Decay widths of bottomonium states in matter – a field theoretic model for composite hadrons

# Decay widths of bottomonium states in matter – a field theoretic model for composite hadrons

Amruta Mishra Department of Physics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi – 110 016, India    S.P. Misra Institute of Physics, Bhubaneswar – 751005, India
###### Abstract

We compute the in-medium partial decay widths of the bottomonium states to open bottom mesons () using a field theoretical model for composite hadrons with quark constituents. These decay widths are calculated by using the explicit constructions for the bottomonium states and the open bottom mesons ( and ), and, the quark antiquark pair creation term of the free Dirac Hamiltonian written in terms of the constituent quark field operators. These decay widths in the hadronic medium are calculated as arising from the mass modifications of the bottomonium states and the and mesons, obtained in a chiral effective model. The decay amplitude in the present model is multiplied with a strength parameter for the light quark pair creation, which is fitted from the observed vacuum partial decay width of the bottomonium state, to . The effects of the isospin asymmetry, the strangeness fraction of the hadronic matter on the decay widths, arising due to the mass modifications due to these effects, have also been studied. There is observed to be appreciable effects from density, and the effects from isospin asymmetry on the parital decay widths of are observed to be quite pronounced at high densities. These effects should show up in the asymmetric heavy ion collisions in Compressed baryonic matter (CBM) experiments planned at the future facility at FAIR. The study of the states will, however, require access to energies higher than the energy regime planned at the CBM experiment. The density effects on the decay widths of the bottomonium states should show up in the production of these states, as well as, in dilepton spectra at the Super Proton Synchrotron (SPS) energies.

## I Introduction

The study of medium modifications of properties of hadrons is a topic of research which has attracted a lot of attention in recent years in strong interaction physics, in particular because of its relevance to the heavy ion collision experiments. Matter at high temperatures and/or densities is produced in ultra-relativistic heavy ion collision experiments and the properties of hadrons in such a medium are modified, consequences of which can show up in the experimental observables of these high energy nuclear collisions.

The open charm (bottom) mesons, () and (), are made up of a heavy charm (bottom) quark (antiquark) and a light (u or d) antiquark (quark) and their mass modifications in the hadronic medium are due to their interaction with the light quark condensate in the QCD sum rule framework haya1 (); qcdsum08 (). The in-medium properties of the open charm mesons have been studied quite extensively by hadronic frameworks, e.g., the quark meson coupling (QMC) model qmc0 (); qmchnm (); qmc1 (); qmc2 () as well as the coupled channel approach ltolos (); ljhs (); mizutani6 (); mizutani8 (); HL (). Within a hadronic framework using pion exchange sudoh (), a study of the open charm and open bottom mesons is observed to lead to an attractive interaction of the and mesons in the nuclear matter, suggesting that these mesons can form bound states with the atomic nuclei. The -nucleon interactions have recently been studied using a description of the hadrons with quark and antiquark constituents dnkrein (), where the field operators of the constituent quarks are written in terms of a constituent quark mass, which arises from dynamical chiral symmetry breaking dnkrein (); hmamspm1 (); hmamspm2 (); spmeffpot (); spmbothcond (); amhm2004 ().

In the effective hadronic model, constructed by generalizing chiral SU(3) model to the charm and bottom sectors, the mass modifications of these open charm mesons amarindam (); amarvind (); amcharmdw () and the open bottom mesons dpambmeson (), arise due to their interactions with the light hadrons, namely the baryons (nucleons and hyperons) and the scalar mesons. On the other hand, the hidden charm and bottom mesons, i.e. the charmonium amcharmdw (); leeko (); kimlee (); charmmass2 () and bottomonium states amdpbottomonium (), have the masses modified in the hadronic medium due to the interactions with the gluon condensates in the medium. The gluon condensates of QCD is mimicked by a scalar dilaton field charmmass2 (); amcharmdw (), within the effective hadronic model, and the medium modifications of the heavy quarkonium states, i.e., the charmonium amcharmdw () and bottomonium states amdpbottomonium (), are studied by medium modification of the dilaton field within the model. Using a field theoretical model for composite hadrons with quark and antiquark constitutents spm781 (); spm782 (); spmdiffscat (), the partial decay widths of the charmonium states to pair, as well as of the decay , in matter have been studied chmdwamspmwg (), using the medium modifications of these hadrons using the effective hadronic model amcharmdw (). These decay widths were compared with the partial decay widths using the model friman (); amcharmdw (), where a light quark aniquark pair is assumed to be created in the state yopr1 (); yopr2 (); yopr3 (); barnes (), and the light quark (antiquark) combines with the heavy charm antiquark (quark) of the decaying charmonium state, to produce the pair. In the present work, we study the medium modification of the decay widths of the bottomonium states to pair in the strange hadronic medium, due to the mass modifications of these hadrons calculated in the effective chiral model amdpbottomonium (); dpambmeson ().

