Interactions of B_{c} Meson in Relativistic Heavy-Ion Collisions

# Interactions of $B_{c}$ Meson in Relativistic Heavy-Ion Collisions

## Abstract

We calculate the dissociation cross-sections of mesons by and mesons including anomalous processes using an effective hadronic Lagrangian. The enhancement of production is expected due to QGP formation in heavy-ion experiments. However it is also expected that the production rate of meson can be affected due to the interaction with comovers. These processes are relevant for the experiments at RHIC. Thermal average cross-sections of are evaluated with a form factor when a cut off parameter in it is 1 and 2 GeV. Using these thermal average cross-sections in the kinetic equation we study the time evolution of mesons due to dissociation in the hadronic matter formed at RHIC.

Keywords: Relativistic heavy ion collisions, Meson-Meson interaction, QGP.

PACS number(s): 25.75.-q, 13.75.Lb, 14.40.Nd

## 1 Introduction

In 1986 Matsui and Satz [1] hypothesized that in a deconfined medium color screening would have dissociated the , resulting in a suppressed yield of . This deconfined state is called Quark-Gluon Plasma (QGP). Thus for the existence of QGP, suppression of could be considered as a probe. Anomalously large suppression of events was observed by NA50 experiment at CERN [2] with moderate to large transfer energy from the Pb+Pb collision at GeV/c. However, this observed suppression may also occur due to absorption by comoving hadrons mainly and , especially if the dissociation cross section is at least few mb [3, 4, 5, 6, 7, 8]. To calculate these cross sections, quark potential models, perturbative QCD [9], QCD sum-rule approach [10, 11] and flavor symmetric effective Lagrangian [12, 13, 14, 15] has been used. Analogous to charmonium, suppression of bottomonium states is also predicted during the formation of QGP [1]. Recently it was observed by CMS in Pb+Pb collisions that excited states of bottomonium are strongly suppressed [16]. To have unambiguous interpretation of the the observed signal, the information of dissociation cross section is also needed [13, 17]. It was suggested that the production rate of heavy mixed flavor hadrons would be affected in the presence of QGP [18, 19]. For calculating the rate of production of these hadrons comprehensive information is required to distinguish QGP affected hadron production and suppression due to dissociation by comovers. It is expected that production could be enhanced in the presence of QGP [19, 20]. QGP contains many unpaired and quarks due to color Debye screening. These unpaired and quarks upon encounter could form or mesons and due to relatively large binding energy, mesons probably survive in QGP [20]. However, observed production rate would also depend upon the dissociation cross section by hadronic comovers.

In Ref. [20] absorption by nucleons was examined with the meson-baryon exchange model. The calculated cross sections were in the range of a few millibarn. Recently in Ref. [21], using the same couplings and hadronic Lagrangian within meson exchange model the dissociation of meson by meson were examined. The range of the resulting cross sections involving the form factors were mb and mb for the processes and , respectively. In Ref. [22], the dissociation of meson by mesons were examined. For the processes and the resultant cross sections with the form factor were in the range of and mb, respectively .
In this paper we investigate the dissociation by  and mesons including anomalous couplings like PVV, PPPV and PVVV which were ignored in the previous studies. Inclusion of these couplings results in opening of new dissociation channels and addition of new processes and extra diagrams. The contribution of anomalous couplings is found to be significant for calculating cross sections of charmonium dissociation with and meson in Ref. [23], mesons in Ref. [24] and dissociation of meson by nucleons in Ref. [25]. We also calculate the thermal average cross sections and study the time evolution of meson at RHIC using a schematic expanding fireball model with an initial abundance determined by the statistical model. The paper is organized as follows. In Sec. 2, the interaction Lagrangian terms which are relevant for the description of the dissociation of by and mesons including anomalous processes are given and also analytical expressions of the amplitudes for the dissociation of meson are reported. In Sec. 3, we calculate the cross sections with and without form factor and thermal average cross sections. In Sec. 4, we study time evolution of the meson abundance at RHIC in a schematic model. In Sec. 5, we present the summary and discussion.

## 2 Interaction Lagrangian and Amplitudes of Bc meson dissociation

### 2.1 Interaction Lagrangian

We consider the following reactions using an effective Hadronic Lagrangian.

