Bc Absorption Cross Sections by Pion in Meson-Exchange Model

Hadronic Cross Sections of Bc Mesons

A thesis submitted for the fulfillment of Ph.D degree in Physics

Faisal Akram

Physics Department University of the Punjab Lahore, Pakistan (2013)

Declaration

The results of the cross sections of Bc meson by pions discussed in the chapter 4 of the thesis have appeared in M.Phil thesis of F. Akram and the paper

 Faisal Akram, Shaheen Irfan, and M.A.K. Lodhi, Hadronic absorption cross sections of Bc, Phys. Rev. C 84, 034901 (2011).

The results of the cross sections of Bc meson by rho meson presented in the chapter 4 of the thesis have appeared in the paper

 Faisal Akram, M.A.K Lodhi, Bc absorption cross sections of rho mesons, Phys. Rev. C 84, 064912 (2011).

The results on K mesons are to be published in the paper

 Faisal Akram, Bc absorption cross sections of K mesons.

The results on the cross sections of Bc meson by nucleons presented in the chapter 5 of the thesis have appeared in the paper

 Faisal Akram, M.A.K Lodhi, Bc absorption cross sections of nucleons, Nucl. Phys. A 877, 95 (2012).

The copyrights of the thesis rest with the author.

Faisal Akram

List of Papers Attached:

Faisal Akram, M.A.K Lodhi, Bc absorption cross sections of rho mesons, Phys. Rev. C 84, 064912 (2011).

Faisal Akram, M.A.K Lodhi, Bc absorption cross sections of nucleons, Nucl. Phys. A 877, 95 (2012).

Faisal Akram, et. al., Dynamical Chiral Symmetry Breaking: Phase diagrams of unquenched QED with 4-fermion contact interaction, Phys. Phys. Rev. D 87, 013011 (2013).

iii

Certificate

It is certified that Faisal Akram s/o Muhammad Akram carried out the work contained in this thesis at Physics Department, Punjab University Lahore under my supervision.

______(Dr. Shaukat Ali) Professor, Physics Department, Punjab University, Lahore.

______(Dr. Mujahid Kamran) Vice Chancellor, Punjab University, Lahore.

Submitted through:

______Chairperson, DPCC, Punjab University, Lahore.

Acknowledgements

I would like to thank my supervisors Dr. Shaukat Ali and Dr. Mujahid Kamran for their support and my collaborator Dr. M.A.K Lodhi for his valuable contribution in the work. I would also like to thank my wife Nargis for her years of patience and never complaining about anything. Thank you Nargis!

Abstract

In this work we calculate Bc absorption cross sections by light mesons and baryons; π, ρ , and K mesons and nucleons. The processes studied are important from the view of relativistic heavy-ion experiments. The formalism used for the calculations is based on an eﬀective hadronic Lagrangian developed by imposing SU(5) symmetry. This approach is similar to other papers in the literature in which absorption cross sections of J/ψ and Υ mesons are calculated using the hadronic Lagrangian based on SU(4)/SU(5) symmetry. The motivation for these calculations comes from the studies of the production of heavy mixed ﬂavor mesons and baryons in quark gluon plasma.

It is expected that Bc production could be enhanced in heavy-ion experiments due to formation of QGP. However, it is also expected that the Bc production rate in heavy-ion collisions will be affected by its interaction with comoving hadrons. Thus we require the knowledge of the cross sections of Bc absorption processes to interpret the observed rates in the experiments. We find that calculated values of cross sections highly depend upon the choice of form factors, related cutoff parameters, and coupling constants. The effect of uncertainties in these parameters on the cross sections is also studied. These results could be useful in calculating the production rate of Bc meson in relativistic heavy-ion collisions. Contents

1 Introduction 12

2 The Hadronic Lagrangian 17 2.1 Irreducible tensors ...... 17 2.2 The Eﬀective Lagrangian ...... 20 2.2.1 Meson-Meson Interaction ...... 20 2.2.2 Baryon-Meson Interaction ...... 24 2.2.3 Anomalous parity interaction ...... 27 2.3 Vertex factors of interactions ...... 27

3 Bc Absorption Processes: Born Diagrams and Amplitudes 39

3.1 Bc absorption processes ...... 39

3.2 Amplitudes of Bc absorpriton by pions: ...... 41

3.3 Amplitudes of Bc absorpriton by rho mesons: ...... 43

3.4 Amplitudes of Bc absorpriton by K mesons ...... 45

3.5 Amplitudes of Bc absorption by nucleons ...... 48 3.6 Eﬀect of anomalous parity interaction ...... 50 3.7 Couplings and Form Factors ...... 52 3.8 Vector-Meson dominance model ...... 54

4 Bc Absorption Cross Sections by π, ρ, and K Mesons 58 4.1 Spin and Isospin averaged cross sections ...... 58

4.2 Cross sections of Bc by π ...... 59 + → 4.2.1 Bc π DB ...... 59

1 + → ∗ ∗ 4.2.2 Bc π D B ...... 60

4.3 Cross sections of Bc by ρ ...... 60 + → ∗ 4.3.1 Bc ρ D B ...... 60 + → ∗ 4.3.2 Bc ρ DB ...... 61

4.4 Cross sections of Bc by K ...... 61 + → + 4.4.1 KBc Ds B ...... 61 + → ∗+ ∗ 4.4.2 KBc Ds B ...... 62 + → 0 4.4.3 KBc DBs ...... 62 + → ∗ ∗0 4.4.4 KBc D Bs ...... 63 4.5 Numerical values of the couplings and other input parameters . . . . . 64

4.6 Results of Bc absorption cross sections by π ...... 65 4.6.1 Results without form factors ...... 65 4.6.2 Form Factors ...... 67

4.6.3 Amplitudes of Bc absorption by pions with form factor . . . . . 69 4.6.4 Results with form factor ...... 70

4.7 Results of Bc absorption cross sections by ρ ...... 72 4.7.1 Results without form factors ...... 72

4.7.2 Amplitudes of Bc absorption by ρ with form factor ...... 74

4.7.3 Results of Bc absorption by ρ with form factor ...... 76

4.8 Results of Bc absorption cross sections by K ...... 79 4.8.1 Results without form factors ...... 79

4.8.2 Amplitudes of Bc absorption by K with form factor ...... 81

4.8.3 Results of Bc absorption by K with form factor ...... 85

5 Bc Absorption Cross Sections by Nucleons. 101

5.1 Cross sections of Bc by N ...... 101 + → 5.1.1 NBc ΛcB ...... 101 + → ∗ 5.1.2 NBc ΛcB ...... 102 − → 5.1.3 NBc DΛb ...... 102 − → ∗ 5.1.4 NBc D Λb ...... 102

2 5.2 Numerical values of input parameters ...... 103 5.3 Results and Discussion ...... 104 5.4 Eﬀect of anomalous parity interaction ...... 107

6 Discussion 117

Appendix A Feynman Rules for Vertex 121

Appendix B Kinematics 123

3 List of Figures

2.1 Vertex factors of the coupling given in Eqs. 2.21a and 2.21b...... 28 2.2 Vertex factors of the couplings given in Eqs. 2.21c, 2.21d, and 2.21e. . . 29 2.3 Vertex factors of the couplings given in Eqs. 2.22a, 2.22b, 2.22c, and 2.22d 30 2.4 Vertex factors of the couplings given in Eqs. 2.22e and 2.22f...... 31 2.5 Vertex factors of the couplings given in Eqs. 2.23a and 2.23b...... 32 2.6 Vertex factors of the couplings given in Eqs. 2.23c and 2.23d...... 33 2.7 Vertex factors of the couplings given in Eqs. 2.23e and 2.23f...... 34 2.8 Vertex factors of the couplings given in Eqs. 2.23g and 2.23h...... 35 2.9 Vertex factors of the couplings given in Eqs. 2.33a and 2.33b...... 36 2.10 Vertex factors of the couplings given in Eqs. 2.33c, 2.33d, and 2.33e. . . 37 2.11 Vertex factors of the couplings given in Eqs. 2.36 ...... 38

+ → 3.1 Born diagrams of Bc absorption process Bc π DB ...... 42 + → ∗ ∗ 3.2 Born diagrams of Bc absorption process Bc π D B ...... 43 + → ∗ 3.3 Born Diagrams of Bc absorption process Bc ρ D B ...... 44 + → ∗ 3.4 Born Diagrams for Bc absorption process Bc ρ DB ...... 44 + → + 3.5 Born Diagrams for Bc absorption process KBc Ds B ...... 45 + → ∗+ ∗ 3.6 Born Diagrams for Bc absorption process KBc Ds B ...... 46 + → 0 3.7 Born Diagrams for Bc absorption process KBc DBs ...... 47 + → ∗ ∗0 3.8 Born Diagrams for Bc absorption process KBc D Bs ...... 48 + → 3.9 Born Diagrams for Bc absorption process NBc ΛcB ...... 49 + → ∗ 3.10 Born Diagrams for Bc absorption process NBc ΛcB ...... 50 − → 3.11 Born Diagrams for Bc absorption process NBc DΛb ...... 51

4 − → ∗ 3.12 Born Diagrams for Bc absorption process NBc D Λb ...... 52 + → ∗ 3.13 Additional diagrams for Bc absorption processes: (2) NBc ΛcB and − → ∗ (4) NBc D Λb, due to anomalous parity interaction...... 53 3.14 Vector Meson Dominance (VMD) model...... 54

+ → 4.1 Bc absorption cross sections for the process Bc π DB without form factor ...... 67 + → ∗ ∗ 4.2 Bc absorption cross sections of the process Bc π D B without form

∗ ∗ factor for three different values four-point coupling gπBcB D = 105, 200, 295 for dotted, solid and dashed curves respectively...... 68 + → 4.3 Bc absorption cross sections of the process Bc π DB with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively...... 71 + → 4.4 Comparison of the cross sections of the process Bc π DB with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 72 + → ∗ ∗ 4.5 Bc absorption cross sections for the process Bc π D B with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively...... 73 + → ∗ ∗ 4.6 Comparison of the cross sections of the process Bc π D B with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 74 + → ∗ ∗ 4.7 Bc absorption cross sections for the process Bc π D B with form fac-

∗ ∗ tor for three diﬀerent values of four-point coupling, gπBcB D = 105, 200, 295 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV...... 75

4.8 Total Bc absorption cross sections by pion for three diﬀerent values

∗ ∗ of four-point coupling, gπBcB D = 105, 200, 295 for dotted, solid and dashed curves respectively. Cutoﬀ parameter is taken 1.5 GeV...... 76

5 + → ∗ 4.9 Bc absorption cross sections for the process Bc ρ D B without form factor...... 77 + → ∗ 4.10 Bc absorption cross sections for the process Bc ρ DB without form factor...... 78 + → ∗ 4.11 Bc absorption cross sections for the process Bc ρ D B with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively...... 79 + → ∗ 4.12 Comparison of the cross sections of the process Bc ρ D B with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 80 + → ∗ 4.13 Bc absorption cross sections for the process Bc ρ DB with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively ...... 81 + → ∗ 4.14 Comparison of the cross sections of the process Bc ρ DB with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 82 + → ∗ 4.15 Bc absorption cross sections of the processes Bc ρ D B for three dif-

∗ ∗ ferent values of the four-point couplings gρBcBD and gρBcB D = 15, 30, 45 for dotted, solid and dashed curves, respectively. Cutoﬀ parameter is taken 1.5 GeV...... 83 + → ∗ 4.16 Bc absorption cross sections of the processes Bc ρ DB for three dif-

∗ ∗ ferent values of the four-point couplings gρBcBD and gρBcB D = 15, 30, 45 for dotted, solid and dashed curves, respectively. Cutoﬀ parameter is taken 1.5 GeV...... 84 + → + 4.17 Bc absorption cross sections of the process KBc Ds B without form factor...... 85 + → 0 4.18 Bc absorption cross sections of the process KBc DBs without form factor...... 86

6 + → ∗+ ∗ 4.19 Bc absorption cross sections of the process KBc Ds B without form factor...... 87 + → ∗ ∗0 4.20 Bc absorption cross sections of the process KBc D Bs without form factor...... 88 + → + 4.21 Bc absorption cross sections of the process KBc Ds B with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively...... 89 + → 0 4.22 Bc absorption cross sections of the process KBc DBs with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively...... 90 + → 0 4.23 Comparison of the cross sections of the process KBc DBs with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 91 + → 0 4.24 Comparison of the cross sections of the process KBc DBs with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 92 + → ∗+ ∗ 4.25 Bc absorption cross sections of the process KBc Ds B with form factors. Lower and upper curves are with cutoff parameter Λ = 1 and 2 GeV respectively...... 93 + → ∗ ∗0 4.26 Bc absorption cross sections of the process KBc D Bs with form factors. Lower and upper curves are with cuttoff parameter Λ = 1 and 2 GeV respectively...... 94 + → ∗+ ∗ 4.27 Comparison of the cross sections of the process KBc Ds B with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and 2 GeV respectively. . . . . 95

7 + → ∗ ∗0 4.28 Comparison of the cross sections of the process KBc D Bs with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively. . . . . 96 + → ∗+ ∗ 4.29 Bc absorption cross sections of the process KBc Ds B with form

∗ ∗ factor for three diﬀerent values of four-point coupling, for gKBcDs B = 149, 283, and 417 for dotted, solid and dashed curve respectively. Cut- oﬀ parameter is taken 1.5 GeV...... 97 + → ∗ ∗0 4.30 Bc absorption cross sections of the process KBc D Bs with form

∗ ∗ factor for three diﬀerent values of four-point coupling, for gKBcD Bs = 149, 283, and 417 for dotted, solid and dashed curve respectively. Cut- oﬀ parameter is taken 1.5 GeV...... 98

+ 4.31 Total Bc absorption cross sections by kaons for three diﬀerent values of

∗ ∗ ∗ ∗ four-point coupling, gKBcDs B = gKBcD Bs = 149, 283, 417 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV. 99

+ 4.32 Total Bc absorption cross sections by anti kaons for three diﬀerent val-

∗ ∗ ∗ ∗ ues of four-point coupling, gKBcDs B = gKBcD Bs = 149, 283, 417 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV...... 100

+ → 5.1 Bc absorption cross sections of the process (i)NBc ΛcB using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively...... 105 + → ∗ 5.2 Bc absorption cross sections of the process (ii) NBc ΛcB using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively...... 106

8 − → 5.3 Bc absorption cross sections of the process (iii) NBc DΛb using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively...... 107 − → ∗ 5.4 Bc absorption cross sections of the process (iv) NBc D Λb using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively...... 108 + → 5.5 Bc absorption cross sections of the process (i)NBc ΛcB using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively...... 109 + → ∗ 5.6 Bc absorption cross sections of the process (ii) NBc ΛcB using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively...... 110 − → 5.7 Bc absorption cross sections of the process (iii) NBc DΛb using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively...... 111 − → ∗ 5.8 Bc absorption cross sections of the process (iv) NBc D Λb using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively...... 112

9 5.9 Bc absorption cross sections of the process (ii) using the values of the couplings given in set 1. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV...... 113

5.10 Bc absorption cross sections of the process (iv) using the values of the couplings given in set 1. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV...... 114

5.11 Bc absorption cross sections of the process (ii) using the values of the couplings given in set 2. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV...... 115

5.12 Bc absorption cross sections of the process (iv) using the values of the couplings given in set 2. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV...... 116

10 List of Tables

3.1 Values of the constants used in VMD model calculations...... 57

4.1 Values of coupling constants required for calculating Bc absorption cross sections by π, ρ and K mesons...... 66

5.1 Two sets of the values of coupling constants used in this paper. . . . . 104 5.2 The peak values of the cross sections of the four processes with and without b-ﬂavor exchange, using coupling values of set 1 and 2. . . . . 109

11 Chapter 1

Introduction

Quantum chromo dynamics (QCD) [1] is currently accepted theory of strong interaction in the Standard Model of the elementary particles. It is a non-abelian gauge theory which is based on the color symmetry group SU(3). In QCD the color charge is considered to be the fundamental property of the quarks and the gluons due to which they interact through strong interaction. Like all non-abelian gauge theories, QCD is also renormalizable and exhibits asymptotic freedom [2] i.e., at large momentum transfer the coupling constant of strong interaction approaches to zero, if the number of quark flavors is less than 33/2. Having the asymptotic freedom in QCD means that we can apply perturbation theory to strong interaction processes at the energy scale greater than QCD scale ΛQCD ≈ 200 MeV where coupling constant of QCD is much less than 1. However at low energy, strong coupling is of order 1 or higher and hence the perturbation theory is not applicable. Hence, at this energy scale one has to apply less reliable and generally more difficult non-perturbative methods like Lattice simulation of QCD [3], Schwinger-Dyson and Bethe-Salpeter approaches [4, 5]. These non-perturbative methods show that QCD has two important features at low energy: i) Confinement [6] and ii) Dynamical breaking of chiral symmetry [7]. QCD Lagrangian possesses chiral symmetry SU(3)L ⊗ SU(3)R if the masses of the light quarks are taken zero. However, the chiral symmetry breaks dynamically as the light quarks gain large dynamical mass due quark condensation. Both the Lattice QCD and Schwinger-Dyson approaches show that the chiral symmetry is restored at certain critical temperature,

12 which converts the large constituent masses of u, d and s quarks to their lower current values. It is also shown by the Lattice QCD that at the critical temperature of about 173 MeV there occur a rise in the effective degree of of freedom of a Hadronic system. This rise in the degree of freedom is associated with the phase transition in which the confined state of matter in converted into its deconfined state called Quark Gluon Plasma (QGP). Presently there are several experimental studies going on to discover this rare state of matter. In these studies the QGP is formed by the collisions of heavy ions. The heavy ion collision experiments involve the processes in which the ions are accelerated to near the velocity of light before smashing upon each other. The available translational energy of the ions is converted into thermal energy and it is expected that the resultant temperature is high enough to produce QGP. Study of the formation of QGP and its properties in heavy ion collision experiments has been an active area of research. Generally there are two experimental probes of studying QGP; the electromagnetic and hadronic probes. Electromagnetic signatures are direct but weak, whereas hadronic signatures are strong but affected by the re-interactions and other non-QGP effects in the period between formation and detection of the hadrons. Strangeness enhancement due to conversion of constituent mass of strange quark into current mass and quarkonia suppression due to color Debye screening are two hadronic tests which are being investigated in relativistic heavy-ion collider (RHIC) at BNL (Brookhaven National Laboratory) and Large hadron collider (LHC) at CERN (Eu- ropean Center of Nuclear Research). As noted above that the hadronic signatures are affected by the final state interactions between the hadrons. Thus, we require the knowledge of relevant interaction cross sections to subtract this effect, otherwise the observed experimental results carry no information about the formation of QGP. Suppression of J/ψ due to color Debye screening in quark gluon plasma was suggested by T. Matsui and H. Satz [8] in 1986. In this case the suppression was first studied in NA50 [9] experiment at CERN and then by PHENIX at BNL [10]. However the observed suppression in J/ψ could be the blend of many other effects occurring in the deconfined and the hadronic states of the matter. Absorption cross sections of J/ψ by light mesons and nucleons, which are required to study the effect interaction with

13 the comovers, are rather well studied as compared to Υ. Empirical studies of J/ψ pho- toproduction from nucleons/nuclei [11] and J/ψ production from nucleon-nucleus [12] interactions show that J/ψ-nucleon cross sections range from ∼ 1 mb to ∼ 7 mb. The- oretically, these cross sections have been calculated using perturbative QCD [13], QCD sum-rule approach [14], quark potential models [15] and meson-baryon exchange models based on the hadronic Lagrangian having SU(4) flavor symmetry [16, 17, 18]. The values of the cross sections obtained from quark potential model and meson-baryon exchange models tend to agree with the empirical values, but are much larger than those from pQCD or QCD sum-rule approach. The higher values of the cross sections suggest that the absorption of J/ψ by the comovers may play significant role. On the other hand it is also suggested in Refs. [22, 23] that at higher collision energy accessi- ble at LHC, large cc pairs production could result into J/ψ enhancement due to heavy quark stopping effect in QGP. Recent studies of J/ψ in ALICE (A Large Ion Collider Experiment) at LHC has hinted at this regeneration effect [24]. Like charmonium, bottomonium states are also affected in the QGP by color Debye screening [8, 19]. In this case the related absorption cross sections are calculated using meson-exchange model in ref. [20]. However, the results of this paper suggest that the effect of Υ absorption by hadronic comovers may be insignificant due to smaller values of the absorption cross sections and higher value of threshold energy. In ref. [20] the absorption cross sections of Υ with pion and rho mesons are calculated without anomalous parity interaction. The results of total cross section may change if anomalous parity interaction is included or cross sections with other light mesons and baryons are also included. Recently the suppression of Υ is observed at CMS [21]. This phenomena could be very important for understanding the properties of the QGP. However, one need to establish whether the observed suppression is due to the formation of QGP or interaction with the comovers, or both. And that requires the exact knowledge of absorption cross sections of Υ.

