Hadronic Absorption Cross Sections of B c Meson

1. Hadronic Absorption Cross Sections of Bc Meson M. A. K. Lodhia,b and Rian Marshalla aDepartment of Physics, Texas Tech University, Lubbock, Texas 79409, USA, bCenter for High Energy Physics and Department of Space Science University of The Punjab, Lahore, Pakistan a.lodhi@ttu.edu 18th International Conference on Few-Body Problems in Physics Santos, Brazil August 21 - 26, 2006

2. Bc Absorption Contents • Introductory Remarks • Meson Exchange Model • Comover Absorption • SU(5) Hadronic Lagrangians • Coupling Constants • Form Factor Cut Off Parameter • Energy Dependence of Cross Section • Concluding Remarks

3. Bc Absorption: Introduction The suggestion of J/ as a signature for quark-gluon plasma (QGP) production [1] is a well studied issue. An anomalously large suppression of J/ in events with moderate to large transfer energy from the Pb + Pb collisions at P-lab = 158 GeV/c in the NA50 experiment at CERN-SPS [2] was interpreted as evidence for holding it true. However, J/ dissociation due to comoving hadrons could contribute significantly to the observed suppression. It is important to distinguish a QGP-reduced charmonium production amplitude from subsequent dissociative scattering, otherwise the interpretation of this signal will be ambiguous. Following the example of charmonium system the interest grew to examine other systems of quarkonia. The case of  production and dissociation has been studied as well [3,4].

4. Bc Absorption: Introduction • The production rate of heavy mixed flavor systems such as Bc would also be affected by the presence of a QGP state [5, 6]. • The idea behind this suggestion is that Bc production could be enhanced due to the presence of unpaired b(b bar) and c(c bar) quarks in the QGP, which upon encounter could form Bc and possibly survive due to a relatively high binding energy. • It is the Bc system that we wish to address; like the other states of quarkonia one must factor in the signal suppression due to absorption by hadronic comovers. • Here we focus on Bc production & absorption in QGP • The same concept is also extended to include heavy flavored baryon production like bc and Ωccc [10].

5. Meson Exchange Model • We use a meson-baryon exchange model, which has been used to study charmonium absorption by pions, rho mesons [11–16] and nucleons [17, 18]. The same technique has also been used for  absorption by pions and rho mesons [4]. In this work we consider absorption by the exchange of charmed hadrons, specifically D mesons, D* mesons and c baryons. We neglect, at least in this initial study, bottom exchange processes as they represent a heavier degree of freedom and therefore are possibly less important.

6. Endothermic Reactions Prior to the deduction of heavy mesons some of them would be destroyed in collisions with other co-moving hadrons. These reactions are endothermic, i.e. their Q-value is negative and there is some threshold bombarding energy below which the reactions could not proceed. Some typical reactions that could take place are: • J/ + π  D + D* • J/ + N c + D • Bc + π  B + D • Bc + N c + B  Heavy mesons detected = Mesons produced - mesons absorbed

7. Co-mover Absorption We examine the Bc absorption by nucleons by considering the following reactions:

9. SU(5) Hadronic LagrangiansThe Lagrangians LD*Nc , LDNc and LBccb are typical fermion - pseudoscalar flavor conserving [17]. The Lagrangians LBcBD* and LBcB*D are taken from an SU(5) flavor-invariant [4]. In these expressions D, B, and N respectively refer to the doublets (D0,D+), (B+,B0) and (p, n). The above diagrams require a total of five Lagrangians, which are given as:

10. SU(5) Lagrangians A general SU(N) flavor-invariant Lagrangian for mesons is constructed by first assuming that strict SU(N) symmetry is observed and treating the vector (V) fields as the gauge fields that mediate interactions with pseudoscalar (P) mesons. This construction leads to several couplings, including a (PPV) coupling of the form: Lint = igTr(µP[P,Vµ], (10) The fields P and Vμ in Eq.(10) denote the respective (NxN) matrices for SU(N). Once one has obtained an expression for Lint the necessary mass terms are simply added.

11. AmplitudesThe amplitudes obtained from these Lagrangians for above diagrams are respectively given as:

12. MassesThe |picm|2 = (1/4s)[s − (m1 − m2)2][s − (m1 + m2)2], where mi stand for the initial state particle masses, s = (p1+p2)2 and p1, p2 are the initial state (nucleon and Bc) four-momenta. The masses used in this work are given as follows:

13. Coupling Constants The coupling constants gD*Nc and gDNc are obtained by using SU(4) symmetry to estimate these coupling constants from the empirical value for gπNN [18]. For gBcB*D and gBcBD* we take the value gBB = 12.6, which was obtained using vector meson dominance (VMD) [4] and then relate using SU(5) symmetry to obtain gBcB*D = gBcBD* . Due to a lack of any plausible similar relations for g Bccb we simply use the value of a similarly flavored system, i.e., we let gBccb = gBcB*D. The values of the coupling constants and the methods for obtaining them are summarized in Table 1.

14. Coupling Constants

15. Form Factor The finite size of the hadrons involved necessitates the introduction of form factors at the vertices. We have chosen a monopole form factor: F = 2/(2 + q2) (16) where  is a cutoff parameter and q2 is the square of the exchange 3-momentum for the system, e.g., q2 = (p1 − p3)2 = (p4 − p2)2. Some fairly suppressive values of  (= 1 GeV) have been chosen for all vertices. Whereas the choice of cutoff is somewhat arbitrary, the values chosen here do have the advantage of allowing direct comparison to the other work [18] involving the study of J/ absorption by nucleons. The form factor then takes the form: F = 1/(1 + q2) (17)

16. Cross Sections -B production

17. Cross Sections -B* production

18. Cross Sections -D production

19. Cross Sections - D* production

20. Discussion & Conclusion • These figures show the Bc absorption cross sections by nucleons as a function of the initial state center-of-mass energy √s. • The processes all peak near threshold. • These cross sections are significant in magnitude for the most part (at threshold), on the order of a few mb. • The cross section estimates are, however, highly dependent on values used for the coupling constants and the choice of form factor, as well as the cutoff value used in the form factor.

21. Discussion & Conclusion • Finally, as pointed out in [5], one might expect a lower absorption rate for Bc in relation to J/ absorption due to the higher binding energy of the Bc system. • One obvious contributor, however, to the (generally) higher cross sections in this work would be the value that we use of 11.9 for gBcB*D and gBcBD* as opposed to the comparable coupling of gDD = 7.64 used in [17, 18]. • This difference of over 50% with respect to gDD alone affects the overall cross section magnitude by a multiplicative factor of about 18. • Furthermore, the introduction of b-flavored exchange processes and anomalous parity interactions [15, 16, 18] would lead to amplitude mixing, again affecting the overall results.

22. Discussion & Conclusion • The cross sections involving form factors range in peak values from roughly 0.1 mb for Bc + Nc + B* to about 7 mb for Bc + N c + B. • It is interesting to note that the relative suppressionof the cross section involving B* creation is due mainly to the inclusion of the form factors. • One can see this by comparing the graphs for B and B* creation without form factors, which corresponds to the (unphysical) limit of point-like hadrons. • Comparison of these two graphs indicates that the difference in the two cross sections, while significant, is nonetheless much smaller when the form factors are not included. • In fact one sees that for B creation the form factors suppress the cross section at threshold by a factor of 6, roughly, whereas B* is suppressed by nearly two orders of magnitude.

23. Thank You