 B+c+π→D+B, B−c+π→¯D+¯B B+c+ρ→D+B, B−c+ρ→¯D+¯B B+c+π→D∗+B, B−c+π→¯D∗+¯B, B+c+ρ→D∗+B, B−c+ρ→¯D∗+¯B, B+c+π→D+B∗, B−c+π→¯D+¯B∗, B+c+ρ→D+B∗, B−c+ρ→¯D+¯B∗, B+c+π→D∗+B∗, B−c+π→¯D∗+¯B∗, B+c+ρ→D∗+B∗, B−c+ρ→¯D∗+¯B∗.
(1)

The processes in the first and second column, and also in the third and fourth column have same cross sections as being charge conjugation of each other. The generic form for the 1st reaction is given as

 B+c+π+→D++B+, B+c+π−→D0+B0, B+c+π0→D++B0 B+c+π0→D0+B+
(2)

For calculating the cross sections of the above reactions, relevant interaction Lagrangian terms are required. The required interaction Lagrangian for normal processes (for which the relevant couplings are dimensionless) are obtained using the method described in Refs. [21, 22] and are given as follows.

 LπDD∗ = igπDD∗D∗μ→τ⋅(¯D∂μ→π−∂μ¯D→π)+hc (3a) LπBB∗ = igπBB∗¯B∗μ→τ⋅(B∂μ→π−∂μB→π)+hc (3b) LBcBD∗ = igBcBD∗¯D∗μ(B−c∂μ¯B−∂μB−c¯B)+hc (3c) LBcB∗D = igBcB∗DB∗μ(B−c∂μD−∂μB−cD)+hc (3d) LπBcD∗B∗ = −gπBcD∗B∗B+c¯B∗μ→τ⋅→π¯D∗μ+hc (3e) LρDD = igρDD(D→τ∂μ¯D−∂μD→τ¯D)⋅→ρμ, (3f) LρBB = igρBB  →ρμ⋅(¯B→τ∂μB−∂μ¯B→τB) (3g) LρD∗D∗ = igρD∗D∗ [→ρμ⋅(∂μD∗ν→τ¯D∗ν−D∗ν→τ∂μ¯D∗ν) +¯D∗μ⋅(D∗ν→τ⋅∂μ→τν−∂μD∗ν→τ⋅→ρν) +D∗μ⋅(→τ⋅→τν∂μ¯D∗ν−→τ⋅∂μ→ρν¯D∗ν)] LρB∗B∗ = igρB∗B∗[→ρμ⋅(∂μ¯B∗ν→τB∗ν−¯B∗ν→τ∂μB∗ν) +B∗μ⋅(¯B∗ν→τ⋅∂μ→ρν−∂μ¯B∗ν→τ⋅→ρν) +¯B∗μ⋅(→τ⋅→ρν∂μB∗ν−→τ⋅∂μ→ρνB∗ν)] LρBcD∗B = gρBcD∗BB+c¯B→τ⋅→ρμ¯D∗μ+hc (3j) LρBcDB∗ = gρBcDB∗B+c¯B∗μ→τ⋅→ρμ¯D+hc (3k)

In addition to the above normal terms there are anomalous terms as well which are required to give a complete description of the hadronic processes. The required interaction Lagrangian for the anomalous processes (for which the relevant couplings are not dimensionless) are obtained using the method described in Ref. [23] and are given as follows.

 LπD∗D∗ = −gπD∗D∗εμναβ[(∂μD∗ν)→τ⋅→π(∂α¯D∗β)] (4a) LπB∗B∗ = gπB∗B∗εμναβ[(∂α¯B∗β)→τ⋅→π(∂μB∗ν)] (4b) LBcD∗B∗ = gBcD∗B∗εμναβ[(∂μD∗ν)(∂αB∗β)Bc−+Bc+(∂α¯B∗β)(∂μ¯D∗ν)] (4c) LρD∗D = −gρD∗Dεμναβ(D∂μρν∂α¯D∗β+∂μD∗ν∂αρβ¯D) (4d) LρB∗B = −gρB∗B εμναβ(B∂μρν∂α¯B∗β+∂μB∗ν∂αρβ¯B) (4e) LπBcD∗B = −igπBcD∗Bεμναβ[D∗μ(∂νB−c)(→τ⋅∂α→π)(∂βB)+¯D∗μ(→τ⋅∂ν→π)(∂αB+c)(∂β¯B)] (4f) LπBcDB∗ = −igπBcDB∗εμναβ[B∗μ(∂νB−c)(→τ⋅∂α→π)(∂βD)+¯B∗μ(∂νB+c)(→τ⋅∂α→π)(∂β¯D)] (4g) LρBcBD = −igρBcBDεμναβ[ρμ(∂νD)(∂αB)(∂βB−c)+ρμ(∂ν¯¯¯¯B)(∂α¯D)(∂βB+c)] (4h) LρBcB∗D∗ = igρBcD∗B∗εμναβ[B∗μρνD∗α(∂βB−c)+¯D∗μρν¯B∗α(∂βB+c)] −ihρBcD∗B∗[B−c(∂μD∗ν)→τ⋅→ραB∗β+B+c(∂μ¯B∗ν)ρα¯D∗β]