In this work we have focused on Bc mesons, which are the bound state of b and c quarks. We calculate the absorption cross sections of Bc meson with light mesons and baryons like π, ρ, and K mesons [25, 26] and nucleons [37] using meson-baryon

14 exchange model based on a hadronic Lagrangian having SU(5). The work is useful for current and future heavy ion studies in RHIC at BNL and LHC in CERN. The motivation for these calculations comes from the studies of the production of heavy mixed ﬂavor mesons and baryons in quark gluon plasma [28, 29, 30]. It is expected that

Bc production could be enhanced in heavy-ion experiments due to formation of QGP [28]. QGP contains many unpaired b(b) and c(c) quarks due to color Debye screening, which upon encounter could form Bc and probably survive in QGP due to relatively high binding energy [31]. Similar eﬀect is also suggested for Ξbc and Ωccc baryons in ref.

[30]. However, it is also expected that the Bc production rate in heavy-ion collisions will be affected by its interaction with comoving hadrons, mainly π, ρ, and K mesons [32, 33] and nucleons [16]. Thus we again require the knowledge of Bc absorption processes to interpret the observed rate in the experiments. To calculate the cross sections of Bc mesons we use an effective Lagrangian based on SU(5) flavor symmetry. The study is similar to other papers in the literature [17, 20, 16, 34, 35] in which SU(3) and SU(4) symmetries are used corresponding to 3 and 4 flavors of the quarks respectively. The SU(5) or even SU(4) symmetries are badly broken and it is inappropriate to develop or use a theory which is based on these symmetries. However, the approach is sound as the symmetry is merely used to construct the possible interactions vertices, between the mesons and baryons, consistent with the laws of conservation of quark flavors and parity in the strong processes. Real implications of the symmetries are given by the various relations through which the couplings of interaction vertices are related. The approach is acceptable as long as we are not using these SU(5) relations. In this work and other similar studies the couplings are either fixed empirically or determined theoretically by using some more fundament theory like QCD or quark potential model. It turns out that we cannot find the values of all required coupling in this way. These couplings are then constrained by the SU(5) relations. That is why the approach is sound but has limitations. Other sources of uncertainty in the approach are the form factors of the interacting mesons or baryons. The knowledge of the form factors is required to include the effect of finite size of hadrons. In principle the form factors of the hadrons can be determined by the theories of interaction of the quark constituents

15 of the hadrons. Considering the difficulties in applying these methods, we prefer to use some phenomenological form in which the cutoff parameters are fixed empirically. This approach is somewhat naive as it neglects the kinematics at each and every vertex. The results of the cross sections of Bc meson by light mesons and nucleons obtained in this work could be useful in determining the production rate of Bc mesons in relativistic heavy-ion collisions. In chapter 2, we derive the effective Lagrangian for meson-meson and meson-baryon interactions. In chapter 3, we produce the diagrams of the different Bc absorption processes by π, ρ, K mesons and nucleons and write the corresponding amplitudes. + → + → In chapter 4, we calculate the cross sections of the processes Bc π DB,Bc π

∗ ∗ + → ∗ + → ∗ + → + + → ∗+ ∗ + → 0 D B ,Bc ρ D B,Bc ρ DB ,KBc Ds B,KBc Ds B , KBc DBs , + → ∗ ∗0 and KBc D Bs . In chapter 5 we calculate of cross sections of Bc absorption + → + → ∗ − → − → processes; (i) NBc ΛcB, (ii) NBc ΛcB (iii) NBc DΛb and (iv) NBc ∗ D Λb, followed by a brief discussion in the chapter 6.

16 Chapter 2

The Hadronic Lagrangian

In this chapter we introduce the effective hadronic Lagrangian which is used for calculating the hadronic cross sections of Bc mesons. In section 2.1, we briefly discuss the concept of irreducible tensors which is used to develop interaction Lagrangian having SU(5) global symmetry. In section 2.2, we derive the interaction Lagrangian for meson-meson and meson-baryon interactions. The symmetry in the effective La- grangian implies that different couplings are related by various SU(5) relations. These symmetry relations between the different couplings are also given in this chapter.

2.1 Irreducible tensors

Let xi represents a vector in n-dimensional vector space defined over the field of complex numbers. Consider a special unitary transformation defined as following

i i j x = Uj x , (2.1) where, U †U = UU † = I and det(U) = 1. Set of all these transformations deﬁne a group called SU(n) group. The complex conjugate of the vector xi is represented by

i ∗ xi = (x ) , which transforms under SU(n) according to

† j xi = U i xj, (2.2)

∗i † j where, U j = U i . These transformations also deﬁne n-dimensional representation of SU(n) which is conjugate of the above deﬁning representation. Both of these represen-

17 tations are the fundamental irreducible representations. Generally SU(n) group has n − 1 fundamental representations of which two are n dimensional and conjugate of one another. A vector |x⟩ in Cn is deﬁne by

i i |x⟩ = x ei = xie .

Following is the transformation of the basis vector under SU(n) obtained by assuming invariance of the vector |x⟩.

i i j e = Uj e , (2.3)

† j ei = U i ej. (2.4)

i † j αβ...γ Thus an upper (lower) index i transform by the matrix Uj (U i ). Let Aij...k is a mixed tensor which deﬁne the components of a vector in a product space according to

| ⟩ αβ...γ i × j k × × A = Aij...k (e e ... e eα eβ ... eγ), (2.5)

i where e and eα etc. represent the basis vectors of n-dimensional fundament repre- αβ...γ sentations of SU(n) respectively. The tensor Aij...k transform in the same way under SU(n) as the direct product of the basis vectors. ( ) ( ) αβ...γ α β γ † l † m † n λµ...ν Aij...k = Uλ Uµ ...Uν . U i U j ...U k Alm...n. (2.6)

It can be seen that set of all such transformations also furnish a representation of SU(n) called product representation, however this representation is reducible unlike αβ...γ the fundamental representations which are irreducible. The tensor Aij...k and the basis i j k tensor (e × e ... e × eα × eβ ... eγ) are accordingly called reducible tensors. It is i ij...k noted that kronecker delta δj and Levi-Civita tensors ϵαβ...γ and ϵ are invariant under SU(n). The invariance of the Levi-Civita tensors implies that a tensor with upper indices can be constructed form a tensor with lower indices and vice versa

i ijk B = ϵ Ajk. (2.7)

So we can conﬁne only to tensors with upper (or lower) indices without lose of general- ity. To decompose the above reducible representation into irreducible representations,

18 we note that operator of permutation of upper (or lower) indices commute with the group transformation. For example; consider the transformation of a tensor of rank 2 given by

ij i j ab A = UaUb A , (2.8)

ij i j ab j i ab i j ab P12A = P12(UaUb A ) = Ua UbA = UaUb P12A . (2.9)

This implies that group transformation preserves the symmetry of the tensor. Hence the tensor with definite permutation symmetry define a reduced invariant subspace with respect to group transformation. It can be shown that these representations are irreducible. Accordingly the tensors with definite permutation symmetry are called irreducible tensors. Independent components of irreducible tensor define the basis for irreducible representation SU(n). These tensors also act as basis for irreducible representations of permutation group, which are represented by the Young diagrams. Correspondingly the irreducible tensors can also be represented by the related Young diagrams. For example; consider the tensor of rank two obtained by the direct product

i i e × ej. For SU(3) group the basis e and ej are represented by the tableaux (10) and (01) and their direct product decompose to the tableaux (11) and (00) which represent 8 and 1 dimensional representations of SU(3). Corresponding irreducible tensors are given by

1 P i(octet) = ei × e − δi (ek × e ), (2.10) j j 3 j k 1 Ai (singlet) = δi (ek × e ). (2.11) j 3 j k

Alternatively the irreducible tensor of adjoint representation can also by obtained as following

i 1 i P = √ (λa) ϕa, (2.12) j 2 j where λa’s are the Gell-Mann matrices and ϕa’s are independent real parameters of i Pj . This result is generally applicable and can be use to ﬁnd the irreducible tensor of adjoint representation of SU(n).

19 2.2 The Eﬀective Lagrangian

In this section, we discuss the eﬀective Lagrangian used for calculating the absorption cross sections of Bc mesons. In section 2.2.1 and 2.2.2, we give meson-meson and meson-baryon interactions terms respectively, which are required to study Bc absorption processes. SU(5) symmetry relations between the couplings are also given in this section.

2.2.1 Meson-Meson Interaction

In this section, we follow ref. [20]. Free Lagrangian of pseudo-scalar (PS) and vector mesons is given by, 1 L = T r(∂ P †∂µP ) − T r(F † F µν), (2.13) 0 µ 2 µν where, Fµν = ∂µVν −∂νVµ; P and Vµ denote pseudo-scalar and vector mesons irreducible tensors obtained by using the generators of SU(5) in the Eq. (2.12). These irreducible tensors transform by the fundamental representations of SU(5) group and are given as following.   0 0 η η η + + + √π + √ + √c + √b π K D B  2 6 12 20    − 0 η η η 0 − 0  π − √π + √ + √c + √b K D B   2 6 12 20   0  √1 − 2η η η − 0 P =  K K − √ + √c + √b D B  , (2.14) 2  6 12 20 s s    0 + + 3η η +  D D D − √ c + √b B   s 12 20 c  − 0 0 − B B B B − √2ηb s c 20   ∗0 ρ0 J/Ψ + ∗+ ∗+ √ + √ω + √ + √Υ ρ K D B  2 6 12 20    − ρ0 J/Ψ ∗0 ∗− ∗0  ρ − √ + √ω + √ + √Υ K D B   2 6 12 20   ∗0  √1 ∗− J/Ψ ∗− ∗0 V =  K K − √2ω + √ + √Υ D B  . (2.15) 2  6 12 20 s s    ∗0 ∗+ ∗+ 3J/Ψ ∗+  D D D − √ + √Υ B   s 12 20 c  ∗− ∗0 ∗0 ∗− B B B B − √2Υ s c 20 Transformation of these tensors under SU(5) symmetry is given by,

i → i † b a Pj UaU j Pb (2.16) i → i † b a Vj UaU j Vb (2.17)

20 where U represents an arbitrary SU(5) transformation in fundament representation. It can be seen that the free Lagrangian given in Eq. 2.13 is invariant under SU(5) transformation. This symmetry is global as the parameters of the transformation do not depend upon the space coordinates. Globally invariant Lagrangian of Eq. 2.13 does not contain any interaction term between PS and vector meson. In order to introduce the interaction we need to apply the gauge principle in which the interaction is introduced by the requirement of having local symmetry in the given Lagrangian. The required locally symmetric Lagrangian can be obtained by applying the following minimal substitution ig ∂ P → D P = ∂ P − [V ,P ], (2.18) µ µ µ 2 µ ig F → F − [V ,V ]. (2.19) µν µν 2 µ ν The free Lagrangian given in Eq. 2.13 is converted to g2 L = L + igT r(∂µP [P,V ]) − T r([P,V ]2) 0 µ 4 µ g2 +igT r(∂µV ν[V ,V ]) + T r([V ,V ]2). (2.20) µ ν 8 µ ν where, L0 is the free Lagrangian and interaction terms describe pseudoscalar-pseudoscalar- vector (PPV), PPVV, VVV and VVVV couplings respectively. It can seen that the mass terms of PS and vector mesons, which break SU(5) symmetry are not present in this symmetric Lagrangian. If the mass terms of mesons are added in ad hoc way, then symmetry is broken explicitly. In order words, SU(5) symmetry exists only in the limit zero mesonic masses. Using the irreducible tensors P and V in diﬀerent interaction terms, we can produce all possible interactions between 24 PS and 24 vector mesons. The couplings of all these interactions are expressed in terms of one universal coupling g. Although the Lagrangian produce a large number of possible interaction between PS and vector mesons, however we require only those couplings which are used in the absorption processes of Bc meson with light PS and vector mesons like, π, ρ, and K. Finding a few required terms out the thousands is a tedious work, which can be done conveniently by using symbolic calculation systems like Mathematica or Maple. Fol- lowing set of interaction terms are required to study the interaction of Bc meson with

21 light mesons [25, 26, 27].

∗µ−→ −→ −→ LπDD∗ = igπDD∗ D τ · (D∂µ π − ∂µD π ) + hc, (2.21a) ∗µ−→ −→ −→ LπBB∗ = igπBB∗ B τ · (B∂µ π − ∂µB π ) + hc, (2.21b)

∗µ − − L ∗ ∗ − BcBD = igBcBD D (Bc ∂µB ∂µBc B) + hc, (2.21c) ∗ µ + + L ∗ ∗ − BcB D = igBcB DB (Bc ∂µD ∂µBc D) + hc, (2.21d) ∗ ∗ + µ−→ −→ L ∗ ∗ − ∗ ∗ · πBcD B = gπBcD B Bc B τ π Dµ + hc. (2.21e)

−→ −→ −→µ LρDD = igρDD(D τ ∂µD − ∂µD τ D) · ρ , (2.22a)

−→ −→ −→µ LρBB = igρBB(B τ ∂µB − ∂µB τ B) · ρ , (2.22b) ( ) ∗ ∗ ∗ν−→ ∗ν−→ −→µ L ∗ ∗ ∗ ∗ − · ρD D = igρD D [ ∂µD τ Dν D τ ∂µDν ρ ∗ ∗ν−→ −→ ∗ν−→ −→ µ + (D τ · ∂µ ρ ν − ∂µD τ · ρ ν) D ( ) ∗ ∗ ∗µ −→ −→ν −→ −→ν +D τ · ρ ∂µD − τ · ∂µ ρ D ], (2.22c) ( ν )ν ∗ ∗ ν−→ ∗ ν−→ ∗ −→µ LρB∗B∗ = igρB∗B∗ [ ∂µB τ B − B τ ∂µB · ρ ( ν )ν ∗ ∗ ν−→ −→ ν−→ −→ ∗µ + B τ · ∂µ ρ ν − ∂µB τ · ρ ν B ∗ µ −→ · −→ν ∗ − −→ · −→ν ∗ +B ( τ ρ ∂µBν τ ∂µ ρ Bν )], (2.22d) ∗ + −→ −→ µ L ∗ ∗ · ρBcD B = gρBcD BBc B τ ρ µD + h.c, (2.22e) ∗ + µ−→ −→ L ∗ ∗ · ρBcDB = gρBcDB Bc B τ ρ µD + h.c. (2.22f)

∗µ L ∗ ∗ − KD Ds = igKD Ds (K∂µDs ∂µKDs)D + h.c, (2.23a)

∗µ L ∗ − ∗ − KDDs = igKDDs (K∂µD ∂µKD)Ds + h.c, (2.23b)

∗µ L ∗ ∗ − KB Bs = igKB Bs (K∂µBs ∂µKBs)B + h.c, (2.23c) ∗µ L ∗ − ∗ − KBBs = igKBBs (K∂µB ∂µKB)Bs + h.c, (2.23d) ∗ + + µ L ∗ − ∗ − BcBsDs = igBcBsDs (Bc ∂µBs ∂µBc Bs)Ds + h.c, (2.23e) ∗ + + µ L ∗ ∗ − BcBs Ds = igBcBs Ds (Bc ∂µDs ∂µBc Ds)Bs + h.c, (2.23f) ∗ ∗ µ + L ∗ ∗ − ∗ ∗ KBcDs B = gKBcDs B B DsµBc K + h.c, (2.23g) ∗ ∗ + µ L ∗ ∗ − ∗ ∗ KBcD Bs = gKBcD Bs KBc BsµD + h.c. (2.23h)

22 where, ( ) ( ) ( ) T 0 + 0 − ∗ ∗0 ∗+ D = D D , D = D D ,Dµ = Dµ Dµ , ( ) ( ) T T + 0 ∗ ∗+ ∗0 B = B B ,Bµ = Bµ Bµ ,

−→ ± 1 π = (π1, π2, π3) , π = √ (π1 ∓ iπ2) 2 −→ ± 1 ρ = (ρ1, ρ2, ρ3) , ρ = √ (ρ1 ∓ iρ2) , 2 ( ) T K = K+ K0 . (2.24)

In all above expressions we follow the convention of representing a ﬁeld by the symbol of the particle which it absorbs. For example, B0 ﬁeld represents absorption of B0 0 particles or creation of B particles. The coupling constants in these interaction terms are related to universal SU(5) coupling constant g as following [25, 26, 27] g g g2 gπDD∗ = gπBB∗ = , gB BD∗ = gB B∗D = √ , gπB D∗B∗ = √ , (2.25a) 4 c c 2 2 c 4 2 g g = g = g ∗ ∗ = g ∗ ∗ = , (2.25b) ρDD ρBB ρD D ρB B 4 g gB BD∗ = gB B∗D = √ , (2.25c) c c 2 2 g2 gρB BD∗ = gρB B∗D = √ , (2.25d) c c 8 2 g gB BD∗ = gB B∗D = gB B D∗ = gB B∗D = √ , (2.25e) c c c s s c s s 2 2 g gKD∗D = gKDD∗ = gKB∗B = gKBB∗ = √ , (2.25f) s s s s 2 2 g2 g ∗ ∗ = g ∗ ∗ = . (2.25g) KBcDs B KBcD Bs 4 Using these SU(5) relations, we can also relate the interaction couplings as following

∗ ∗ ∗ ∗ ∗ ∗ gπBcD B = 2gπDD gBcB D = 2gπBB gBcBD , (2.26a)

∗ ∗ ∗ ∗ gρBcBD = gρBcB D = gρDDgBcB D = gρBBgρBcBD , (2.26b) √ ∗ ∗ ∗ gKD Ds = gKDDs = 2gπDD , (2.26c) √ ∗ ∗ ∗ gKB Bs = gKBBs = 2gπBB , (2.26d)

∗ ∗ ∗ ∗ ∗ ∗ gKBcDs B = 2gKDDs gBcB D = 2gKB Bs gBcBsDs , (2.26e)

∗ ∗ ∗ ∗ ∗ ∗ gKBcD Bs = 2gKBBs gBcBD = 2gKD Ds gBcBs Ds . (2.26f)

23 As discussed earlier that the SU(5) symmetry is explicitly broken by the mass terms of PS and vector mesons. Therefore above relations, which are the direct consequence of SU(5) symmetry, may not be applicable. Later on we will use these relations only to constraint the values of those couplings which cannot be ﬁxed empirically. In this case, it is not entirely unreasonable to use these relations even if the SU(5) symmetry is badly broken.

