Bled Workshops in Physics Vol. 20, No. 1 p. 89 A Proceedings of the Mini-Workshop Electroweak Processes ofHadrons Bled, Slovenia, July 15-19, 2019 A phenomenological lower bound for the E+c+ mass Mitja Rosina Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box 2964, 1001 Ljubljana, Slovenia and J. Stefan Institute, 1000 Ljubljana, Slovenia Abstract. We show that a simple interpolation between mesonic binding energies can give a good semiquantitative binding energy of the cc diquark and the baryon. The mass of the baryon is almost insensitive to widely different choices of the constituent quark masses. 1 Introduction After the discovery of the E++ baryon at LHCb, there is a strong interest to verify whether the quark models which have been successful for light and single-heavy hadrons apply also to double-heavy hadrons; in particular, how rich spectrum we can expect. It is important to check whether we may use the same effective quark-quark interaction (apart from the colour factor and the mass-dependent spin-spin term): Vuu = Vcu = Vcc = Vce = Vbu = Vbb = Vbb. For this purpose it is instructive to study some phenomenological models even if the results are only semiquantitative. The present study is based on two assumptions: (1) The quark-quark interaction in colour-triplet state is half the quark-antiquark interaction in colour-singlet state. (2) The ccu baryon can be treated as a two-body system, the cc diquark plus the u quark, similar to the cu or bu meson. These assumptions have been made already by several authors, for example [1,2]. The purpose of this presentation is to show a nice trick how to obtain easily the binding energies of meson-like systems by a simple interpolation between mesonic data [3]. 2 The cc diquark interpolated between mesons We compare the nonrelativistic Schrodinger equations for an (ab) meson in the colour singlet state and for an (ab) diquark in a colour antitriplet state (with twice weaker interaction): 2m, + V ab alb ^ = Eab ^ = F(mab 2 P 90 Mitja Rosina 2m; + Vab ab 2m ab + 2 Vab P2 2(mab/2) + V alb = Eab ^ = 1 F(1 mab)^. Here the reduced masses are mab = ma mb/ (ma + mb) and mab = ma mb/(ma + mb), respectively. The binding energy F(m) is a smooth function of m as illustrated in Fig. 1. Phenomenological binding energies of mesons are obtained from experimental meson masses M and model vales of constituent quark masses: Eab = Mab — ma — mb. The diquark masses are then predicted (Table 1). The trick is to take for the diquark binding energy 2F( 1 mab), according to the above Schrodinger equation. The constituent quark masses in Fig. 1 and Table 1 are taken from Bhaduri [4]: mu,d,s,c,b = 337, 337, 600, 1870, 5259 MeV, and in Table 1 also from Karliner and Rosner [1]: mu,d,s,c,b = 310, 310, 483,1663, 5004 MeV. 2 2 P P \F{m) 0.0 [GeV] -0.1 -0.2 -0.3 -0.4 -0.5 0.5 1.0 1.5 2.0 2.5 rri [GeV] Fig. 1. The meson binding energy F(m), multiplied by 1, as a function of the reduced mass m = mag. The diquark binding; energies lF(2mab) are then predicted by interpolation. (From [3]). 3 The binding of tine E++ baryon The (cc)u baryon is treated as a two-body system. The reduced mass is m = mu Mcc/(mu + Mcc), where Mcc = 2mc + 2F(2mcc) and the binding energy between the u quark and the (cc) diquark E(cc)u = F(m) is obtained by interpolation in Fig. 1 or Table 1. The mass of the E+++ baryon is then M(ic)u = Mcc+mu + E(cc)u =3605 (3596) MeV for the choice of constituent masses of Badhuri (or Karliner-Rosner). A phenomenological lower bound for the mass 91 Table 1. The interpolation between mesons. The tilde means spin average, A is the difference between the vector and scalar mesons, m is the reduced mass for mesons and half the reduced mass for diquarks; F is the meson or baryon binding energy and twice the diquark binding energy. Reduced masses refer to the constituent quark masses of Bhaduri [4] or Karliner-Rosner [1], respectively. Energies and masses are in MeV. In the 6th and 9th column are predictions for the diquark and double heavy baryons. Meson mass A m Bha F mass predict m Kar-Ros F mass predict D 1973 141 286 -234 261 0 B 5314 46 317 -282 292 0 Ds 2076 144 454 -394 374 -70 BS 5403 48 539 -456 440 -84 i 3069 113 935 -671 832 -257 Y 9445 61 2630 -1073 2502 -563 cc 467 -405 3538 416 -80 3286 bb 1315 -819 10108 1251 -383 9817 (cc)u 308 -268 3605 283 0 3596 (bb)u 317 -282 10163 301 -4 10123 4 The hyperfine correction So far, spin averages were taken for the diquark and baryon binding energies. The hyperfine splitting is obtained from the experimental differences between vector and scalar mesons. The cc diquark (S = 1) is therefore heavier by (1/4)A(^)/2 = 113 MeV/8 = 14 MeV. (The extra (1/2) comes from the fact, that the potential in cc colour triplet state is twice weaker than in mesons.) On the other hand, the (cc)u (S = 2) baryon is lighter by « A(D) (1870/3552) = -74 MeV. (The latter factor takes into account that the spin-spin interaction is inversely proportional to both masses, so instead of the u quark mass in the D meson one takes the (cc) mass. Also, it is convenient that the reduced mass of (cc)u is close to that of D and Ds mesons, so the interpolation is trivial.) The result for the E++ mass is then 3545 MeV (Badhuri quark masses) or 3539 MeV (Karliner-Rosner quark masses). 5 A note on the binding energy of the DD* dimeson We cannot estimate the binding energy of the DD* dimeson in the same way since the (cc)ub ("tetraquark" or "atomic" or " He-like") configuration is about 100 MeV above the D+D* threshold [3]. This is then only a minor configuration, the main configuration is a DD* "molecule", with a covalent bond like the H2 92 Mitja Rosina molecule. In the restricted 4-body space with the two c quarks far apart and a general wavefunction of U and d the energy is also above the D+D* threshold, as presented by several authors. Only combining both types of configurations brings the energy below the threshold, as shown by Janc and Rosina [5-7]. In the nonrelativistic calculation with the one-gluon exchange potential (including the chromomagnetic term) plus the linear confining potential they obtain the binding energy (DD*) - (D + D*) = - 2.7 MeV . The model parameters (Grenoble AL1) [8] fitted all relevant mesons and baryons and a rich 4-body space was used (Gaussian expansion at optimized distances, with 3 types of Jacobi coordinates). We pose an important question ("to be discussed at the next Bled Workshop") whether the pion and sigma clouds between the u and d antiquarks can increase binding, in analogy with the deuteron. Is there a double counting? Would it be necessary to refit the model parameter so much that this extra binding would be compensated? If, however, the binding really becomes much stronger, at least below -6 MeV, the DDn decay channel would be closed, the DD*system would live longer and would be easier to be recognized in experiment. 6 Conclusion The phenomenological binding energies of the cc diquark and the E++ baryon can be obtained by interpolation between the mesonic data. The mass of the E+++ baryon is a lower bound, further corrections (eg. the Coulomb energy and the finite size of the cc diquark) would raise it, possibly close to the experimental value. It is instructive to see that the final result depends only very weakly on the choice of quark constituent masses. In the binding energy, larger constituent masses (larger by as much as 200 MeV) are compensated by a stronger attractive potential. References 1. M. Karliner and J. L. Rosner, Phys. Rev. D 90 (2014) 094007. 2. M. Karliner and J. L. Rosner, Bled Workshops in Physics 20, No. 1 (2019) , these Proceedings; also available at http://www-f1.ijs.si/BledPub. 3. D. Janc and M. Rosina, Few-Body Systems 31 (2001) 1-11; also available at arXiv:hep-ph/0007024v3. 4. R. K. Bhaduri L. E. Cohler, Y. Nogami, Nuovo Cim. A65 (1981) 376. 5. D. Janc and M. Rosina, Few-Body Systems 35 (2004) 175-196; also available at arXiv:hep-ph/0405208v2. 6. M. Rosina and D. Janc, Bled Workshops in Physics 5, No. 1 (2004) 74; also available at http://www-f1.ijs.si/BledPub. 7. M. Rosina, Bled Workshops in Physics 18, No. 1 (2017) 82; also available at http://www-f1.ijs.si/BledPub. 8. B. Silvestre-Brac, Few-Body Systems 20 (1996) 11.