In Eqs. (3) and (4) represents Pauli spin matrices, and and represent isospin triplets,

 →π=(π1,π2,π3),   →ρ=(ρ1,ρ2,ρ3),

while vector and pseudoscalar charm and bottom meson doublets are given as

 ¯D∗μ=(¯D∗0μ,D∗−μ)T , ¯D=(¯D0,D−)T, D=(D0,D+), B∗μ=(B∗+μ,B∗0μ)T, ¯B=(B−,¯B0), B=(B+,B0)T.

### 2.2 Amplitudes for Bc meson dissociation

For calculating the cross section for meson dissociation by and mesons, we use the effective Lagrangian given in Eqs. (3) and (4). In this paper we are only reporting the scattering amplitudes of anomalous processes and of additional diagrams which are dependent on the anomalous couplings. Absorption amplitudes of other diagrams which depend only on normal couplings are given in Refs. [21, 22]. Diagrams of the process are shown in Fig. 1 (2a to 2c) and the amplitudes of the diagrams are

 M2a = gπD∗D∗gBcBD∗  εμνασpμ3(p3−p1)β−it−m2D∗(gασ−(p1−p3)α(p1−p3)σm2D∗) (−p2−p4)νεβD∗(p3), M2b = gπBB∗gBcB∗D∗ εμνασpμ3(p1+p4)ν−iu−m2B∗(gασ−(p1−p4)α(p1−p4)σm2B∗) (p3−p2)βεβD∗(p3), M2c = −igπBcBD∗ εμναβpα1pμ4pν2εβD∗(p3). (5c) And the full amplitude is written as M2=M2a+M2b+M2c. (5d)

Diagrams of the process are shown in Fig. 1 (3a to 3c) and the amplitudes of the diagrams are

 M3a = gπD∗DgBcB∗D∗  εμνασpμ4(p4−p2)β−it−m2D∗(gασ−(p1−p3)α(p1−p3)σm2D∗) (p1+p3)νεβB∗(p4), M3b = gπB∗B∗gBcB∗D εμνασpμ4(p4−p1)β−iu−m2B∗(gασ−(p1−p4)α(p1−p4)σm2D∗) (−p2−p3)νεβB∗(p4), M3c = −igπBcDB∗  εμναβpα2pμ3pν1εβB∗(p4). (6c) And the full amplitude is written as M3=M3a+M3b+M3c. (6d)

Diagrams of the process are shown in Fig. 1 (4a to 4e). The amplitudes of diagram 4d and 4e which depend on anomalous couplings are

 M4d = gπD∗D∗gBcB∗D∗εσλαβεσλγζpγ4(p3−p1)μ−it−m2D∗ (gαβ−(p1−p3)α(p1−p3)βm2D∗)pζ3(p4−p2)νεμD∗(p3)ενB∗(p4), M4e = gπB∗B∗gBcD∗B∗εσλαβεσλγζ(p4−p1)νpγ3−iu−m2B∗ (gαβ−(p1−p4)α(p1−p4)βm2B∗)(p3−p2)μpζ4εμD∗(p3)ενB∗(p4). And the full amplitude is written as M4=M4a+M4b+M4c+M4d+M4e. (7c)

Now we report the absorption amplitudes of the anomalous processes of by . Diagrams of the process are shown in Fig. 2 (5a to 5c). The amplitudes of these diagrams are