2.2.2 Baryon-Meson Interaction

In this section we follow ref. [25, 37]. In SU(5) quark model, spin half even par-

P 1 + ity baryons (anti-baryons) (J = 2 ) are represented by the multiplets of 40 (40)- dimensional representation of SU(5) group, which is defined by 1100 (0011) Young tableau. Mesons on other hand are represented by 24-dimensional represent of SU(5) group which is defined by the Young tableau 1001. An SU(5) invariant Lagrangian defining Baryon-Baryon-Pseudoscalar meson (BBP) or Baryon-Baryon-Vector meson (BBV) coupling must be singlet. Therefore, we need to find singlet (1-dimensional representation) in the direct product of 40, 40 and 24-plets of anti-baryons, baryons and mesons respectively. The above direct product can produce singlet if the product 40 ⊗ 24 contains at least one 40-dimensional representation or the product 40 ⊗ 24 contains 40 representation. Through young diagrams we find that

40 ⊗ 24 = 450 ⊕ 210 ⊕ 175 ⊕ 40 ⊕ 40 ⊕ 35 ⊕ 10. (2.27)

Appearance of two 40-dimensional representations in the above product implies that BBP and BBV couplings can be constructed in two possible ways as in case of SU(3) and SU(4). Following the case of SU(3) and SU(4) [18, 36], we use the following SU(5) invariant Lagrangian for BBP and BBV couplings.

L ∗αµν 5 β ∗αµν 5 β PBB = gP (aϕ γ Pα ϕβµν + bϕ γ Pα ϕβνµ), (2.28a) L ∗αµν β ∗αµν β VBB = igV (cϕ γVα ϕβµν + dϕ γVα ϕβνµ), (2.28b)

β β where the Greek indices run from 1 to 5. The irreducible tensors Pα and Vα are deﬁned by the pseudo-scalar and vector meson matrices given in Eqs. 2.14 and 2.15

24 αµν ∗αµν P 1 + and the irreducible tensor ϕ (ϕ ) deﬁnes the J = 2 baryons (anit-baryons) which belongs to 40(40)-plet states. The tensor ϕαµν (ϕ∗αµν), satisﬁes the conditions

P 1 + ϕµνλ + ϕλµν + ϕνλµ = 0 and ϕµνλ = ϕνµλ. The relations deﬁning J = 2 baryons in terms of the elements of ϕµνλ for u, d, s and c quarks are given in ref. [36] and for b quarks, they are given in ref. [37]. Here we give all relations taken from these references. For the baryons with u, d, and s quarks

p = ϕ112, n = ϕ221, √ + 0 − Σ = ϕ113, Σ = 2ϕ123, Σ = ϕ223, √ √ 0 − Ξ = 2ϕ331, Ξ = 2ϕ332, √ 2 Λ = (ϕ − ϕ ). (2.29) 3 321 312

For the baryons with one c quark √ ++ + 0 Σc = ϕ114, Σc = 2ϕ124, Σc = ϕ224, √ √ + 0 Ξ = 2ϕ134, Ξ = 2ϕ234, √ c c √ 2 2 +′ − 0′ − Ξc = (ϕ413 ϕ431), Ξc = (ϕ423 ϕ432), 3 √ 3 2 Λ+ = (ϕ − ϕ ), Ω0 = ϕ (2.30) c 3 421 412 c 334.

For the baryons with two c quarks

++ + + Ξcc = ϕ441, Ξcc = ϕ442, Ωcc = ϕ443. (2.31)

25 For the baryons with b quarks √ + 0 − Σb = ϕ115, Σb = 2ϕ125, Σb = ϕ225, √ √ 0 − Ξ = 2ϕ135, Ξ = 2ϕ235, √ b b √ 2 2 ′0 − ′− − Ξb = (ϕ513 ϕ531) , Ξb = (ϕ523 ϕ532) , √ 3 3 2 Λ0 = (ϕ − ϕ ) , Ω− = ϕ , Ω+ = ϕ , b 3 521 512 b 335 ccb 445 0 − Ξbb = ϕ155, Ξb = ϕ552,

− o Ωbb = ϕ553, Ωbbc = ϕ554, √ √ + 0 Ξ = 2ϕ145, Ξ = 2ϕ245, √ cb cb √ 2 2 ′+ − ′0 − Ξcb = (ϕ514 ϕ541) , Ξcb = (ϕ524 ϕ542) , 3 √ 3 2 Ω0 = ϕ , Ω′0 = (ϕ − ϕ ) . (2.32) cb 345 cb 3 543 534

P 1 + Using these deﬁnitions of possible J = 2 baryons (anti-baryons) in SU(5) invariant Lagrangian of BBP and BBV interaction, we can produce all possible couplings. Once again thousands of terms are produced. However, we require only those which are used in Bc absorption cross sections by nucleons. These terms are given by

L 5 5 DNΛc = igDNΛc (Nγ ΛcD + DΛcγ N), (2.33a) ∗ µ ∗ µ L ∗ ∗ D NΛc = gD NΛc (Nγ ΛcD + D Λcγ N), (2.33b) L 5 5 BNΛb = igBNΛb (Nγ ΛbB + BΛbγ N), (2.33c) ∗ ∗µ µ L ∗ ∗ B NΛb = gB NΛb (NγµΛbB + B ΛbγµN), (2.33d) L 5 + 5 − BC ΛcΛb = igBC ΛcΛb (Λcγ ΛbBc + Λbγ ΛcBc ), (2.33e)

Where, ( ) ( ) ( ) T 0 + 0 − ∗ ∗0 ∗+ D = D D , D = D D ,Dµ = Dµ Dµ , ( ) ( ) T T ∗ B = B+ B0 ,B = B∗+ B∗0 ,   µ µ µ p N =   . (2.34) n

26 The Lagrangian of Eq. 2.28a deﬁnes all BBP couplings in terms of the universal coupling gP and the constants a and b. Similarly the Lagrangian of Eq. 2.28b deﬁnes all BBV couplings in terms of the universal coupling gV and the constants c and d.

For the coupling constants gπNN , gρNN , gKNΛ, gK∗NΛ and given in the Eqs. 2.33a to 2.33e, we obtain the following results.

√1 − 5 3 − gπNN = gP (a b), gBcΛcΛb = gP (a b), (2.35a) 2 4 √ 4 3 6 − gKNΛ = gDNΛc = gBNΛ = gP (b a), (2.35b) b 8 √ 1 5 3 6 gρNN = √ gV (c − d), gK∗NΛ = gD∗NΛ = gB∗NΛ = gV (d − c).(2.35c) 2 4 c b 8 As discussed above that SU(5) ﬂavor symmetry in badly broken due to large variation in the related quark masses. Thus, we expect that these symmetry relations are also violated. It is noted that SU(4) ﬂavor symmetry also produces the same relations as given in Eqs. 2.35 for couplings of the hadrons containing u, d, s and c quarks [18].

2.2.3 Anomalous parity interaction

In this work we have also studied the effect of the PVV coupling, called anomalous parity interaction, in Bc absorption processes by nucleons. The effective Lagrangian density defining the anomalous interaction of mesons is discussed in [38]. These terms are obtained by the gauged Wess-Zumino action [39]. Here, we report the relevant interaction terms of the Lagrangian density as following. [ ( )( )] ( ) ∗ ∗ µ ∗ν α ∗β − + α β µ ν L ∗ ∗ ∗ ∗ BcB D = gBcB D εαβµν (∂ D ) ∂ B Bc + Bc ∂ B ∂ D , (2.36) where εαβµν is totally antisymmetric Levi-Civita tensor of rank four. The coupling con- − ∗ ∗ 1 ∗ stant gBcB D , which has the dimension of GeV , can be approximated by gBcB D/M D ∗ in heavy quark mass limit [40]. Where M D is the average mass of D and D mesons.

2.3 Vertex factors of interactions

To write the amplitudes of the Bc absorption processes by light mesons and nucleons, we require the expressions of vertex factors for diﬀerent couplings. General procedure

27 of deriving the vertex factor is described in Appendix A. In this section, we present the vertex factors of all the couplings given by the equations 2.21, 2.22, 2.23, 2.33, and 2.36 ∗ ¯ ¯ ∗ Dµ D Dµ D

k1 k2 k1 k2

(πDD¯ ∗) (πDD¯ ∗)

k3 k3

π π

−igπDD∗ (k3 − k2)µ igπDD∗ (k3 − k2)µ

∗ ¯ ¯∗ Bµ B Bµ B

k1 k2 k1 k2

(πBB¯ ∗) (πBB¯∗)

k3 k3

π π

−igπBB∗ (k3 − k2)µ igπBB∗ (k3 − k2)µ

Figure 2.1: Vertex factors of the coupling given in Eqs. 2.21a and 2.21b.

28 ¯∗ ¯ ∗ Bµ D Bµ D

k1 k2 k1 k2

+ ¯ ¯∗ − ∗ (Bc DB ) (Bc DB )

k3 k3

+ − Bc Bc

− ∗ − ∗ − igBcDB (k2 k3)µ igBcDB (k2 k3)µ

¯ ∗ ¯ ∗ Dµ B Dµ B

k1 k2 k1 k2

+ ¯ ¯ ∗ − ∗ (Bc BD ) (Bc BD )

k3 k3

+ − Bc Bc

∗ − − ∗ − igBcBD (k2 k3)µ igBcBD (k2 k3)µ

¯∗ ¯ ∗ ∗ ∗ Bµ Dµ Bµ Dµ

+ ¯∗ ¯ ∗ − ∗ ∗ (πBc B D ) (πBc B D )

π + π − Bc Bc

−igµν −igµν

Figure 2.2: Vertex factors of the couplings given in Eqs. 2.21c, 2.21d, and 2.21e.

29 D D¯ B B¯

k1 k2 k1 k2

(ρDD¯) (ρBB¯)

k3 k3

ρµ ρµ − − − igρDD¯ (k2 k1)µ igρBB¯(k2 k1)µ

∗ ¯ ∗ ∗ ¯ ∗ Dµ Dν Bµ Bν

k1 k2 k1 k2

(ρD∗D¯ ∗) (ρB∗B¯∗)

k3 k3

ρλ ρλ

−igρD∗D¯ ∗ (k1 − k2)λgµν + (k3 − k1)ν gµλ + (k2 − k3)µgνλ

igρB∗B¯∗ (k1 − k2)λgµν + (k3 − k1)ν gµλ + (k2 − k3)µgνλ

Figure 2.3: Vertex factors of the couplings given in Eqs. 2.22a, 2.22b, 2.22c, and 2.22d

30 ¯ ¯ ∗ ∗ B Dν B Dν

+ ¯ ¯ ∗ − ∗ (ρBc BD ) (ρBc BD )

ρµ + ρµ − Bc Bc

−igµν −igµν

¯∗ ¯ ∗ Bν D Bν D

+ ¯∗ ¯ − ∗ (ρBc B D) (ρBc B D)

ρµ + ρµ − Bc Bc

−igµν −igµν

Figure 2.4: Vertex factors of the couplings given in Eqs. 2.22e and 2.22f.

31 ¯ ∗ ∗ ¯ Dµ Ds Dµ Ds

k2 k3 k2 k3

¯ ¯ ∗ ¯ ∗ (KDsD ) (KDsD )

k1 k1

K¯ K

− ∗ − ∗ − igKDsD (k3 k1)µ igKDsD (k3 k1)µ

¯ ∗ ¯ ∗ D Dsµ D Dsµ

k2 k3 k2 k3

¯ ¯ ∗ ¯ ∗ (KDDs) (KDDs)

k1 k1

K¯ K

∗ − − ∗ − igKDDs (k2 k1)µ igKDDs (k2 k1)µ

Figure 2.5: Vertex factors of the couplings given in Eqs. 2.23a and 2.23b.

32 ¯∗ ∗ ¯ Bµ Bs Bµ Bs

k2 k3 k2 k3

¯∗ ¯ ¯ ∗ (KBsB ) (KBsB )

k1 k1

K K¯

∗ − − ∗ − igKBsB (k3 k1)µ igKBsB (k3 k1)µ

¯ ∗ ¯∗ B Bsµ B Bsµ

k2 k3 k2 k3

¯ ∗ ¯ ¯∗ (KBBs ) (KBBs )

k1 k1

K K¯

− ∗ − ∗ − igKBBs (k2 k1)µ igKBBs (k2 k1)µ

Figure 2.6: Vertex factors of the couplings given in Eqs. 2.23c and 2.23d.

33 ¯∗ ¯ ∗ Bsµ Ds Bsµ Ds

k3 k2 k3 k2

+ ¯ ¯∗ − ∗ (Bc DsBs ) (Bc DsBs )

k1 k1

+ − Bc Bc

− ∗ − ∗ − igBcDsBs (k2 k1)µ igBcDsBs (k2 k1)µ

¯ ∗ ¯ ∗ Dsµ Bs Dsµ Bs

k2 k3 k2 k3

+ ¯ ¯ ∗ − ∗ (Bc BsDs) (Bc BsDs)

k1 k1

+ − Bc Bc

∗ − − ∗ − igBcBsDs (k3 k1)µ igBcBsDs (k3 k1)µ

Figure 2.7: Vertex factors of the couplings given in Eqs. 2.23e and 2.23f.

34 ¯∗ ¯ ∗ ∗ ∗ Bsµ Dν Bsµ Dν

¯ + ¯∗ ¯ ∗ − ∗ ∗ (KBc Bs D ) (KBc Bs D )

¯ K + K − Bc Bc

− ∗ ∗ − ∗ ∗ igKBcBs D gµν igKBcBs D gµν

¯∗ ¯ ∗ ∗ ∗ Bµ Dsν Bµ Dsν

¯ + ¯∗ ¯ ∗ − ∗ ∗ (KBc B Ds) (KBc B Ds )

¯ K + K − Bc Bc

− ∗ ∗ − ∗ ∗ igKBcB Ds gµν igKBcB Ds gµν

Figure 2.8: Vertex factors of the couplings given in Eqs. 2.23g and 2.23h.

35 ¯ ¯ Λc D Λc D

¯ ¯ ¯ (NΛcD) (NΛcD)

N¯ N − 5 − 5 gDNΛc γ gDNΛcγ

¯ ∗ ¯ ∗ Λc Dµ Λc Dµ

¯ ¯ ∗ ¯ ∗ (NΛcD ) (NΛcD )

N¯ N

∗ µ ∗ µ igD NΛc γ igD NΛc γ

Figure 2.9: Vertex factors of the couplings given in Eqs. 2.33a and 2.33b.

36 ¯ ¯ Λb B Λb B

¯ ¯ ¯ (NΛbB) (NΛbB)

N¯ N − 5 − 5 gBNΛb γ gBNΛb γ

¯∗ ¯ ∗ Λb Bµ Λb Bµ

¯ ∗ ¯ ¯∗ (NΛbB ) (NΛbB )

N¯ N

∗ µ ∗ µ igB NΛbγ igB NΛb γ

¯ ¯ Λb Λc Λb Λc

¯ − ¯ ¯∗ (ΛbΛcBc ) (NΛbB )

− + Bc Bc − 5 − 5 gΛbΛcBc γ gΛbΛcBc γ

Figure 2.10: Vertex factors of the couplings given in Eqs. 2.33c, 2.33d, and 2.33e.

37 ¯∗ ¯ ∗ ∗ ∗ B Dν B Dν

k2 k3 k2 k3

+ ¯∗ ¯ ∗ − ∗ ∗ (Bc B D ) (Bc B D )

k1 k1

+ − Bc Bc

αµβν αµβν −iǫ k2αk3β −iǫ k2αk3β

Figure 2.11: Vertex factors of the couplings given in Eqs. 2.36

38 Chapter 3

Bc Absorption Processes: Born Diagrams and Amplitudes

In this chapter, we produce the Born diagrams of the absorption processes of Bc mesons by π, ρ, and K mesons using the eﬀective Hadronic Lagrangian discussed in the last chapter. Vertex factors of diﬀerent couplings along with the Feynman rules are used to the write amplitudes of the diagrams. The role of form factor is also discussed.

3.1 Bc absorption processes

Absorption of the Bc mesons in heavy-ion collision experiments can in principle occur with all possible light and heavy mesons or baryons. However, the production of light mesons especially pions, rho, K mesons, and nucleons is much higher than the heavy mesons and baryons. Thus, the absorption by the heavy particles can be neglected for calculating Bc production rate in QGP. We, therefore, identify the following processes which could signiﬁcantly aﬀect the yield of Bc mesons.

39 Bc absorpriton by pions:

+ → Bc π DB, − → Bc π DB, + → ∗ ∗ Bc π D B , − → ∗ ∗ Bc π D B . (3.1)

First and second processes have the same cross sections as they are charge conjugation of each other. Similarly third and fourth processes also have the same cross sections due to charge conjugation symmetry.

Bc absorption by rho mesons:

+ → ∗ Bc ρ D B, − → ∗ Bc ρ D B, + → ∗ Bc ρ DB , − → ∗ Bc ρ DB . (3.2)

In this case, the cross sections of ﬁrst and second processes are same and similarly the cross sections of third and fourth processes are also same due to charge conjugation symmetry.

Bc absorption by K mesons:

+ → + KBc Ds B, + → ∗+ ∗ KBc Ds B , + → 0 KBc DBs , + → ∗ ∗0 KBc D Bs . (3.3)

In this case, all the processes are distinct and their cross sections are not related by charge conjugation symmetry.

40 Bc absorption by nucleons:

+ → NBc ΛcB, + → ∗ NBc ΛcB , − → NBc DΛb, − → ∗ NBc D Λb. (3.4)

In this case again all the processes are distinct and their cross sections cannot be related by charge conjugation symmetry. In writing these processes, we have neglected the possibility of creating quark anti-quark pairs through which we can write many more 3-body processes. Generally for a 3-body channels the available phase space shrinks and consequently the cross sections of the processes are lower as compared to 2-body processes. It is, therefore, good approximation to neglect all three body processes at first stage. In writing the above processes, we have also assumed only PPV, PPVV, VVV, and VVVV couplings for meson-meson processes and BBP and BBV couplings for baryon-meson processes. These are the only possible interactions given by the effective Lagrangian based on SU(5) symmetry. However, if we also include the anomalous parity interaction PVV coupling then we can obtain many more diagrams or processes. All these diagrams or processes produced by anomalous parity interaction are neglected in this work for meson-meson processes. However, the effect of anomalous interaction is included and studied in case of baryon-meson processes.

