by a factor f ′, i.e., κ′ → f ′ κ′. As expected from the 1987 papers [5,6], stability should be reached for large f ′, when the chromomagnetic interac- tion dominates. Stability is reached for f ′ ∼ 3 in the bottom case and f ′ ∼ 1.8 in the charm sector, as expected from the 1/(mq M Q ) dependence of the chromomagnetic interaction. This means that the short-range correlation is significantly weakened in the pen- taquark as compared to its value in baryons, and that the config- uration favoring the chromomagnetic binding is not optimal when the chromoelectric and kinetic terms are included. As a further check, one can also play with the central po- tential. For instance, if the Coulomb term in (3) is reduced by a factor of 10, namely κ → κ/10, and the threshold and pentaquark energies are recalculated, then the pentaquark remains unbound, but the critical factor for forcing binding is slightly reduced, to f ′ ∼ 2.6 in the bottom sector. As already noticed in [9], this indi- cates that there is somewhat a conflict between the chromoelectric and the chromomagnetic contributions: the chromoelectric forces 250 J.-M. Richard et al. / Physics Letters B 790 (2019) 248–250 Table 2 Threshold masses for c̄ssud and b̄ssud, and pentaquark theoretical estimate, 5q. All masses are in GeV. D̄ s � 3.116 B s � 6.515 J = 1/2 D̄ � 3.243 B � 6.674 5q 3.174 5q 6.567 favor some internal configuration that is nearly orthogonal to the one optimizing the chromomagnetic term. 3.2. Results for Q̄ ssud The calculations described above are now repeated for the pen- taquark with strangeness S = −2. The results are shown in Table 2. The same tests as for S = −1 have been carried out, which con- firm the absence of binding for this kind of modeling, indepen- dently from the details of the tuning of the parameters. The factor f ′ which ensures binding when multiplying the chromomagnetic term, is now about 2.6 in the bottom sector, i.e., somewhat smaller than in the case of strangeness S = −1 [10], but still indicating that the pentaquark with one heavy antiquark and strangeness S = −2 is rather far from stability in this class of potential models. 3.3. Other flavor configurations Some calculations were carried out for the case of hidden strangeness, namely s̄uuds. In this case, the ordering of the thresh- olds are inverted as compared to the values observed in Table 1. For Q = s, K � is below ηp, while for Q = c, D̄� is above D̄ s p. No bound states were found. As a curiosity, we have run the case of a fictitious charm quark, say c′, of mass 1 GeV, i.e., intermediate between the ac- tual charm quark and the strange quark. No fine tuning is done to fix that mass. It is now observed that the thresholds c̄′ u + uds and c̄′s + uud are nearly degenerate. This provides, in principle, the opportunity to gain some attraction by mixing the configu- rations corresponding to each threshold. However, one does not get binding, because the coupling between the two configurations c̄ ′u + uds and c̄ ′s + uud is not strong enough. 4. Outlook Various configurations have been studied for the heavy pen- taquark systems Q̄ qqqq with Q = c or b and qqqq = uuds, ddsu and ssud, and spin s = 1/2 or s = 3/2, in the framework of a con- ventional constituent model. No bound state has been obtained, nor any indication for some narrow resonance in the continuum. Our results are on the line of the recent experimental findings of the LHCb collaboration [15]. To perform exploratory studies of systems with more than three-quarks it is of basic importance to work with models that correctly describe the two- and three-quark problems which thresholds are made of. Therefore, varying the parameters do not significantly affect our results, as we have checked, because the induced changes in the multiquark and threshold energies are sim- ilar. Obviously, a multiquark state contains color configurations that are not present asymptotically in the thresholds and this is the basic ingredient that may drive to a bound state. As already em- phasized in Ref. [13], and shown in Figs. 2 or 3 of this reference, there is a strong competition between the color-spin configura- tions favored by the chromoelectric terms and the ones favored by the chromomagnetic terms, and this mismatch spoils the possible binding of pentaquarks with anticharm or beauty. We have explored different possibilities for the mass m Q of the heavy quark. Increasing the mass of the heavy quark, separates the two thresholds as seen in Table 1 when comparing the results for charm and bottom cases. A large m Q induces a large chromoelec- tric attraction in the Q̄ s pair, but the same attraction is present in the lowest threshold, B s p. On the contrary, a mass m Q � 1 GeV makes the two thresholds, B� and B s p, nearly degenerate. Then, one may expect a favorable mixing. However, there is a conflict be- tween the color-spin configurations favored by the chromoelectric terms and the chromomagnetic ones. For the resonances, our conclusion is based on the content of the variational wavefunctions, which are found to consist mainly of two color-singlets in the channel corresponding to the lowest threshold. For the light pentaquark states and for the hidden-flavor sector Q̄ Q qqq, the method of real scaling has been used, which is rather demanding in terms of computation [16–18]. Certainly, a critical comparison of the different methods of handling resonances is in order. 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