 M5a = gBcBD∗gρD∗Dεσναβpν1(−p2−p4)σ−it−m2D∗ (gαβ−(p1−p3)α(p1−p3)βm2D∗)(p3−p1)μεμρ(p1), M5b = gρB∗BgBcB∗Dεσναβpσ1−iu−m2B∗(−p3−p2)ν (gαβ−(p1−p4)α(p1−p4)βm2B∗)(p4−p1)μεμρ(p1), M5c = −igρBcBDεμναβpν2pα3pβ4εμρ(p1). (8c) And the full amplitude is written as M5=M5a+M5b+M5c. (8d)

Diagrams of the process are shown in Fig. 2 (6a to 6d). The amplitudes of the anomalous diagram 6d is given as

 M6d = gρB∗BgBcB∗D∗εδγσλεδγαβpσ1pλ4(p3−p1)μ−it−m2B∗ (gαβ−(p1−p3)α(p1−p3)βm2B∗)(p4−p2)νεμρ(p1)ενD∗(p4). And the full amplitude is written as M6=M6a+M6b+M6c+M6d. (9b)

Diagrams of the process are shown in Fig. 2 (7a to 7d). The amplitude of the anomalous diagram 7d is given as

 M7d = gρD∗DgBcB∗D∗εσλγδεγδαβpσ1pλ4(p3−p1)μ−it−m2D∗ (gαβ−(p1−p3)α(p1−p3)βm2D∗)(p2−p4)νεμρ(p1)ενD∗(p4). And the full amplitude is written as M7=M7a+M7b+M7c+M7d. (10b)

Diagrams of the process are shown in Fig. 2 (8a to 8e), the amplitudes of these diagrams are

 M8a = gρD∗DgBcB∗D\largeεμναβpα1(p3−p1)βit−m2D(p4−2p2)λεμρ(p1) ενD∗(p3)ελB∗(p4), M8b = (11b) M8c = gρD∗D∗gBcB∗D∗εσδαβpδ4−it−m2D∗[(2p3−p1)μgσν+(2p1−p3)νgμσ +(−p3−p1)σgμν](gαβ−(p1−p3)α(p1−p3)βm2D∗)(p4−p2)λ εμρ(p1)ενD∗(p3)ελB∗(p4), M8d = gρB∗B∗gBcB∗D∗εσδαβpδ3−iu−m2B∗[(−2p4+p1)λgσμ+(p1+p4)σgμλ +(p4−2p1)μgσλ](gαβ−(p1−p4)α(p1−p4)βm2B∗)(p2−p3)ν εμρ(p1)ενD∗(p3)ελB∗(p4), M8e = (−igρBcB∗D∗εμνλβpβ2+ihρBcB∗D∗εμνλβpβ4)εμρ(p1)ενD∗(p3)ελB∗(p4). (11e) And the full amplitude is written as M8=M8a+M8b+M8c+M8d+M8e. (11f)

We define the four-momenta of the incoming particles as and and those of the final particles as and , which then defines and . Here , , and represent the , , and mesons masses, respectively. The polarization vector of a vector meson with momentum is represented by . After averaging (summing) over initial (final) spins and including isospin factor, we calculate the cross sections by using the total amplitudes specified in above equations. The isospin factor for calculating these cross section is 2 for all the processes.

## 3 Dissociation Cross-Sections of Bc Meson

### 3.1 Numerical values of input parameters

Numerical values of all the meson masses are taken from Particle Data Group [26]. Estimation of the coupling constants of effective Lagrangian is required for calculating the cross sections. To fix the couplings for the normal processes, we follow the methods of Refs. [12, 27]; we refer to Ref. [12] for details. In a similar way we have determined the couplings for the anomalous interactions, which are reported in this paper whereas normal couplings are given in Refs. [21, 22]. The coupling which has a dimension of is fixed by applying the heavy quark spin symmetry. We follow Ref. [23] in which this coupling is given as

 gD∗D∗π=gD∗Dπ¯¯¯¯¯¯MD≈9.08\textmdGeV−1 (12)

where represents the average mass of and .
For couplings, we can apply the VMD (Vector Meson Dominance) model [12] to the radiative decays of into , i.e., . We use the same method as in ref. [23]; this leads to

 gρD∗D=2.82\textmdGe