3.2 Amplitudes of Bc absorpriton by pions:

In this section, we produce the Born diagrams and write the amplitudes of the processes of Bc absorption by pions giving in Eq. 3.1. In Fig. 3.1 we show the diagrams of the + → process Bc π DB. Scattering amplitudes of these diagrams are given as following ( ) − (p −p )µ(p −p )ν ∗ ∗ i µν 1 3 1 3 M = g g (p + p ) 2 g − 2 (−p − p ) ,(3.5a) 1a πDD BcBD 1 3 µ t−m ∗ m ∗ 4 2 ν D ( D ) − (p −p )µ(p −p )ν ∗ ∗ i µν − 1 4 1 4 − − M1b = gπBB gBcB D(p1 + p4)µ − 2 g 2 ( p3 p2)ν.(3.5b) u mB∗ mB∗

41 D B D B

p p 3 4 p p 3 4

D* B* q=p −p q=p −p 1 3 1 4

p p p p 1 2 1 2

π B π B (a)c (b) c

+ → Figure 3.1: Born diagrams of Bc absorption process Bc π DB

Total amplitude of the process is given by

M1 = M1a + M1b. (3.6)

+ → ∗ ∗ Born diagrams of the process Bc π D B are given in Fig. 3.2. The related scattering amplitudes of these diagrams are given by

i µ ν M = −g ∗ g ∗ (2p − p ) (p − p + p ) ε (p )ε (p ), (3.7a) 2a πDD BcB D 1 3 µ − 2 2 1 3 ν r 3 s 4 t mD i µ ν M = −g ∗ g ∗ (2p − p ) (p − p + p ) ε (p )ε (p ), (3.7b) 2b πBB BcBD 1 4 µ − 2 2 1 4 ν r 3 s 4 u mB µ ν − ∗ ∗ M2c = igπBcB D gµνεr (p3)εs (p4). (3.7c)

And total amplitude of the process is given by,

M2 = M2a + M2b + M2c. (3.8)

In this case, two and three diagrams contribute in the total amplitude of ﬁrst and second processes respectively.

42 D* B* D* B* D* B*

p p 3 4 p p p p 3 4 3 4

D B q=p −p q=p −p 1 3 1 4 p p p p p p 1 2 1 2 1 2

π B π B π B (a) c (b) c (c) c

+ → ∗ ∗ Figure 3.2: Born diagrams of Bc absorption process Bc π D B

3.3 Amplitudes of Bc absorpriton by rho mesons:

In this section, we produce the Born diagrams and write the amplitudes of the processes of Bc absorption by rho meson giving in Eq. 3.2. In Fig. 3.3 we show the diagrams of + → ∗ the process Bc ρ D B.

Scattering amplitudes of these diagrams are given as following

i µ ν M = g g ∗ (2p − p ) (p − 2p ) ε (p )ε (p ), (3.9a) 3a ρDD BcBD 3 1 µ − 2 4 2 ν r 1 s 4 [ t mB ]

M3b = gρD∗D∗ gB BD∗ (2p4 − p1) gβν + (2p1 − p4) gµβ + (−p4 − p1) gµν c( µ ) ν β −i (p − p )α (p − p )β αβ − 1 4 1 4 − − µ ν g ( p3 p2) εr (p1)εs (p4),(3.9b) − ∗ 2 α u mD mD∗

∗ µ ν M3c = igρBcBD gµνεr (p1)εs (p4). (3.9c)

Total amplitude is given by

M3 = M3a + M3b + M3c. (3.10)

+ → ∗ Born diagrams of the process Bc ρ DB are given in Fig. 3.4.

43 (a) (b) (c) ∗ B D∗ B D∗ B D

p3 p4 p3 p4 p3 p4 B D∗

q = p1 − p3 q = p1 − p4 p p 1 2 p1 p2 p1 p2

+ + + ρ Bc ρ Bc ρ Bc

+ → ∗ Figure 3.3: Born Diagrams of Bc absorption process Bc ρ D B

(a) (b) (c) ∗ D B∗ D B∗ D B

p3 p4 p3 p4 p3 p4 D B∗

q = p1 − p3 q = p1 − p4 p p 1 2 p1 p2 p1 p2

+ + + ρ Bc ρ Bc ρ Bc

+ → ∗ Figure 3.4: Born Diagrams for Bc absorption process Bc ρ DB

Scattering amplitudes of these diagrams are given as following

i µ ν M = g g ∗ (2p − p ) (p − 2p ) ε (p )ε (p ), (3.11a) 4a ρDD BcB D 3 1 µ − 2 4 2 ν r 1 s 4 [ t mD ]

∗ ∗ ∗ − − − − M4b = gρB B gBcB D (2p4 p1)µ gβν + (2p1 p4)ν gµβ + ( p4 p1)β gµν ( ) − α β i αβ (p1−p4) (p1−p4) µ ν g − 2 (−p − p ) ε (p )ε (p ), (3.11b) m ∗ 3 2 α r 1 s 4 u − mB∗ B

∗ µ ν M4c = igρBcB Dgµνεr (p1)εs (p4). (3.11c)

Total amplitude is given by

M4 = M4a + M4b + M4c. (3.12)

44 In this case three diagrams contribute in the total amplitudes for both of the processes.

3.4 Amplitudes of Bc absorpriton by K mesons

In this section, we produce the Born diagrams and write the amplitudes of the processes of Bc absorption by K mesons giving in Eq. 3.3. In Fig. 3.5 we show the diagrams of + → + the process KBc Ds B,. ( ) + a + (b) Ds B Ds B

p3 p4 p3 p4

∗ ∗ D Bs

q = p3 − p1 q = p4 − p1

p1 p2 p1 p2

+ + K Bc K Bc

+ → + Figure 3.5: Born Diagrams for Bc absorption process KBc Ds B

Scattering amplitudes of these diagrams are given as following

( ) µ ν −i µν (p1−p3) (p1−p3) M5a = −gKD∗D gB BD∗ (p1+p3)µ g − (p4+p2)ν , (3.13a) s c t−m2 ∗ m2 ∗ D ( D ) − (p −p )µ(p −p )ν − ∗ ∗ i µν − 1 4 1 4 M5b = gKBB gB D B (p1+p4)µ g (p3+p2)ν . (3.13b) s c s s u−m2 ∗ m2 ∗ Bs Bs

Total amplitude is given by

M5 = M5a + M5b. (3.14)

+ → ∗+ ∗ Born diagrams of the process KBc Ds B are given in Fig. 3.6

45 ∗ +∗ (a) ∗ +∗ (b) ∗ + (c) ∗ Ds B Ds B Ds B

p3 p4 p3 p4 4 p3 p D Bs

q = p3 − p1 q = p4 − p1 p1 p2 p1 p2 p1 p2

+ + + K Bc K Bc K Bc

+ → ∗+ ∗ Figure 3.6: Born Diagrams for Bc absorption process KBc Ds B

Scattering amplitudes of these diagrams are given as following

i µ ν M = −g ∗ g ∗ (p − 2p ) (p − p − p ) ε (p )ε (p ), (3.15a) 6a KDDs BcB D 3 1 µ − 2 1 3 2 ν r 3 s 4 t mD i µ ν M = −g ∗ g ∗ (p − 2p ) (p − p − p ) ε (p )ε (p ),(3.15b) 6b KB Bs BcBsDs 4 1 ν u − m2 1 4 2 µ r 3 s 4 Bs µ ν − ∗ ∗ M6c = igKBcDs B gµνεr (p3)εs (p4). (3.15c)

Total amplitude is given by

M6 = M6a + M6b + M6c. (3.16)

+ → 0 Born diagrams of the process KBc DBs are given in Fig. 3.7

Scattering amplitudes of these diagrams are given as following

( ) − (p −p )µ(p −p )ν − ∗ ∗ i µν − 1 3 1 3 M7a = gKDD gB B D (p1+p3)µ g (p4+p2)ν , (3.17a) s c s s t−m2 ∗ m2 ∗ Ds ( Ds ) µ ν −i µν (p1−p4) (p1−p4) M7b = −g ∗ g ∗ (p1+p4)µ g − (p3+p2)ν . (3.17b) KB Bs BcB D u−m2 m2 B∗ B∗

Total amplitude is given by

M7 = M7a + M7b. (3.18)

+ → ∗ ∗0 Born diagrams of the process KBc D Bs are given in Fig. 3.8

46 (a) 0 (b) 0 D Bs D Bs

p3 p4 p3 p4

∗ ∗ Ds B

q = p3 − p1 q = p4 − p1

p1 p2 p1 p2

¯ + ¯ + K Bc K Bc

+ → 0 Figure 3.7: Born Diagrams for Bc absorption process KBc DBs

Scattering amplitudes of these diagrams are given as following

i µ ν M = −g ∗ g ∗ (p − 2p ) (p − p − p ) ε (p )ε (p ),(3.19a) 8a KD Ds BcBs Ds 3 1 µ t − m2 1 3 2 ν r 3 s 4 Ds i µ ν M = −g ∗ g ∗ (p − 2p ) (p − p − p ) ε (p )ε (p ), (3.19b) 8b KBBs BcBD 4 1 ν − 2 1 4 2 µ r 3 s 4 u mB µ ν − ∗ ∗ M8c = igKBcD Bs gµνεr (p3)εs (p4). (3.19c)

Total amplitude is given by

M8 = M8a + M8b + M8c. (3.20)

In case of absorption by K mesons, two diagrams contribute to the total amplitudes of the first and third processes and three diagrams contribute to the total amplitudes of second and fourth processes. It noted that in writing these amplitudes we assume contact interaction at each vertex. Mesons and baryons are composite objects which have finite size, so we need to include their form factors as well to correctly calculate the cross sections of different Bc absorption processes. The discussion on the form factors and their inclusion in the scattering amplitudes is given in the next chapter.

47 ∗ (a) 0∗ ∗ (b) 0∗ ∗ (c) 0∗ D Bs D Bs D Bs

p3 p4 p3 p4 4 p3 p Ds B

q = p3 − p1 q = p4 − p1 p1 p2 p1 p2 p1 p2

¯ + ¯ + ¯ + K Bc K Bc K Bc

+ → ∗ ∗0 Figure 3.8: Born Diagrams for Bc absorption process KBc D Bs

3.5 Amplitudes of Bc absorption by nucleons

In this section, we produce the Born diagrams and write the amplitudes of the processes of Bc absorption by nucleons giving in Eq. 3.4. In Fig. 3.9 we show the diagrams of + → the ﬁrst process NBc ΛcB.

Scattering amplitudes of these diagrams are given as following ( ) − − µ − ν i µν (p1 p3) (p1 p3) M = −g ∗ g ∗ (−p − p ) g − 9a D NΛc BcBD 4 2 µ − 2 2 t mD∗ mD∗ × u (p )γ u (p ), Λc 3 ν N 1 ( ) (p − p ).γ + m M = g g u (p )γ5 i 1 4 Λb γ5u (p ). (3.21) 9b BNΛb BcΛcΛb Λc 3 u − m2 N 1 Λb Total amplitude is given by

M9 = M9a + M9b. (3.22)

+ → ∗ Born diagrams of the second process NBc ΛcB are shown in Fig. 3.10

Scattering amplitudes of the related diagrams are given by

i 5 µ M = ig g ∗ (p − 2p ) u (p )γ u (p )ε ∗ (p ), (3.23) 10a DNΛc BcB D 4 2 µ − 2 Λc 3 N 1 B 4 t ( mD ) − 5 (p1 p4).γ + mΛb µ M = −ig ∗ g u (p )γ i γ u (p )ε ∗ (p )(3.24). 10b B NΛb BcΛcΛb Λc 3 u − m2 µ N 1 B 4 Λb

48 (1a) (b) Λc B Λc B

p3 p4 p3 p4

∗ D Λb

q = p1 − p3 q = p1 − p4

p1 p2 p1 p2

+ + N Bc N Bc

+ → Figure 3.9: Born Diagrams for Bc absorption process NBc ΛcB

Total amplitude is given by

M10 = M10a + M10b. (3.25)

− → Born diagrams of the third process NBc DΛb are shown in Fig. 3.11

And scattering amplitudes are given by ( ) (p − p ).γ + m M = g g u (p )γ5 i 1 3 Λc γ5u (p ), (3.26) 11a DNΛc BcΛcΛb Λb 4 t − m2 N 1 Λ(c ) − − µ − ν i µν (p1 p4) (p1 p4) M = −g ∗ g ∗ (−p − p ) g − 11b B NΛb BcB D 3 2 µ − 2 2 u mB∗ mB∗ × uΛb (p4)γνuN (p1). (3.27)

Total amplitude is given by

M11 = M11a + M11b. (3.28)

− → ∗ Shown in Fig. 3.12 are the Born diagrams of the fourth process NBc D Λb.

49 (a) (b) ∗ ∗ Λc B Λc B

p3 p4 p3 p4

D Λb

q = p1 − p3 q = p1 − p4

p1 p2 p1 p2

+ + N Bc N Bc

+ → ∗ Figure 3.10: Born Diagrams for Bc absorption process NBc ΛcB

Scattering amplitude of the diagram shown in Fig. 3.12 are given by ( ) − 5 (p1 p3).γ + mΛc µ M = −ig ∗ g u (p )γ i γ u (p )ε ∗ (p )(3.29), 12a D NΛc BcΛcΛb Λb 4 t − m2 µ N 1 D 3 Λc i 5 µ M = ig g ∗ (p − 2p ) u (p )γ u (p )ε ∗ (p ). (3.30) 12b BNΛb BcBD 3 2 µ − 2 Λb 4 N 1 D 3 u mB Total amplitude is given by

M12 = M12a + M12b. (3.31)

3.6 Eﬀect of anomalous parity interaction

The diagrams of the Figs. 3.9-3.12 are produced using PPV, BBP and BBV couplings deﬁned in Eqs. 2.33. However, if the PVV coupling of Bc meson due to anomalous parity interaction is also included then two additional diagrams shown in Fig. 3.13 ∗ + → ∗ − → are introduced for the processes NBc ΛcB and NBc D Λb respectively. The

50 (a) (b) D¯ Λb D¯ Λb

p3 p4 p3 p4

∗ Λc B

q = p1 − p3 q = p1 − p4

p1 p2 p1 p2

− − N Bc N Bc

− → Figure 3.11: Born Diagrams for Bc absorption process NBc DΛb scattering amplitudes of the diagrams of Fig. 3.13 are given by ( ) − − − αµβν i (p3 p1)ν(p3 p1)λ M = g ∗ g ∗ ∗ ε (p ) (p − p ) g − 2c D NΛc BcB D 4 α 3 1 β − 2 νλ 2 t mD∗ mD∗ λ µ ×u (p )γ u (p )ε ∗ (p ), (3.32a) Λc 3 N 1 B 4 ( ) − − − ανβµ i (p1 p4)ν(p1 p4)λ M = −g ∗ g ∗ ∗ ε (p ) (p − p ) g − 4c B NΛb BcB D 3 β 1 4 α − 2 νλ 2 u mB∗ mB∗ λ µ × ∗ uΛb (p4)γ uN (p1)εD (p3). (3.32b)

In order to produce the diagrams of all 12 Bc absorption processes by pions, rho mesons, K mesons, and nucleons we have used the Hadronic Lagrangian derived in the previous chapter. The eﬀective Lagrangian does not produce any PVV coupling. However, if PVV interaction is also included then we have several other proceses and diagrams contributing to the processes under study. In this work, we have neglected this interaction for studying Bc absorption processes by π, ρ, and K mesons.

51 (a) (b) ∗ ∗ D¯ Λb D¯ Λb

p3 p4 p3 p4

Λc B

q = p1 − p3 q = p1 − p4

p1 p2 p1 p2

− − N Bc N Bc

− → ∗ Figure 3.12: Born Diagrams for Bc absorption process NBc D Λb 3.7 Couplings and Form Factors

It can be seen in the expressions of the amplitudes of the Bc absorption processes given in the previous sections, we require the values of the several different couplings to calculate the cross sections. In Eqs. 2.25 the values of required PPV, PPVV, VVV, and VVVV couplings are expressed in terms of universal coupling g of the effective hadronic Lagrangian introduced in the previous chapter. Whereas in Eqs. 2.35, we write the values of the required BBP and BBV couplings in terms of universal coupling gP and gV and constants a, b, c, and d. These relations are the consequence of SU(5) symmetry, which is badly broken. Thus we do not expect these relations to be valid. In that case, we would fix the values of these coupling empirically or using the fundament theory of strong interaction. In this way, we can fix the values of many required coupling, but some still remain unknown and we have no other way except to use the SU(5) relations to constrain these unknown couplings. The detail discussion on the numerical values of these couplings through empirical results and by SU(5) relations is given in the next chapter.

52 (a) (b) ∗ ¯ ∗ Λ Λc B D b

p3 p4 p3 p4 D∗ B∗

q = p3 − p1 q = p1 − p4

p1 p2 p1 p2

+ − N Bc N Bc

+ → ∗ Figure 3.13: Additional diagrams for Bc absorption processes: (2) NBc ΛcB and − → ∗ (4) NBc D Λb, due to anomalous parity interaction.

In addition to the couplings, we also require the knowledge of the form factors to include the effect of finite size of the hadrons. In principle the form factors of the hadrons can be determined by the theories of the interaction of the quark constituents of the hadrons. Alternatively the effect of the form factor can also be introduced with in hadronic Lagrangian by the loop corrections to the tree level diagrams. Considering the difficulties in applying these methods, we prefer to use some phenomenological form. One form which is extensively used in the study of hadronic cross section is following monopole form factor [17] Λ2 f(q) = , (3.33) Λ2 + q2 where Λ is cutoff parameter which could have different values for different vertices. The value of cutoff for a given interaction vertex can be determined by the theories of interaction of quark constituents of the hadron or by the empirical means. Above form applies on a 3 point vertex, whereas for a 4 point vertex the following form is used. ( )( ) Λ2 Λ2 f(q) = 1 2 , (3.34) 2 ⟨ ⟩2 2 ⟨ ⟩2 Λ1 + q Λ2 + q

53 where ⟨q⟩ is the average value of the squared three momenta transfer in t and u channels.

In this case two different cutoff parameters Λ1 and Λ2 are used. Another monopole form [41] used in the study of hadronic cross sections is given by Λ2 − m2 f(t) = , (3.35) Λ2 − t where m is mass of exchange particle and Λ is again the cutoff parameter. In some studies [35] the following more advanced form has also been used. ( ) nΛ4 n f(q) = , (3.36) nΛ4 + (q2 − m2)2 where m and q are mass and four momentum of the exchange particle and Λ is the cutoff parameter. This form factor satisfies the correct normalization f(q2 = m2) = 1. If n → ∞, then above form is converted into Gaussian function. However, the condition of current conservation is violated if this form factor is used in the scattering amplitude, unlike the monopole form factor of Eq. 3.33 which conserve it. In this work we have applied the form factors of Eq. 3.33 and 3.34 to include the effect of finite size of hadrons at three and four point interaction vertices. The discussion on the values of the corresponding cutoff parameters is given in the next chapter.

3.8 Vector-Meson dominance model

φ φ

γ γ V gV φφ

φ φ

Figure 3.14: Vector Meson Dominance (VMD) model.

As mentioned in the last section that diﬀerent couplings required to calculate Bc absorption cross sections are preferably ﬁxed emprically or by using microscopic theories.

54 In this regard vector meson dominance (VMD) model can also be used to ﬁnd the values of few PPV (or VVV) couplings. In VMD the elastic form factor of pseudo-scalar (or vector) meson φ is modeled according to the Fig. 3.14. This allow as to write the following equations ∑ γ g qF (t) = − V V φφ , (3.37) φ t − m2 V =ρ,ω,ψ,Υ V where Fφ(t) is elastic form factor of φ meson of charge q, gV φφ is coupling of φ meson with vector meson V , and γV is photon-vector meson mixing amplitude and can be + − determined from the decay width ΓV ee of V → e e .

2 αγV ΓV ee = 3 . (3.38) 3mV

In the limit of t → 0; Fφ(t) become 1, which convet the equation 3.37 into following

∑ γ g q = V V φφ . (3.39) m2 V =ρ,ω,ψ,Υ V

For example: lets apply VMD on the e−D+ → e−D+. In this case q = e; hence equation 3.39 can be written as

γψgψD+D− γρgρD+D− γωgωD+D− 2 + 2 + 2 = e. (3.40) mψ mρ mω

Applying the equation 3.39 on charm quark of D+. In this case the interaction of photon with the charm quark can occur only through vector meson J/ψ. Thus we have γψgψD+D− 2 2 = e. (3.41) mψ 3 And hence equation 3.40 implies that

γρgρD+D− γωgωD+D− 1 2 + 2 = e. (3.42) mρ mω 3

Similarly for the process e−D0 → e−D0, the charge q = 0. Thus, VMD equation 3.39 can be written as following

γψg 0 γρg 0 0 ψD0D ρD0D γωgωD0D 2 + 2 + 2 = 0. (3.43) mψ mρ mω

55 Again for the interaction of photon with charm quark we have

γψgg 0 ψD0D 2 2 = e. (3.44) mψ 3 And it implies that γρg 0 0 γωg 0 2 ρD D ωD0D − 2 + 2 = e. (3.45) mρ mω 3 SU(2) isospin symmetry in D meson sector implies

+ − − 0 ≡ gρD D = gρD0D gρDD,

+ − 0 ≡ gωD D = gωD0D gωDD, (3.46)

+ − 0 ≡ gψD D = gψD0D gψDD.

In these notations we can write the above equations as following

γψgψDD 2 γρgρDD γωgωDD 1 −γρgρDD γωgωDD −2 2 = e, 2 + 2 = e, 2 + 2 = e. (3.47) mψ 3 mρ mω 3 mρ mω 3 These realtions yield ( ) ( ) ( ) 1/2 1/2 1/2 2 4πmψ 1 4πmρ 1 4πmω gψDD = α , gρDD = α , gωDD = − α . 3 3Γψee 2 3Γρee 6 3Γωee (3.48) Using the experimental values of the masses, decay constants, and α given in the Table 3.1, we obtain the following results

gψDD = gψD∗D∗ = 7.5, gρDD = gρD∗D∗ = 2.5, gωDD = gωD∗D∗ = −2.8 (3.49)

Similarly, we also apply the VMD on e−B+ → e−B+ and e−B0 → e−B0, and obtain the results ( ) ( ) ( ) 1/2 1/2 1/2 1 4πmΥ 1 4πmρ 1 4πmω gΥBB = α , gρBB = α , gωBB = α . 3 3ΓΥee 2 3Γρee 6 3Γωee Using the experimental values of the masses, and decay constant given in the Table 3.1, we obtain the following result

gΥBB = gΥB∗B∗ = 13.3 (3.50)

For other couplings, VMD obviously implies that

gρDD = gρD∗D∗ = gρBB = gρB∗B∗ , gωDD = gωD∗D∗ = −gωBB = −gωB∗B∗ . (3.51)

56 Constant Value

mρ 0.776 GeV

mω 0.783 GeV

mψ 3.097 GeV

mΥ 9.460 GeV −3 Γρee 7.02 10 MeV −4 Γωee 6.0 10 MeV −3 Γψee 5.4 10 MeV −3 ΓΥee 1.314 10 MeV α 1/137

Table 3.1: Values of the constants used in VMD model calculations.

57 Chapter 4

Bc Absorption Cross Sections by π, ρ, and K Mesons

In this chapter we calculate Bc absorption cross sections by π, ρ and K mesons using the amplitudes given in the chapter 3. Numerical values of the required coupling are also calculated along with the cutoﬀ parameters of the form factors used in the calculations.

Sections 4.2, 4.5, and 4.6, which cover Bc absorption cross sections by pions are based on ref. [25]. Sections 4.3 and 4.7, in which Bc absorption cross sections by rho mesons are calculated, are based on ref. [26].

4.1 Spin and Isospin averaged cross sections

General formula for the diﬀerential cross section of a two-body process is given by

dσ I0 2 = 2 M , (4.1) dt 64πspi,cm where the bar over M represents averaging (summing) over initial (ﬁnal) spins, I0 is isospin factor which is required to produce isospin averaged value of the cross section, and pi,cm is 3 momentum of initial particles in c.m frame given by

[s − (m + m )2][s − (m − m )2] p2 = 1 2 1 2 (4.2) i,cm 4s

58 Total cross section is obtained by integrating over the variable t with lower and upper limits given by [ ] m2 − m2 − m2 − m2 2 t = 1 3 √ 2 4 − (p − p )2 (4.3) 0 2 s 1cm 3cm [ ] m2 − m2 − m2 − m2 2 t = 1 3 √ 2 4 − (p + p )2 (4.4) 1 2 s 1cm 3cm Using the formula given is Eq. 4.1, we can write the diﬀerential cross sections of the

8 processes of Bc absorption by π, ρ and K mesons, whose amplitude are given in the previous chapter

4.2 Cross sections of Bc by π

+ → 4.2.1 Bc π DB

+ → The spin and isospin average diﬀerential cross section of the process Bc π DB is given by dσ1 2 ∗ = 2 M1M1 (4.5) dt 64πspi,cm where M1 is given by the Eq. 3.6. Contraction of Lorentz indices in M1a and M1b gives the results ( ) (p2−p2)(p p +p p −p p −p p ) ∗ ∗ 1 1 3 1 2 1 4 2 3 3 4 M = g g 2 p1p2+p1p4+p2p3+p3p4− , (4.6) 1a πDD BcBD t−m ∗ m2 D ( D∗ )

(p2−p2)(p p +p p −p p −p p ) ∗ ∗ 1 − 1 4 1 2 1 3 2 4 3 4 M1b = gπBB gB B D 2 p1p2+p1p3+p2p4+p3p4 , (4.7) c u−m ∗ m2 ∗ Bs Bs where i factor in the amplitudes is discarded because it gets eliminated due to square modulus in the diﬀerential cross section. Amplitudes after contraction are given in term of invariant scalar products of 4-momentum of the particles, which can be written in term of Mandelstam variables. These relations are summarized in the Appendix B. After applying these relations and putting the values of the masses of the particles and couplings whose numerical values are calculated in section 4.5, we obtain the following result for M1 ( ) 147.56(s − t) − 5421.01 104.72s + 52.36t − 4378.01 M = + 1 43.95 − s − t − 28.40 t − 4.0401

59 + → ∗ ∗ 4.2.2 Bc π D B

+ → ∗ ∗ The spin and isospin average diﬀerential cross section of the process Bc π D B is given by ( )( ) dσ2 2 µν ∗µ′ν′ − p3µp3µ′ − p4νp4ν′ = 2 M2 M2 gµµ′ 2 gνν′ 2 , (4.8) dt 64πspi,cm m3 m4

µν where M2 is given by µν µν µν µν M2 = M2a + M2b + M2c , (4.9)

And,

µν µ i ν M = −g ∗ g ∗ (2p − p ) (p − p + p ) , (4.10) 2a πDD BcB D 1 3 − 2 2 1 3 t mD µν µ i ν M = −g ∗ g ∗ (2p − p ) (p − p + p ) , (4.11) 2b πBB BcBD 1 4 − 2 2 1 4 u mB µν µν − ∗ ∗ M2c = igπBcB D g . (4.12)

In this case the contraction of the Lorentz indices is done after taking the square modulus. The resultant expression of diﬀerential cross section is very lengthy and cannot be written here.

4.3 Cross sections of Bc by ρ

+ → ∗ 4.3.1 Bc ρ D B

+ → ∗ The spin and isospin average diﬀerential cross section of the process Bc ρ D B is given by ( )( ) dσ3 2 µν ∗µ′ν′ − p1µp1µ′ − p4νp4ν′ = 2 M3 M3 gµµ′ 2 gνν′ 2 , (4.13) dt 192πspi,cm m1 m4

µν where M3 is given by µν µν µν µν M3 = M3a + M3b + M3c , (4.14)

60 And,

µν µ i ν M = g g ∗ (2p − p ) (p − 2p ) , (4.15) 3a ρDD BcBD 3 1 − 2 4 2 [ t mB ] µν µ βν ν µβ β µν ∗ ∗ ∗ − − − − M3b = gρD D gBcBD (2p4 p1) g + (2p1 p4) g + ( p4 p1) g ( ) − − −i (p1 p4)α (p1 p4)β α × gαβ − (−p3 − p2) , (4.16) − ∗ 2 u mD mD∗ µν ∗ µν M3c = igρBcBD g . (4.17)

Again the resultant expression obtained after contraction is very long.

+ → ∗ 4.3.2 Bc ρ DB

+ → ∗ The spin and isospin average diﬀerential cross section of the process Bc ρ DB is given by ( )( ) dσ4 2 µν ∗µ′ν′ − p1µp1µ′ − p4νp4ν′ = 2 M4 M4 gµµ′ 2 gνν′ 2 , (4.18) dt 192πspi,cm m1 m4 µν where M4 is given by µν µν µν µν M4 = M4a + M4b + M4c , (4.19) And,

µν µ i ν M = g g ∗ (2p − p ) (p − 2p ) , (4.20a) 4a ρDD BcB D 3 1 − 2 4 2 [ t mD ] µν µ βν ν µβ β µν ∗ ∗ ∗ − − − − M4b = gρB B gBcB D (2p4 p1) g + (2p1 p4) g + ( p4 p1) g ( ) − − −i (p1 p4)α (p1 p4)β α gαβ − (−p3 − p2) , (4.20b) − ∗ 2 u mB mB∗ µν ∗ µν M4c = igρBcB Dg . (4.20c)

In this case again the resultant expression after contractions is very long.

4.4 Cross sections of Bc by K

+ → + 4.4.1 KBc Ds B

+ → + The spin and isospin average diﬀerential cross section of the process KBc Ds B is given by dσ5 1 ∗ = 2 M5M5 , (4.21) dt 64πspi,cm

61 where M5 is given by Eq. 3.14. Contraction of Lorentz indices in M5a and M5b gives the results ( ) 2− 2 − − 1 (p1 p3)(p1p2+p1p4 p2p3 p3p4) M5a = −gKD∗D gB BD∗ p1p2+p1p4+p2p3+p3p4− , (4.22) s c t−m2 ∗ m2 ∗ D ( D ) (p2−p2)(p p +p p −p p −p p ) − ∗ ∗ 1 − 1 4 1 2 1 3 2 4 3 4 M5b = gKBB gB D B p1p2+p1p3+p2p4+p3p4 , (4.23) s c s s u−m2 ∗ m2 ∗ Bs Bs where i factor in the amplitudes is again discarded. Amplitudes after contraction are given in term of invariant scalar products of 4-momentum of the particles, which can be written in term of Mandelstam variables. After applying these relations and putting the values of the masses of the particles we obtain the following result for M5 ( ) 208.681(s − 1.t − 33.5988 74.0482(2s + t − 81.9559 M = + 5 42.2 − s − t t − 4.0401

+ → ∗+ ∗ 4.4.2 KBc Ds B

+ → ∗+ ∗ The spin and isospin average diﬀerential cross section of the process KBc Ds B is given by ( )( ) dσ6 1 µν ∗µ′ν′ − p3µp3µ′ − p4νp4ν′ = 2 M6 M6 gµµ′ 2 gνν′ 2 , (4.24) dt 64πspi,cm m3 m4 µν where M6 is given by µν µν µν µν M6 = M6a + M6b + M6c , (4.25) And,

µ i ν M = −g ∗ g ∗ (p − 2p ) (p − p − p ) , (4.26) 6a KDDs BcB D 3 1 − 2 1 3 2 t mD ν i µ M = −g ∗ g ∗ (p − 2p ) (p − p − p ) , (4.27) 6b KB Bs BcBsDs 4 1 u − m2 1 4 2 Bs µν − ∗ ∗ M6c = igKBcDs B g . (4.28)

+ → 0 4.4.3 KBc DBs

+ → 0 The spin and isospin average diﬀerential cross section of the process KBc DBs is given by dσ7 1 ∗ = 2 M7M7 , (4.29) dt 64πspi,cm

62 where M7 is given by Eq. 3.18. Contraction of Lorentz indices in M1a and M2a gives the results

( ) (p2−p2)(p p +p p −p p −p p ) − ∗ ∗ 1 − 1 3 1 2 1 4 2 3 3 4 M7a = gKDD gB B D p1p2+p1p4+p2p3+p3p4 (4.30) s c s s t−m2 ∗ m2 ∗ Ds ( Ds ) 2− 2 − − 1 (p1 p4)(p1p2+p1p3 p2p4 p3p4) M7b = −g ∗ g ∗ p1p2+p1p3+p2p4+p3p4− (4.31) KB Bs BcB D u−m2 m2 B∗ B∗

Where i factor in the amplitudes is again discarded. Amplitudes after contraction are given in term of invariant scalar products of 4-momentum of the particles, which can be written in term of Mandelstam variables. After applying these relations and putting the values of the masses of the particles we obtain the following result for M7 ( ) 208.681(s − t − 36.2725) 74.0482(2s + t − 79.82) M = + 7 43.68 − s − t t − 4.46054

+ → ∗ ∗0 4.4.4 KBc D Bs

+ → ∗ ∗0 The spin and isospin average diﬀerential cross section of the process KBc D Bs is given by ( )( ) dσ8 1 µν ∗µ′ν′ − p3µp3µ′ − p4νp4ν′ = 2 M8 M8 gµµ′ 2 gνν′ 2 , (4.32) dt 64πspi,cm m3 m4

µν where M8 is given by µν µν µν µν M8 = M8a + M8b + M8c , (4.33)

And,

µ i ν M = −g ∗ g ∗ (p − 2p ) (p − p − p ) , (4.34) 8a KD Ds BcBs Ds 3 1 t − m2 1 3 2 Ds ν i µ M = −g ∗ g ∗ (p − 2p ) (p − p − p ) , (4.35) 8b KBBs BcBD 4 1 − 2 1 4 2 u mB µν − ∗ ∗ M8c = igKBcD Bs g . (4.36)

Calculating diﬀerential cross sections using the given expression of amplitude is a tedious task as it requires to apply contractions over several Lorentz indices and evalu- ating the Dirac traces in case of baryon interactions. The calculations are especially

63 very tedious when three or more diagrams are contributing. In this case symbolic com- putational software can be very useful. In this work we use a Mathematica package ‘FeynCalc‘ [58] for evaluation of contractions and the traces. The package also allows to apply the invariant scalar products of four momentum of initial and ﬁnal particles in terms of Mandelstam variables s and t.

4.5 Numerical values of the couplings and other input parameters

Numerical values of the masses of mesons and baryons are taken from Particle Data

Group [47]. The coupling constant gπDD∗ can be determined directly from the decay ∗ width of the strong process D → Dπ [48, 49]. The resultant value of the gπDD∗ is 4.4. However, recent measurement of the decay width at CLEO gives 17.9 [50]. This value, which is about 4 times greater than given in Refs. [48, 49], can significantly enhance the values of calculated cross sections. However, we prefer to use the value 4.4 until we have another experimental confirmation of this result. The coupling gπBB∗ can be determined from the decay width of the strong decay process B∗ → Bπ. So far we do not have any experimental result on this decay width. However, various theoretical results based on heavy quark effective theories and QCD sum rule are available. Heavy

mB quark symmetries [49, 51, 52] imply that gπBB∗ ≈ gπDD∗ = 12.4 and from light-cone mD

QCD sum rule [49], we obtain gπBB∗ = 10.3. In this work, we use the value obtained

∗ ∗ from the former method. The values of the couplings gBcBD and gBcB D can be ﬁxed by using gΥBB = 13.3, which is obtained using vector meson dominance (VMD) model 2 in Ref. [20] and SU(5) symmetry result g ∗ = g ∗ = √ g [31]. In this BcBD BcB D 5 ΥBB

∗ ∗ way we obtain gBcBD = gBcB D = 11.9. There is no empirically result available for

∗ ∗ the four-point coupling gπBcB D , so we use SU(5) symmetry relations, which implies

∗ ∗ ∗ ∗ ∗ ∗ gπBcD B = 2gπDD gBcB D = 2gπBB gBcBD . These two identities give the values of 105 and 295. Considering the large variation in the results, we use the average value 200. However, we also study in this work the eﬀect of this variation on the cross sections of the related processes.

64 The values of the couplings gρDD, gρBB, gρD∗D∗ and gρB∗B∗ are calculated using vector- meson dominance model (VMD) in Ref. [17, 20]. These values are given by

gρDD = gρBB = gρD∗D∗ = gρB∗B∗ = 2.52, (4.37)

It is an interesting result that these VMD values are consistent with the SU(5) symmetry relations given in Eq. 2.25b. Again we do not have any empirical result of the

∗ ∗ four-point contact couplings gρBcBD and gρBcB D; thus we use the SU(5) symmetry relations of Eq. 2.26b, which implies

∗ ∗ gρBcBD = gρBcB D = 30, (4.38)

∗ ∗ Unlike the result on the coupling gπBcB D , SU(5) symmetry relations provide ﬁxed

∗ ∗ ∗ ∗ ∗ values of gρBcBD and gρBcB D. The couplings gKD Ds and gKDDs are ﬁxed using gπDD

∗ ∗ and SU(5) symmetry relation of Eq. (2.26c). The couplings gKB Bs and gKBBs are ﬁxed using gπBB∗ and SU(5) symmetry relation of Eq. (2.26d). There is no empirically ﬁtted

∗ ∗ ∗ ∗ value available for the four-point couplings gKBcDs B and gKBcD Bs , thus we use SU(5) symmetry relations given by Eqs. (2.26e) and (2.26f). These relations give two values of 149 and 417, whereas their mean value is 283. The values of coupling constants used in this paper and methods for obtaining them are summarized in Table 4.1.

4.6 Results of Bc absorption cross sections by π

4.6.1 Results without form factors

+ → Shown in Fig. 4.1 is the Bc absorption cross section of the process Bc π DB as √ a function of total center of mass (c.m) energy s. The process is endothermic with √ threshold energy of 7.15 GeV. Thus the cross section is zero if s < 7.15 as shown in the Fig. 4.1. The figure shows that initially cross section increases rapidly and then decreases slowly beyond 8 GeV energy. Average cross section away form the threshold is about 45 mb. This value is considerably high and could drastically affect the production rate of Bc meson in heavy-ion experiments. However, in these results we have assumed contact interaction at the vertices. We expect significant change in

65 Coupling Constant Value Method of Derivation ∗ gπDD∗ 4.4 D decay width

gπBB∗ 12.4 Heavy quark symmetries

gπBB∗ 10.3 light-cone QCD-sum rule

∗ ∗ gBcBD and gBcB D 11.9 VMD, SU(5) symmetry

∗ ∗ gπBcB D 105 to 295 SU(5) symmetry

gρDD, gρBB, gρD∗D∗ , gρB∗B∗ 2.52 VMD

gρB D∗B, gρB DB∗ 30 SU(5) symmetry c c √ gKD∗D , gKDD∗ 4.4 2 gπDD∗ , SU(5) symmetry s s √ ∗ ∗ ∗ gKB Bs , gKBBs 12.4 2 gπBB , SU(5) symmetry

∗ ∗ gBcBsDs , gBcBs Ds 11.9 VMD, SU(5) symmetry

∗ ∗ ∗ ∗ gKBcDs B and gKBcD Bs 149 to 417 SU(5) symmetry

Table 4.1: Values of coupling constants required for calculating Bc absorption cross sections by π, ρ and K mesons. the cross sections when the form factors are included. In Fig. 4.2, we present the + → ∗ ∗ Bc absorption cross section of the process Bc π D B as a function of total c.m √ energy s. This process is also endothermic with threshold energy of 7.34 GeV. The

∗ ∗ cross section of this process depend upon the four-point contact coupling gπBcB D , whose value is not ﬁxed empirically or by the microscopic theory. Thus we have to use SU(5) symmetry relations in order to ﬁx its value. As discussed in the section 4.1, SU(5) relations give two possible values 105 and 295. Accordingly we produce the cross sections for these values as well as their mean 200. Dotted, solid and dashed curves in

∗ ∗ the Fig. 4.2 are for gπBcB D = 105, 200, and 295 respectively. The ﬁgure shows that the cross sections increase very rapidly for the values 105 and 295, which is unrealistic.

∗ ∗ However, if we use gπBcB D = 200, the average of the two extreme values the variation in the cross section is some what realistic. In this work, we treat this variation in the 4-point contact coupling as an uncertainty and study its eﬀect on the cross sections

66 60

40 L mb

H 30 Σ 20

0 4 6 8 10 12 14 s HGeVL

+ → Figure 4.1: Bc absorption cross sections for the process Bc π DB without form factor when the form factor are included.

4.6.2 Form Factors

In the previous section the cross section are reported assuming the contact interaction at the vertices. However, the mesons and baryons have finite size and we must include its effect by the form factors. In section 3.7, we have discussed different forms of phenomenological form factor used in study of hadronic cross sections. In this work we use the monopole form factors given in Eqs. 3.33 and 3.34. For convenience we note them again as following

Λ2 f(q) = 2 2 , (4.39) (Λ + q )( ) Λ2 Λ2 f(q) = 1 2 , (4.40) 2 ⟨ ⟩2 2 ⟨ ⟩2 Λ1 + q Λ2 + q

67 HaL 40

30 L mb

H 20 Σ 10

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗ Figure 4.2: Bc absorption cross sections of the process Bc π D B without form

∗ ∗ factor for three different values four-point coupling gπBcB D = 105, 200, 295 for dotted, solid and dashed curves respectively. where q2 is squared three momentum transfer in c.m frame and ⟨q⟩2 is the average value of the squared three momentum transfers in t and u channels. The first form factor, which applies on 3-point interaction, require only one cutoff parameter Λ, whereas the second form factor applies on 4-point interaction and require two parameters Λ1 and

Λ2. The values of the cutoff parameters are not same for different interaction vertices. It is noted that like form factors the cutoff parameters in the given form factor can also be determined by the theories of the interaction of the quark constituents of the interacting hadrons [54] or by the Loop corrections to tree level diagrams with in the effective hadronic Lagrangian. Alternatively, the cutoff parameters can also be fixed empirically by studying hadronic scattering data in meson or baryon exchange models. Such studies show that the cutoff parameters varies from 1 to 2 GeV for the vertices

68 connecting light hadrons (π, K, ρ, N etc) [55]. However, due to limited empirical information about the scattering data of charmed and bottom hadrons, we do not have any direct knowledge of the empirical values of the related cutoff parameters. In this case we can estimate cutoff parameters by relating them with the inverse root mean square (rms) size of the hadrons. In Ref. [56] the Cutoff parameters for the meson- meson vertices are determined by the ratio of size of nucleons to the pseudo scalar mesons. rN rN ΛD = ΛN , ΛB = ΛN (4.41) rD rB

The values of the ratios rN /rD = 1.35 and rN /rB = 1.29 are determined by the quark potential model for D and B mesons respectively [56]. The Cutoﬀ parameter ΛN for the nucleon-meson vertex can be obtained from the empirical data of nucleon-nucleon systems. In Ref. [56] ΛN = 0.94 GeV, is ﬁxed from the empirical value of the binding energy of deuterium, whereas nucleon-nucleon scattering data gives ΛπNN = 1.3 GeV and ΛρNN = 1.4 GeV [57]. A variation of 0.9 to 1.4 GeV in ΛN produces a variation of

1.2 to 1.8 GeV in ΛD and ΛB. Based on these results we take all the cutoﬀ parameters to be the same for simplicity and vary them on the scale 1 to 2 GeV to study the uncertainties in cross sections due to the cutoﬀ parameter.

4.6.3 Amplitudes of Bc absorption by pions with form factor

+ → Given in Eqs. 3.6 and 3.8 are scattering amplitudes of the processes Bc π DB and + → ∗ ∗ Bc π D B respectively. In these expressions form factor of interacting mesons are not included. In this section, we rewrite these amplitude after including the monopole + → form factors used in this work. Two amplitudes of the process Bc π DB are given by ( ) ( ) 2 2 µ ν Λ −i µν − (p1−p3) (p1−p3) − − M1a = gπDD∗ gB BD∗ (p1+p3)µ g ( p4 p2)ν (4.42a) c Λ2+q2 t−m2 ∗ m2 ∗ ( 1 ) D ( D ) 2 2 µ ν Λ −i µν (p1−p4) (p1−p4) M = g ∗ g ∗ (p1+p4)µ g − (−p3−p2)ν (4.42b) 1b πBB BcB D Λ2+q2 u−m2 m2 2 B∗ B∗ 2 2 where q1 and q2 are given by 2 − 2 − q1 = (E1cm E3cm) t (4.43) 2 − 2 − q2 = (E1cm E4cm) u (4.44)

69 + → ∗ ∗ And three amplitudes of the process Bc π D B are given by

( ) 2 Λ2 i µ ν M = −g ∗ g ∗ (2p1−p3)µ (p2−p1+p3)ν εr (p3)ε (p4) (4.45a) 2a πDD BcB D Λ2+q2 t−m2 s ( 1 ) D 2 Λ2 i µ ν M = −g ∗ g ∗ (2p1−p4)µ (p2−p1+p4)ν εr (p3)ε (p4) (4.45b) 2b πBB BcBD Λ2+q2 u−m2 s ( 2 ) B 2 2 Λ µ ν − ∗ ∗ M2c = igπBcB D 2 2 gµνεr (p3)εs (p4) (4.45c) Λ + q3

2 2 2 where q1 and q2 are same as in Eqs. 4.43 and 4.44, whereas q3 is given by 1 [ ] q2 = (E − E )2 − t + (E − E )2 − u (4.46) 3 2 1cm 3cm 1cm 4cm

These new amplitudes are used again in the formula given in Eqs. 4.5 and 4.8, to determine diﬀerential and total cross sections. The results are reported in the next sections.

4.6.4 Results with form factor

+ → Shown in Fig. 4.3 is the Bc absorption cross section of the process Bc π DB as a √ function of total c.m energy s. In this case the form factor is included. Lower and upper curves correspond to Λ = 1 and 2 GeV respectively. The cross section again increases rapidly near the threshold energy 7.15 GeV and then becomes almost constant + → after decreasing slightly near 8 GeV. The figure shows that for Bc π DB process the cross section roughly varies from 2 to 7 mb, when the cutoff parameter is between 1 to 2 GeV. In order to compare these results with those obtained without form factor, we combine all the plots in Fig. 4.4. This figure shows that substantial suppression is produced when the form factor in include. Suppression due to form factor at cutoff

Λ = 1 and 2 GeV is roughly by factor 11 and 3 respectively. Fig. 4.5 shows the Bc + → ∗ ∗ absorption cross sections of the process Bc π D B as a function of total c.m energy √ s. The monopole form factor is included and the value of the 4-point contact coupling

∗ ∗ gπBcB D = 200 . The cross section of the process roughly varies between 0.2 to 2 mb,

∗ ∗ when the cutoﬀ parameter is varied from 1 to 2 GeV at gπBcB D = 200. Comparison of the cross sections with and without form factor is given in the Fig. 4.6. Suppression due to form factor at cutoﬀ Λ = 1 and 2 GeV is approximately by factor 45 and 7

70 20

15 L mb

H 10 Σ

0 4 6 8 10 12 14 s HGeVL

+ → Figure 4.3: Bc absorption cross sections of the process Bc π DB with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively. respectively. Relatively high suppression in this process is mainly due to larger values of the masses of ﬁnal particles D∗ and B∗. It is noted in Fig. 4.2 that values of + → ∗ ∗ the cross sections of the process Bc π D B without form factor highly depend

∗ ∗ upon the choice of the value of 4-point coupling gπBcB D . In order to see the effect of its value on the cross sections with form factor, we produce three plots of the cross ∗ ∗ + → ∗ ∗ sections of the process Bc π D B in Fig. 4.7 for gπBcB D = 105, 200, 295 at cutoff Λ = 1.5 GeV. The figure shows that the value of the contact coupling significantly affects the cross section only near the threshold energy (7.34 GeV). At energy greater than 8.5 GeV the values of cross sections become independent of the value of coupling

∗ ∗ gπBcB D . This show ours results may be uncertain near threshold, but little away from it the results are robust. The eﬀect of uncertainty in the four point contact coupling is further marginalized in the total absorption cross section for Bc + π due to relatively small values of the cross sections of the second process. This is shown in Fig. 4.8 in

71 60 50 40 L mb

H 30 Σ 20 10 0 4 6 8 10 12 14 s HGeVL

+ → Figure 4.4: Comparison of the cross sections of the process Bc π DB with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively. which the plots of total absorption cross sections are given for three diﬀerent values of

∗ ∗ gπBcB D = 105, 200, 295.

4.7 Results of Bc absorption cross sections by ρ

4.7.1 Results without form factors

+ → ∗ Shown in Fig. 4.9 is the Bc absorption cross section of the process Bc ρ D B as a √ function of total center of mass energy s. The process is endothermic with threshold √ energy of 7.29 GeV. Thus the cross section is zero if s < 7.29 as shown in the Fig. 4.9. The ﬁgure shows that initially cross section increases rapidly and then decreases slowly and becomes almost constant beyond 9 GeV energy. Average cross section away form

72 4

3 L mb

H 2 Σ

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗ Figure 4.5: Bc absorption cross sections for the process Bc π D B with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

the threshold is about 5 mb. In Fig. 4.10, we present the Bc absorption cross section of √ + → ∗ the process Bc ρ DB as a function of total center of mass energy s. This process is also endothermic with threshold energy of 7.2 GeV. The ﬁgure shows that initially cross section again increases rapidly and then decreases slowly and becomes almost constant beyond 8 GeV energy. Average cross section away form the threshold is about 0.65 mb. In these results we have assumed contact interaction at the vertices. We expect signiﬁcant change in the cross sections when the form factors are included. The cross

∗ section of these processes depend upon the four-point contact couplings gρBcBD and

∗ gρBcB D respectively, whose values are not known empirically or by the microscopic theory. Thus we have to use SU(5) symmetry relations in order to ﬁx its value. As

∗ ∗ discussed in the section 2.2.1, SU(5) relations implies gρBcBD = gρBcB D = 30. As SU(5) is badly broken, so we consider this value unreliable. The eﬀect of variation in the 4-point contact coupling on the cross sections is studied in next section after

73 30 25 20 L mb

H 15 Σ 10 5 0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗ Figure 4.6: Comparison of the cross sections of the process Bc π D B with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively. including the form factors.

4.7.2 Amplitudes of Bc absorption by ρ with form factor

In the last section the cross section are reported assuming the contact interaction at the + → ∗ + → ∗ vertices. In this section we report the result of processes Bc ρ D B and Bc ρ DB with form factors. In this case we again use the same monopole for factor with the cutoﬀ parameters estimated from the sizes of the interacting mesons. Given in Eqs. + → ∗ + → ∗ 3.10 and 3.12 are scattering amplitudes of the processes Bc ρ D B and Bc ρ DB respectively, without inclusion of the form factor. Here, we rewrite these amplitude after including the monopole form factors used in this work. Three amplitudes of the

74 4

3 L mb

H 2 Σ

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗ Figure 4.7: Bc absorption cross sections for the process Bc π D B with form factor

∗ ∗ for three diﬀerent values of four-point coupling, gπBcB D = 105, 200, 295 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV.

+ → ∗ process Bc ρ D B are given by ( ) 2 2 Λ i µ ν M = g g ∗ (2p − p ) (p − 2p ) ε (p )ε (p ), (4.47a) 3a ρDD BcBD 2 2 3 1 µ − 2 4 2 ν r 1 s 4 Λ + q1 t mB ( ) [ ] Λ2 2 ∗ ∗ ∗ − − − − M3b = gρD D gBcBD 2 2 (2p4 p1)µ gβν + (2p1 p4)ν gµβ + ( p4 p1)β gµν ( Λ + q2 ) −i (p − p )α (p − p )β αβ − 1 4 1 4 − − µ ν g ( p3 p2) εr (p1)εs (p4), (4.47b) − ∗ 2 α u mD mD∗ ( ) Λ2 2 ∗ µ ν M3c = igρBcBD gµν 2 2 εr (p1)εs (p4). (4.47c) Λ + q3

75 20

15 L mb

H 10 Σ 5

0 4 6 8 10 12 14 s HGeVL

Figure 4.8: Total Bc absorption cross sections by pion for three diﬀerent values of

∗ ∗ four-point coupling, gπBcB D = 105, 200, 295 for dotted, solid and dashed curves respectively. Cutoﬀ parameter is taken 1.5 GeV.

2 2 2 where q1, q2, and q3 are given by equations 4.43, 4.44, and 4.46 respectively. The + → ∗ amplitudes of the second process Bc ρ DB are given by ( ) 2 2 Λ i µ ν M = g g ∗ (2p − p ) (p − 2p ) ε (p )ε (p ), (4.48a) 4a ρDD BcB D 2 2 3 1 µ − 2 4 2 ν r 1 s 4 Λ + q1 t mD ( ) [ ] Λ2 2 M = g ∗ ∗ g ∗ (2p − p ) g + (2p − p ) g + (−p − p ) g 4b ρB B BcB D Λ2 + q2 4 1 µ βν 1 4 ν µβ 4 1 β µν ( 2 ) − α β i αβ (p1−p4) (p1−p4) µ ν g − 2 (−p − p ) ε (p )ε (p ), (4.48b) m ∗ 3 2 α r 1 s 4 u − m ∗ B B ( ) Λ2 2 ∗ µ ν M4c = igρBcB D 2 2 gµνεr (p1)εs (p4). (4.48c) Λ + q3

4.7.3 Results of Bc absorption by ρ with form factor

+ → ∗ Shown in Fig. 4.11 is the Bc absorption cross section of the process Bc ρ D B as √ a function of total c.m energy s. In this case the form factor is included. Lower

76 Ρ Bc D B 10

L 6 mb H

Σ 4

0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.9: Bc absorption cross sections for the process Bc ρ D B without form factor. and upper curves correspond to Λ = 1 and 2 GeV respectively. The cross section again increases rapidly near the threshold energy 7.29 GeV and then becomes almost + → ∗ constant after decreasing slightly near 8 GeV. The figure shows that for Bc ρ D B process the cross section roughly varies from 0.6 to 3 mb, when the cutoff parameter is between 1 to 2 GeV. In order to compare these results with those obtained without form factor, we combine all the plots in Fig. 4.12. This figure shows that substantial suppression is produced when the form factor in include. Suppression due to form factor at cutoff Λ = 1 and 2 GeV is roughly by factor 6.5 and 1.6 respectively. + → ∗ Fig. 4.13 shows the Bc absorption cross sections of the process Bc ρ DB as a √ function of total c.m energy s. The cross section of the process roughly varies between 0.05 to 0.3 mb, when the cutoff parameter is varied from 1 to 2 GeV. Comparison of

77 Ρ Bc DB 2.0

1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.10: Bc absorption cross sections for the process Bc ρ DB without form factor. the cross sections with and without form factor is given in the Fig. 4.14. Suppression due to form factor at cutoﬀ Λ = 1 and 2 GeV is approximately by the factors of 10 and 2.5 respectively. The scale of this variation is about 3 times smaller than the calculated variation of the total absorption cross section by pions in Ref. [25], using the same model.

It is noted that the Bc absorption cross sections by ρ mesons depend upon the four-

∗ ∗ point contact couplings gρBcBD and gρBcB D, whose values are fixed through the SU(5) symmetry. This symmetry is broken by the mass terms of the vector mesons in the Lagrangian. Thus, it is expected that estimated values of these couplings are less reliable. In order to study the role of these couplings in fixing the values of the cross + → ∗ sections, we calculate the absorption cross sections of the processes Bc ρ D B and ∗ + → ∗ ∗ Bc ρ DB for three different values of four-point couplings gρBcBD and gρBcB D =

78 Ρ Bc D B 10

L 6 mb H

Σ 4

0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.11: Bc absorption cross sections for the process Bc ρ D B with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

15, 30, 45. The results given in Fig. 4.15 and 4.16 show that for both of the processes the effect of uncertainty in the four-point couplings is marginal. Thus, any change in the values of these four-point couplings cannot significantly affect the results reported in this work.

4.8 Results of Bc absorption cross sections by K

4.8.1 Results without form factors

Shown in Fig. 4.17 and 4.18 are the Bc absorption cross sections of the processes (1) + → + + → 0 KBc Ds B and (3) KBc DBs respectively as a function of total center of mass √ energy s.

79 Ρ Bc D B 10

L 6 mb H

Σ 4

0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.12: Comparison of the cross sections of the process Bc ρ D B with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

The processes are endothermic with threshold energy of 7.247 and 7.233 GeV respectively. The ﬁgures shows that initially cross sections increases rapidly and then decreases slowly beyond 8 GeV energy. Average cross section away form the threshold is about 90 mb for both of the processes. The value is considerably high. However, in these results we have assumed contact interaction at the vertices. We expect signiﬁcant suppression in the cross sections when the form factors are included. In Fig. 4.19 and + → ∗+ ∗ 4.20, we present the Bc absorption cross sections of the processes (2) KBc Ds B √ + → ∗ ∗0 and (4) KBc D Bs as a function of total center of mass energy s. These processes are also endothermic with threshold energy of 7.437 and 7.423 GeV respectively. The

∗ ∗ cross sections of these processes depend upon the four-point contact coupling gKBcDs B

80 Ρ Bc DB 1.0

0.8

L 0.6 mb H

Σ 0.4

0.2

0.0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.13: Bc absorption cross sections for the process Bc ρ DB with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively

∗ ∗ and gKBcD Bs , whose values are not fixed empirically or by the microscopic theory. Thus we have to use SU(5) symmetry relations in order to fix its value. As discussed in the section 4.5, SU(5) relations give two possible values 149 and 417, where as their mean value is 283. The figures shows that the cross sections of both of the processes are increasing function of energy. Initial the increase in very rapid, which is followed

∗ ∗ ∗ ∗ by a moderate increase beyond 8 GeV for gKBcDs B = gKBcD Bs = 283.

4.8.2 Amplitudes of Bc absorption by K with form factor

Given in Eqs. 3.14, 3.16, 3.18, and 3.20 are scattering amplitudes of the processes (1) + → + + → ∗+ ∗ + → 0 + → ∗ ∗0 KBc Ds B, (2) KBc Ds B , (3) KBc DBs , and (4) KBc D Bs respectively. In these expressions the forms factor of interacting mesons are not included. In this section, we rewrite these amplitude after including the monopole form factors

81 Ρ Bc DB 2.0

1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.14: Comparison of the cross sections of the process Bc ρ DB with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

+ → + used in this work. Two amplitudes of the process KBc Ds B are given by

( ) ( ) 2 2 µ ν Λ −i µν (p1−p3) (p1−p3) M = −g ∗ g ∗ (p1+p3)µ g − (−p4−p2)ν , (4.49a) 5a KD Ds BcBD Λ2+q2 t−m2 m2 1 D∗ D∗ ( ) ( ) 2 2 − (p −p )µ(p −p )ν − ∗ ∗ Λ i µν − 1 4 1 4 − − M5b = gKBB gB D B (p1+p4)µ g ( p3 p2)ν ,(4.49b) s c s s Λ2+q2 u−m2 ∗ m2 ∗ 2 Bs Bs

2 2 where q1 and q2 are given by Eqs. 4.43 and 4.44. And three amplitudes of the process + → ∗+ ∗ KBc Ds B are given by

( ) 2 Λ2 i µ ν M = −g ∗ g ∗ (p3−2p1)µ (p1−p3−p2)ν εr (p3)ε (p4), (4.50a) 6a KDDs BcB D Λ2+q2 t−m2 s ( 1 ) D 2 Λ2 i µ ν M = −g ∗ g ∗ (p4−2p1)ν (p1−p4−p2)µεr (p3)ε (p4), (4.50b) 6b KB Bs BcBsDs Λ2+q2 u−m2 s ( ) 2 Bs 2 2 µ −ig ∗ ∗ Λ g ε (p )εν (p ). M6c = KBcDs B 2 2 µν r 3 s 4 (4.50c) Λ +q3

82 Ρ Bc D B 10

L 6 mb H

Σ 4

0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.15: Bc absorption cross sections of the processes Bc ρ D B for three

∗ ∗ diﬀerent values of the four-point couplings gρBcBD and gρBcB D = 15, 30, 45 for dotted, solid and dashed curves, respectively. Cutoﬀ parameter is taken 1.5 GeV.

2 2 2 where q1, q2, and q3 are given by Eqs. 4.43, 4.44, and 4.46 respectively. These new amplitudes are used again in the formula given in Eqs. 4.5 and 4.8, to determine diﬀerential and total cross sections. The results are reported in the next sections. Two + → 0 amplitudes of the process KBc DBs are given by

( ) ( ) 2 2 − (p −p )µ(p −p )ν − ∗ ∗ Λ i µν − 1 3 1 3 M7a = gKDD gB B D (p1+p3)µ g (p4+p2)ν , (4.51) s c s s Λ2+q2 t−m2 ∗ m2 ∗ 1 D D ( ) s ( s ) 2 2 µ ν Λ −i µν (p1−p4) (p1−p4) M = −g ∗ g ∗ (p1+p4)µ g − (p3+p2)ν , (4.52) 7b KB Bs BcB D Λ2+q2 u−m2 m2 2 B∗ B∗

These new amplitudes are used again in the formula given in Eqs. 4.5 and 4.8, to determine diﬀerential and total cross sections. And three amplitudes of the process

83 Ρ Bc DB 2.0

1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

+ → ∗ Figure 4.16: Bc absorption cross sections of the processes Bc ρ DB for three

∗ ∗ diﬀerent values of the four-point couplings gρBcBD and gρBcB D = 15, 30, 45 for dotted, solid and dashed curves, respectively. Cutoﬀ parameter is taken 1.5 GeV.

+ → ∗+ ∗ KBc Ds B are given by

( ) 2 Λ2 i µ ν M = −g ∗ g ∗ (p3−2p1)µ (p1−p3−p2)ν εr (p3)ε (p4), (4.53a) 8a KD Ds BcBs Ds Λ2+q2 t−m2 s ( )1 Ds 2 Λ2 i µ ν M = −g ∗ g ∗ (p4−2p1)ν (p1−p4−p2)µεr (p3)ε (p4), (4.53b) 8b KBBs BcBD Λ2+q2 u−m2 s ( ) 2 B 2 2 µ −ig ∗ ∗ Λ g ε (p )εν (p ). M8c = KBcD Bs 2 2 µν r 3 s 4 (4.53c) Λ +q3

These new amplitudes are used again in the formula given in Eqs. 4.5 and 4.8, to determine diﬀerential and total cross sections. The results are reported in the next sections.

84 120 H L 100 1 KBc Ds B

80 L mb

H 60 Σ 40

0 4 6 8 10 12 14 s HGeVL

+ → + Figure 4.17: Bc absorption cross sections of the process KBc Ds B without form factor.

4.8.3 Results of Bc absorption by K with form factor

+ → + Shown in Figs. 4.21 and 4.22 are the cross sections of the processes (1) KBc Ds B + → 0 and (3) KBc DBs respectively with monopole form factor with cutoff parameter + → + varied form 1 to 2 GeV. Fig. 4.21 shows that for the process (1) KBc Ds B the cross section roughly varies from 3 to 12 mb in the most part of the energy scale1, when the cutoff parameter is between 1 to 2 GeV. The reduction in the cross section due to form factor at Λ = 1 and 2 GeV, is roughly by the factors of 30 and 7.5, respectively. + → 0 Whereas, for the process (3) KBc DBs the cross section roughly varies from 3 to 11 mb in the most part of the energy scale, when the cutoff parameter is between 1 to 2 GeV as show in Fig. 4.22. The reduction in the cross sections due to form factor at Λ = 1 and 2 GeV, is roughly by the factors 30 and 8, respectively. The comparison of the cross sections of these processes with and without form factors is given the Figs. √ 1These approximate variations are defined for s ≥ 8 GeV

85 120 H3L KBDB0 100 c s 80 L mb

H 60 Σ 40 20 0 4 6 8 10 12 14 s HGeVL

+ → 0 Figure 4.18: Bc absorption cross sections of the process KBc DBs without form factor.

4.23 and 4.24. + → Fig. 4.25 and 4.26 shows the Bc absorption cross sections of the processes (2) KBc √ ∗+ ∗ + → ∗ ∗0 Ds B and (4) KBc D Bs as a function of total c.m energy s. The cross sections of these processes roughly varies from 0.2 to 2 mb, when the cutoff parameter is between 1 to 2 GeV and four-point couplings are taken 283. Suppression due to form factor at cutoff Λ = 1 and 2 GeV is roughly by factors 150 and 15 respectively. The comparison of the cross sections of these processes with and without form factor is also given in Figs. 4.27 and 4.28. Relatively high suppression in these processes is mainly due to large values of mass of final particles. It is noted that Bc absorption cross sections + → ∗+ ∗ + → ∗ ∗0 of the processes (2) KBc Ds B and (4) KBc D Bs depend upon the four-

∗ ∗ ∗ ∗ point contact couplings gKBcDs B and gKBcD Bs , whose values are fixed through SU(5) symmetry. It is found in the section 4.5 that although SU(5) symmetry uniquely fixes their values, but the difference in the values of the couplings gπDD∗ and gπBB∗ produces two values 149 and 417 of the four-point contact couplings. In this paper, we treat this

86 50 H2L KBc Ds B 40

L 30 mb H

Σ 20

0 4 6 8 10 12 14 s HGeVL

+ → ∗+ ∗ Figure 4.19: Bc absorption cross sections of the process KBc Ds B without form factor. variation as uncertainty in these couplings and study its eﬀect on the cross sections of the processes (2) and (4). In order to study this eﬀect, we plot the cross sections of

∗ ∗ ∗ ∗ these processes for three different values of gKBcDs B = gKBcD Bs = 149, 283, and 417, in Figures 4.29 and 4.30. These figures indicate that the values of the contact couplings significantly affect the cross sections only near the threshold energies. However, it is observed that the effect of uncertainty in the four-point contact couplings is marginal on the total cross section due to relatively small value of the cross sections of the processes (2) and (4). This is shown in the Fig. 4.31 and 4.32, in which total absorption cross

+ + sections for K + Bc and K + Bc are plotted respectively for three diﬀerent values of

∗ ∗ ∗ ∗ gKBcDs B = gKBcD Bs = 149, 283, 417.

87 50 0 H4L KBc D Bs 40

L 30 mb H

Σ 20

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗0 Figure 4.20: Bc absorption cross sections of the process KBc D Bs without form factor.

88 40 H1L KBc Ds B 30 L mb

H 20 Σ

0 4 6 8 10 12 14 s HGeVL

+ → + Figure 4.21: Bc absorption cross sections of the process KBc Ds B with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

89 30 H3L KBDB0 25 c s 20 L mb

H 15 Σ 10 5 0 4 6 8 10 12 14 s HGeVL

+ → 0 Figure 4.22: Bc absorption cross sections of the process KBc DBs with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

90 120 H1L KBDB 100 c s 80 L mb

H 60 Σ 40 20 0 4 6 8 10 12 14 s HGeVL

+ → 0 Figure 4.23: Comparison of the cross sections of the process KBc DBs with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

91 120 0 100 H3L KBc DBs 80 L mb

H 60 Σ 40 20 0 4 6 8 10 12 14 s HGeVL

+ → 0 Figure 4.24: Comparison of the cross sections of the process KBc DBs with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

92 5 H2L KBc Ds B 4

L 3 mb H

Σ 2

0 4 6 8 10 12 14 s HGeVL

+ → ∗+ ∗ Figure 4.25: Bc absorption cross sections of the process KBc Ds B with form factors. Lower and upper curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

93 5 0 H4L KBc D Bs 4

L 3 mb H

Σ 2

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗0 Figure 4.26: Bc absorption cross sections of the process KBc D Bs with form factors. Lower and upper curves are with cuttoﬀ parameter Λ = 1 and 2 GeV respectively.

94 50 H2L KBc Ds B 40

L 30 mb H

Σ 20

0 4 6 8 10 12 14 s HGeVL

+ → ∗+ ∗ Figure 4.27: Comparison of the cross sections of the process KBc Ds B with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

95 50 0 H4L KBc D Bs 40

L 30 mb H

Σ 20

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗0 Figure 4.28: Comparison of the cross sections of the process KBc D Bs with and without form factor. Solid and dashed curves represents cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and 2 GeV respectively.

96 5 H2L KBc Ds B 4

L 3 mb H

Σ 2

0 4 6 8 10 12 14 s HGeVL

+ → ∗+ ∗ Figure 4.29: Bc absorption cross sections of the process KBc Ds B with form fac-

∗ ∗ tor for three diﬀerent values of four-point coupling, for gKBcDs B = 149, 283, and 417 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV.

97 5 0 H4L KBc D Bs 4

L 3 mb H

Σ 2

0 4 6 8 10 12 14 s HGeVL

+ → ∗ ∗0 Figure 4.30: Bc absorption cross sections of the process KBc D Bs with form fac-

∗ ∗ tor for three diﬀerent values of four-point coupling, for gKBcD Bs = 149, 283, and 417 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV.

98 30 Total KB cross section 25 c 20 L mb

H 15 Σ 10 5 0 4 6 8 10 12 14 s HGeVL

+ Figure 4.31: Total Bc absorption cross sections by kaons for three diﬀerent values

∗ ∗ ∗ ∗ of four-point coupling, gKBcDs B = gKBcD Bs = 149, 283, 417 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV.

99 30 Total KB cross section 25 c 20 L mb

H 15 Σ 10 5 0 4 6 8 10 12 14 s HGeVL

+ Figure 4.32: Total Bc absorption cross sections by anti kaons for three diﬀerent values

∗ ∗ ∗ ∗ of four-point coupling, gKBcDs B = gKBcD Bs = 149, 283, 417 for dotted, solid and dashed curve respectively. Cutoﬀ parameter is taken 1.5 GeV.

100 Chapter 5

Bc Absorption Cross Sections by Nucleons.

In this chapter we calculate Bc absorption cross sections by nucleons using the amplitudes given in chapter 3. Numerical values of the required coupling are also calculated. Monopole form factors are included and the eﬀect of cutoﬀ parameters on cross sections is studied. In this chapter we follow the ref. [37].

5.1 Cross sections of Bc by N

+ → 5.1.1 NBc ΛcB

+ → The spin and isospin average diﬀerential cross section of the process NBc ΛcB is given by dσ 1 ∑ 9 = M M ∗, (5.1) dt 128πsp2 9 9 i,cm Fermion spin where M9 is given by Eq. 3.22.

101 + → ∗ 5.1.2 NBc ΛcB

+ → ∗ The spin and isospin average diﬀerential cross section of the process NBc ΛcB is given by ( ) dσ 1 ∑ p p 10 = M µ M ∗ν g − 4µ 4ν , (5.2) dt 128πsp2 10 10 µν m2 i,cm Fermion spin 4 µ where M10 is given by µ µ µ M10 = M10a + M10b (5.3)

And,

µ µ i 5 M = ig g ∗ (p − 2p ) u (p )γ u (p ), (5.4) 10a DNΛc BcB D 4 2 − 2 Λc 3 N 1 t ( mD ) − µ 5 (p1 p4).γ + mΛb µ M = −ig ∗ g u (p )γ i γ u (p ). (5.5) 10b B NΛb BcΛcΛb Λc 3 u − m2 N 1 Λb

− → 5.1.3 NBc DΛb

+ → − → The spin and isospin average diﬀerential cross section of the process NBc NBc

DΛb is given by dσ 1 ∑ 11 = M M ∗ , (5.6) dt 128πsp2 11 11 i,cm Fermion spin where M11 is given by Eq. 3.28.

− → ∗ 5.1.4 NBc D Λb − → ∗ The spin and isospin average diﬀerential cross section of the process NBc D Λb is given by ( ) dσ 1 ∑ p p 12 = M µ M ∗ν g − 3µ 3ν , (5.7) dt 128πsp2 12 12 µν m2 i,cm Fermion spin 3 µ where M10 is given by µ µ µ M12 = M12a + M12b (5.8)

And, ( ) − µ 5 (p1 p3).γ + mΛc µ M = −ig ∗ g u (p )γ i γ u (p ), (5.9) 12a D NΛc BcΛcΛb Λb 4 t − m2 N 1 Λc µ µ i 5 M = ig g ∗ (p − 2p ) u (p )γ u (p ). (5.10) 12b BNΛb BcBD 3 2 − 2 Λb 4 N 1 u mB

102 It is noted that in the study of the processes of the Eq. 3.4, we do not include any diagram in which Σc or Σb particle is exchanged. These diagram require BcΣbΛc and ∗ ∗ BcΣcΛb couplings in addition to DNΣc, BNΣb, D NΣc and B NΣb couplings. These couplings of Bc meson violate isospin (I) and also not produced by the SU(5) invariant

Lagrangian given in Eq. 2.28. Therefore, It is a good approximation to neglect Σc or

Σb exchange diagrams for the processes given in Eq. 3.4.

5.2 Numerical values of input parameters

∗ ∗ The values of the couplings gBcBD and gBcB D are ﬁxed by using gΥBB = 13.3, which is obtained using vector meson dominance (VMD) model in ref. [20] and SU(5) symmetry

2 result g ∗ = g ∗ = √ g [31]. In this way we obtain g ∗ = g ∗ = 11.9. BcBD BcB D 5 ΥBB BcBD BcB D

∗ The couplings gDNΛc and gD NΛc can be ﬁxed by using SU(5)/SU(4) symmetry relations

∗ ∗ gKNΛ = gDNΛc and gK NΛ = gD NΛc given in Eqs. 3.9 & 3.10 and the empirical values of the couplings gKNΛ and gK∗NΛ given in ref. [53]. In this way we obtain the following results,

∗ gDNΛc = 13.1, gD NΛc = 4.3 (5.11)

Whereas, the QCD sum-rule approach gives the following values of these couplings [54].

| | | ∗ | gDNΛc = 7.9, gD NΛc = 7.5 (5.12)

Due to significant difference between the values given in Eqs. 5.11 & 5.12, we use both of them separately to study their effect on the calculated cross sections. However, It is noted that the values given in Eq. 5.11 are less reliable due to the effect of breaking of SU(5)/SU(4) flavor symmetries. There are no empirically fitted values available for

∗ the couplings gBcΛcΛb , gBNΛb and gB NΛb , thus we use SU(5) symmetry relations given in Eqs. 2.35, which implies,

2 gB Λ Λ = −√ gDNΛ , gBNΛ = gDNΛ , gB∗NΛ = gD∗NΛ (5.13) c c b 6 c b c b c

∗ − The values of gDNΛc and gD NΛc in Eq. 5.11 give gBcΛcΛb = 10.7, gBNΛb = 13.1

∗ − and gB NΛb = 4.3, whereas the values in Eq. 5.12 give gBcΛcΛb = 6.5, gBNΛb = 7.9

103 Set 1 Set 2 Coupling Constant Value Method of derivation Value Method of derivation

∗ ∗ gBcBD & gBcB D 11.9 VMD, SU(5) 11.9 VMD, SU(5)

gDNΛc 13.1 gKNΛ, SU(4)/SU(5) 7.9 QCD sum-rule

∗ ∗ gD NΛc 4.3 gK NΛ, SU(4)/SU(5) 7.5 QCD sum-rule

gBNΛb 13.1 gDNΛc , SU(5) 7.9 gDNΛc , SU(5)

∗ ∗ ∗ gB NΛb 4.3 gD NΛc , SU(5) 7.5 gD NΛc , SU(5)

gBcΛcΛb -10.7 gDNΛc , SU(5) -6.5 gDNΛc , SU(5)

Table 5.1: Two sets of the values of coupling constants used in this paper.

∗ ∗ and gB NΛb = 7.5. Where, we choose the sign of the couplings gDNΛc & gD NΛc in accordance with the Eq. 5.11. Two sets of the values of the coupling constants used in this work and methods of obtaining them are summarized in Table 5.1.

5.3 Results and Discussion

Shown in Figs. 5.1, 5.2, 5.3, and 5.4 are the Bc absorption cross sections by nucleons for the four processes given in Eq. 3.4, as function of total center of mass energy. These cross sections are obtained using the values of couplings given in set 1. Solid and dashed curves in these ﬁgures represent cross sections without and with form factors. Form factors are included to account the ﬁnite size of interacting hadrons. We use following monopole form factor at all three point vertices.

Λ2 f = (5.14) 3 Λ2 + q2

Where, Λ is cutoff parameter and q2 is squared three momentum transfer in c.m frame. We take all the cutoff parameters same for simplicity and vary them on the scale 1 to 2 GeV to study the uncertainties in cross sections due to cutoff parameter. The + → + → ∗ − → figures show that, for the processes (i) NBc ΛcB, (ii) NBc ΛcB , (iii) NBc − → ∗ − − DΛb, (iv) NBc D Λb the cross sections roughly vary 2 5 mb, 0.05 0.3 mb, 0.1 − 2 mb, and 0.1 − 1 mb respectively in the most part of the energy scale, when the

104 50 HiL NBc cB 40

L 30 mb H

Σ 20

0 4 6 8 10 12 14

+ → Figure 5.1: Bc absorption cross sections of the process (i)NBc ΛcB using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively. cutoff parameter Λ is between 1 − 2 GeV. Relatively high suppression due to cutoff in the processes 2 and 4 is due to higher values of the masses of vector mesons D∗ and B∗. In Figs. 5.5, 5.6, 5.7, and 5.8 we present the plots of cross sections using the values of couplings given in the set 2. Again the solid and dashed curves in these figures represent cross sections without and with form factors respectively. These cross sections roughly vary 5−15 mb, 0.05−0.2 mb, 0.1−1 mb and 0.1−0.6 mb in the most part of energy scale, for the processes (i) to (iv), when the cutoff Λ is between 1 − 2 GeV. In table 5.2, we present the comparison of peak values of the cross sections for two sets of the coupling values. These results show that the values of set 2 increase the peak values of the cross sections of the process 1 by the factor of ∼ 1.5 and decrease of the process 3 by the factor ∼ 4, whereas the peak values of the processes 2 and 4 almost

105 8 +®L * HiiL NBc cB 6

0 4 6 8 10 12 14

+ → ∗ Figure 5.2: Bc absorption cross sections of the process (ii) NBc ΛcB using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively. remain unchanged. In order to study the effect of b-flavored hadron exchange, we also present in the table 5.2, the comparison of the peak values with and without b-flavor exchange diagrams. The results show that the b-flavor exchange between interacting hadrons significantly increases the peak values of the cross sections of the first three processes for the coupling set 1 and processes (ii) and (iii) for the set 2. Generally, the contribution of a diagram at tree level depend upon the mass of the exchange particle, coupling product and the form of amplitude. Higher value of the mass of the exchange particle tends to decrease the contribution of a diagram. However, in some case the other factors like the higher value of the coupling product or the form of amplitude may significantly increase the contribution even when mass of the exchange particle is increased. In our case, two contributing amplitudes for each process have different

106 30 HiiiL NB Db 25 c 20 L mb

H 15 Σ 10 5 0 4 6 8 10 12 14 s HGeVL

− → Figure 5.3: Bc absorption cross sections of the process (iii) NBc DΛb using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively. form and contain diﬀerent coupling product. Thus, the mere fact that the mass of bottom-hadron is higher than charm-hadron does not imply that contribution of the bottom-exchange diagrams is lesser than charm-exchange diagrams.

5.4 Eﬀect of anomalous parity interaction

Shown in Figs. 5.9, 5.10, 5.11, and 5.12 are the cross sections of the processes ∗ + → ∗ − → (ii) NBc ΛcB ,and (iv) NBc D Λb with anomalous interaction for the both coupling sets. The results show that the cross section of the process (ii) is increased by ∼ 0.2 mb and ∼ 0.5 mb for the coupling sets 1 and 2 respectively, in most part of the energy range. Figs. 5.10 and 5.12 also show that the eﬀect of anomalous interaction

107 20 -® *L HivL NBc D b 15

0 4 6 8 10 12 14 s HGeVL

− → ∗ Figure 5.4: Bc absorption cross sections of the process (iv) NBc D Λb using the values of the couplings given in set 1. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoff parameter Λ = 1 and Λ = 2 GeV respectively. on cross section of the process (iv) is negligible for the both coupling sets. Although the anomalous interaction significantly increases the cross section of the process (ii), but this effect is marginal on the total cross section (i.e., the sum of processes (i) and (ii)) due to relatively small value of the cross section of the process (ii).

108 50 HiL NBc cB 40

L 30 mb H

Σ 20

0 4 6 8 10 12 14

+ → Figure 5.5: Bc absorption cross sections of the process (i)NBc ΛcB using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively.

Set 1 Set 2 Λ = 1 GeV Λ = 2 GeV Λ = 1 GeV Λ = 2 GeV with b-exchange 6 mb 19 mb 10 mb 28 mb NB+ → Λ B c c without b-exchange 3 mb 7 mb 9 mb 23 mb with b-exchange 0.25 mb 0.8 mb 0.3 mb 0.75 mb NB+ → Λ B∗ c c without b-exchange 0.08 mb 0.6 mb 0.03 mb 0.22 mb with b-exchange 6 mb 14 mb 1.5 mb 4.2 mb NB− → DΛ c b without b-exchange 4.5 mb 10.5 mb 0.6 mb 1.5 mb

∗ with b-exchange 0.6 mb 2 mb 0.7 mb 2.1 mb NB− → D Λ c b without b-exchange 0.6 mb 2 mb 0.7 mb 2.3 mb

Table 5.2: The peak values of the cross sections of the four processes with and without b-ﬂavor exchange, using coupling values of set 1 and 2.

109 8 +®L * HiiL NBc cB 6

0 4 6 8 10 12 14

+ → ∗ Figure 5.6: Bc absorption cross sections of the process (ii) NBc ΛcB using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively.

110 30 HiiiL NB Db 25 c 20 L mb

H 15 Σ 10 5 0 4 6 8 10 12 14 s HGeVL

− → Figure 5.7: Bc absorption cross sections of the process (iii) NBc DΛb using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively.

111 20 -® *L HivL NBc D b 15

0 4 6 8 10 12 14 s HGeVL

− → ∗ Figure 5.8: Bc absorption cross sections of the process (iv) NBc D Λb using the values of the couplings given in set 2. Solid and dashed curves represent cross sections without and with form factor respectively. Lower and upper dashed curves are with cutoﬀ parameter Λ = 1 and Λ = 2 GeV respectively.

112 2.0 Coupling Set 1 HiiL NBc cB 1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

Figure 5.9: Bc absorption cross sections of the process (ii) using the values of the couplings given in set 1. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV.

113 2.0 Coupling Set 1 HivL NBc D b 1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

Figure 5.10: Bc absorption cross sections of the process (iv) using the values of the couplings given in set 1. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV.

114 2.0 Coupling Set 2 HiiL NBc cB 1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

Figure 5.11: Bc absorption cross sections of the process (ii) using the values of the couplings given in set 2. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV.

115 2.0 Coupling Set 2 HivL NBc D b 1.5 L mb

H 1.0 Σ 0.5

0.0 4 6 8 10 12 14 s HGeVL

Figure 5.12: Bc absorption cross sections of the process (iv) using the values of the couplings given in set 2. Solid and dashed curves represent the cross sections with and without anomalous diagrams respectively, and dotted curves represent the contribution from anomalous diagrams alone, i.e., without including the contribution from the interference terms. Cutoﬀ parameter Λ is taken 1.5 GeV.

116 Chapter 6

Discussion

In this work, we present the results of cross sections of Bc absorption processes by light mesons and baryons; π, ρ , and K mesons and nucleons. The calculations are based on use of an effective hadronic Lagrangian developed by imposing SU(5) symmetry. The approach has been frequently used to calculate hadronic cross sections of the other mesons and baryons using similar effective Lagrangian. The hadronic Lagrangian based on SU(4) or SU(5) symmetries is developed by imposing the gauge symmetry, which is explicitly broken by the mass terms. Thus the symmetry exists only in limit of zero hadronic masses. Due to symmetry breaking, the corresponding SU(5) symmetry relations cannot be used to determine the values of the coupling constants of three or four-point interaction vertices. It can be seen that the empirical or theoretical values of the couplings also violate SU(5) symmetry relations. In this respect the role of SU(5) symmetry is not more than consistently producing possible interactions between the hadrons in which isospin, strange, charm and bottom quantum numbers remain conserved. Real use of symmetry, therefore, is implied when we use relations between different couplings constants appearing in the effective Lagrangian. Consequently, in this and other similar works the values of the required couplings are preferably fixed empirically or using the microscopic theories. Unfortunately, in this way we cannot fix all possible required couplings. Thus, one is bound to use the symmetry relations to find the values of such couplings by using the values of known couplings. Generally this method may not produce unique value of the couplings because of using broken

117 symmetry relations. In this work we treat this variation as uncertainty and study its effect on the total cross sections of the processes. That is why the approach is sound despite these limitations. Another source of uncertainty in the cross sections comes from the form factors. In the first place the nature of the form factor for different interaction vertices used in the work is unknown. There are different forms which are applied in the literature. Some of these forms are summarized in the section 3.7. In this

2 Λ2 work we use the monopole form factor f(q ) = q2+Λ2 , in which the cutoff parameter is unknown for the required couplings. We constrain it by the size of interacting mesons and find that 1 to 2 GeV is an appropriate range in which it could be varied to study the effect of its uncertainty on the cross sections. Same variation in the cutoff is also assumed in other studies of hadronic cross sections in meson-exchange model. Using these ingredients, we calculate the cross sections of 12 different Bc absorption processes by π, ρ , and K mesons and nucleons. Calculated cross sections + → are found to be in range 2 to 7 mb and 0.2 to 2 mb for the processes Bc π DB and + → ∗ ∗ Bc π D B respectively, when the cutoff parameter Λ in the monopole form factor is

∗ ∗ ∗ varied from 1 to 2 GeV. The values of the required couplings gπDD , gπBB , gBcBD , and

∗ ∗ ∗ gBcB D are ﬁxed phenomenologically. However, the 4-point coupling gπBcD B , which is also required for the calculations, cannot be ﬁx without using the SU(5) symmetry relations. The symmetry relations show that the value of this coupling could vary over a wide range. In this work we have treated this variation as an uncertainty and

find that the total cross section for Bc + π is almost independent of the value of this coupling especially when we are little away from the threshold energy. This mean that the results of Bc absorption cross sections by pion will not affected even if we find a better estimate of the 4-point coupling. + → ∗ + → ∗ For the processes Bc ρ D B and Bc ρ DB the cross sections are in range 0.6 − 3 mb and 0.05 − 0.3 mb respectively. In this case we require the couplings

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ gρDD, gρBB, gρD D , gρB B , gρBcBD , and gρBcB D in addition to gBcBD , and gBcB D which are already fixed for absorption cross sections by pions. The required three-point couplings are fixed by VMD model, whereas the 4-point couplings are again fixed by

∗ SU(5) symmetry relations. In this case the symmetry produce unique value gρBcBD =

118 ∗ gρBcB D = 30. However, we again vary these four couplings in the range 15 to 45 and ﬁnd that corresponding variation in the cross section is negligible. The scale of this variation in the cross sections is about 3 times smaller than the calculated variation of the total absorption cross section by pions.

For Bc absorption processes by K meson, we ﬁnd that the cross sections vary in the − − − − + → ranges 3 12 mb, 0.2 2 mb, 3 11 mb, and 0.2 2 mb for the processes KBc + + → ∗+ ∗ + → 0 + → ∗ ∗0 Ds B,KBc Ds B , KBc DBs , and KBc D Bs respectively, when the monopole form factor is included. The scale of these variations are comparable to the

Bc absorption cross section by pions calculated using the same model. The couplings

∗ ∗ ∗ ∗ ∗ required to calculate these cross section are gKD Ds , gKDDs , gKB Bs , gKBBs , gBcBsDs ,

∗ ∗ ∗ ∗ ∗ ∗ ∗ gBcBs Ds and gKBcDs B and gKBcD Bs including gBcBD , and gBcB D. In this case 4- point couplings are again not uniquely fixed by SU(5) symmetry. Based on the values obtained from SU(5) symmetry relations we consider a large variation in the these coupling and study its effect on the cross sections. We find that that the cross sections + → ∗+ ∗ + → ∗ ∗0 of the processes KBc Ds B and KBc D Bs significantly depend upon the values of 4-couplings. However, the effect is marginal on total cross sections due to + → + small values of the cross sections of these processes as compared to KBc Ds B and + → ∗+ KBc Ds B.

We have also used this approach to calculate Bc absorption cross sections by nucleons.

∗ ∗ In this case the requires couplings are gDNΛc , gD NΛc , gBNΛb , gB NΛb , and gBcΛcΛb in

∗ ∗ ∗ addition to gBcBD , and gBcB D. The coupling constants gDNΛc and gD NΛc can either be

ﬁxed by SU(5) (or SU(4)) ﬂavor symmetry and empirical values of the couplings gKNΛ and gK∗NΛ or using QCD-sum rules. We have calculated absorption cross sections using

∗ both set of values for comparison. Whereas, for the coupling constants gBNΛb , gB NΛb and gBcΛcΛb no empirical values are available, so we use SU(5) symmetry relations. These estimates are less reliable as the SU(5) flavor symmetry is broken. It is noted that + → + → ∗ for the processes (i) NBc ΛcB and (ii) NBc ΛcB only the b-flavor exchange diagrams depend upon these three couplings. Thus the effect of these couplings on the cross sections of the first two processes is less significant when the contribution of b-flavor exchange diagram is small or negligible as in the case of process (i) for

119 − → coupling values of the set 2. However, for the processes (iii) NBc DΛb and (iv) − → ∗ NBc D Λb, the amplitudes of both c and b-flavor exchange diagrams depends upon these couplings. Thus, any change in the values of these couplings could significantly change the cross sections of these processes irrespective of the relative contribution of the two diagrams. We conclude that a more rigorous study on these couplings could further improve our results. The anomalous parity interaction is found to be + → ∗ significant only for the process (ii) NBc ΛcB . The effect, however, itself marginal on the total value of the cross section (i.e., sum of the processes (i) and (ii)) due to lesser contribution from that process. The results reported in this study could be useful in calculating the production rate of Bc meson in relativistic heavy ion collisions.

120 Appendix A

Feynman Rules for Vertex

Consider a general form of interaction Lagrangian containing scalr, vector and Dirac field † LI (x) = LI (ϕ, χ, χ , ψ, ψ) (A.1) where ϕ(χ) represents a set of real (complex) boson fields that may be scalar or vector and ψ represents a set of fermion fields. In the first step LI (x) is written in the following form in which different field are shown by different indices and space-time coordinates ∫ L 4 4 I (x) = d x1d x2...αi1...im...in...ip...iq...(x; x1, ..., xm, ...xn, ...xp, ...xq, ...) × ψi1 (x1)...ψim (xm)...ϕin (xn)..., χip (xp)...χiq (xq)... (A.2)

In order to ﬁnd the vertex factor of the couplng for using in the scattering amplitude in momentum space, we ﬁnd the fourier transformation of

αi1...im...in...ip...iq...(x; x1, ..., xm, ...xn, ...xp, ...xq, ...) as following ∫ d4k d4k 1 2 ik1(x1−x) ik1(x2−x) αi i ...(x; x1, x2, ...) = ...e e ... × αei i ...(k1, k2...). (A.3) 1 2 (2π)4 (2π)4 1 2

For every 4-derivative in α there appear a factor ikµ. The vertex factor for Amplitude is given by ∑ − p e I(k1, k2, ...) = i ( 1) αi1i2...(k1, k2...) (A.4) where summation is taken over all possible permutations of the indices of the ﬁeld of identical particles. The factor (−1)p is result of combinations of (−1) contributed by

121 every permutation in which two fermions are exchanged. For complex fields, momenta are taken to flow inward (outward) corresponding to χ(χ†) and similarly for fermion fields ψ(ψ¯). Whereas, real fields are represented by undirected line. For example consider the following VVV coupling ( ) ∗ ∗ ∗ν ∗ν µ L ∗ ∗ ∗ ∗ − · ρD D = igρD D [ ∂µD Dν D ∂µDν ρ ∗ ∗ν ∗ν µ + (D ∂µρν − ∂µD ρν) D ( ) ∗ ∗ ∗µ ν − ν +D ρ ∂µDν ∂µρ Dν ], (A.5)

µ ν λ In the ﬁrst step we represent the vector ﬁelds by ρ (x1),D (x2), and D (x3) and write the Lagrangian as following ∫ ( ) ( ) 4 4 4 (x ) (x ) (x ) (x ) L ∗ ∗ ∗ ∗ 2 − 3 3 − 1 ρD D = igρD D d x1d x2d x3[ ∂µ ∂µ gνλ + ∂ν ∂ν gµλ ( ) (x1) − (x2) − − − + ∂λ ∂λ gµν]δ(x x1)δ(x x2)δ(x x3) (A.6)

µ ν λ .ρ (x1)D (x2)D (x3), (A.7)

Thus, ( ) ( ) (x2) (x3) (x3) (x1) α(x; x1, x2, x3) = igρD∗D∗ [ ∂ − ∂ gνλ + ∂ − ∂ gµλ ( µ ) µ ν ν (x1) − (x2) − − − + ∂λ ∂λ gµν]δ(x x1)δ(x x2)δ(x x3) (A.8)

Its Fourier transformation is given by

e − ∗ ∗ − − − α(k1, k2, k3) = gρD D [(k2 k3)µ gνλ + (k3 k1) νgµλ + (k1 k3)λ gµν] (A.9)

Hence, the vertex factor of the given coupling is

− ∗ ∗ − − − I = igρD D [(k2 k3)µ gνλ + (k3 k1)ν gµλ + (k1 k3)λ gµν].

When this factor is used for a VVV vertex in Feynman diagram, we also have to invert the sign of momentu if it is directed outward.

122 Appendix B

Kinematics

General formula for the diﬀerential cross section of a two-body process is given by

dσ 1 2 = 2 M , (B.1) dt 64πspcm where the bar over M represents averaging (summing) over initial (ﬁnal) spins, and pcm is 3 momentum of initial particles in c.m frame, which is given by [s − (m + m )2][s − (m − m )2] p2 = 1 2 1 2 (B.2) cm 4s Total cross section is obtained by integrating over the variable t with lower and upper limits given by [ ] m2 − m2 − m2 − m2 2 t = 1 3 √ 2 4 − (p − p )2 (B.3a) 0 2 s 1cm 3cm [ ] m2 − m2 − m2 − m2 2 t = 1 3 √ 2 4 − (p + p )2 (B.3b) 1 2 s 1cm 3cm Energies of the initial and ﬁnal particles in c.m in term of Mandelstam variable s are given by s + m2 − m2 E = √1 2 , (B.4) 1cm 2 s s − m2 + m2 E = √1 2 , (B.5) 2cm 2 s s + m2 − m2 E = √3 4 , (B.6) 3cm 2 s s − m2 + m2 E = √3 4 , (B.7) 4cm 2 s

123 which can be used to calculated 3-momenta of ith particle by

2 2 − 2 picm = Eicm mi (B.8)

Various invariant products of 4-momenta can also be written in terms of Mandelstam variables as following

1 p · p = (s − m2 − m2), (B.9) 1 2 2 1 2 1 p · p = (s − m2 − m2), (B.10) 3 4 2 3 4 1 p · p = (m2 + m2 − t), (B.11) 1 3 2 1 3 1 p · p = (m2 + m2 − t), (B.12) 2 4 2 2 4 1 p · p = (m2 + m2 − u), (B.13) 1 4 2 1 4 1 p · p = (m2 + m2 − u), (B.14) 2 3 2 2 3

2 2 2 2 − − where u = m1 + m2 + m3 + m4 s t. Using these relations the diﬀerential cross section given by Eq. B.1 can be written in terms of the independent Mandelstam variables s and t. Whereas, total cross section is function of s only, which is obtained by integrating with respect to t over the limits given by Eqs. B.3.

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