The outline of the paper is as follows: In section II, we describe briefly the field theoretical model for the hadrons with quark constitutents, which is used in the present work to calculate the partial decay widths of the bottomonium states to open bottom mesons ( pair). The decay widths are calculated by using explicit constructions of the bottomonium states (, , and ) and the and mesons in terms of the quark and antiquark constituents. We then calculate the matrix element of the S-matrix in the lowest order to compute the decay widths of the bottomonium states to ( or ) pair. The matrix element, however, is multiplied with a parameter, which is fitted from the observed vacuum decay width of . In the present work, the partial decay widths for the decay of the bottomonium states, , =1,2,3,4, to , are calculated using the field theoretic model for composite hadrons and their medium modifications have been studied as arising from the changes in the masses of these mesons in the hadronic medium. In section III, we briefly describe the effective hadronic model, which has been used to investigate the masses of the open bottom mesons ( and ) and of the states. The in-medium masses of the and mesons in the strange hadronic medium arise due to their interactions with the baryons and the scalar mesons dpambmeson (). On the other hand, the mass modifications of the bottomonium states amdpbottomonium () airse due to the medium modification of the scalar dilaton field, which is incorporated in the effective hadronic framework, to simulate the scale symmetry breaking of QCD through scalar gluon condensate. In section IV, we discuss the results obtained in the present investigation. Using the explicit constructions for the bottomonium states (, N=1,2,3,4) and using the quark pair creation term of the free Dirac Hamiltonian written in terms of the constituent quark field operators, the decay widths of the bottomonium states to pair, are calculated within the present model. In section V, we summarize the results for the medium modifications of these decay widths, and discuss possible outlook.

## Ii The model for composite hadrons

The model used in the present work for calculating the partial decay widths of the bottomonium states to , describes the hadrons comprising of the quark and antiquark constituents. In the present section, we shall very briefly describe the model so as to apply the same for investigating these decay widths.

The field operator for a constituent quark for a hadron at rest at time, t=0, is written as

 ψ(x,t=0) = (1) ≡ Q(x)+~Q(x).

In the above, and are the two component quark annihilation and antiquark creation operators. The operator annihilates a quark with spin and momentum , whereas, creates an antiquark with spin and momentum , and these operators satisfy the usual anticommutation relations

 {qr(k),qs(k′)†}={~qr(k),~qs(k′)†}=δrsδ(k−k′). (2)

In equation (1), and are given as

 U(\bf k)=(f(|k|)\boldmathσ⋅kg(|k|)),V(\bf k)=(% \boldmathσ⋅kg(|k|)f(|k|)), (3)

where the functions and satisfy the constraint spm781 (),

 f2+g2k2=1, (4)

as obtained from the equal time anticommutation relation for the four-component Dirac field operators. These functions, for the case of free Dirac field of mass , are given as,

 f(|k|)=(k0+M2k0)1/2,g(|k|)=(12k0(k0+M))1/2, (5)

where . In the above, is the constituent quark mass, which is calculated from dynamical chiral symetry breaking and in general, can be momentum dependent dnkrein (); hmamspm1 (); hmamspm2 (); spmeffpot (); spmbothcond (). Using a four point interaction for the quark operators, as in Nambu Jona Lasinio model, the constituent quark mass turns out to be momentum independent amhm2004 (). Also, a recent study dnkrein () shows the momentum dependence of calculated within a color confining model, to be appreciable only at high momenta. In the present work of the study of decay widths of the bottomonium states to open bottom mesons, we shall assume the constituent quark mass to be momentum independent. We shall also take the approximate forms (with a small momentum expansion) for the functions and of the field operator as given by and chmdwamspmwg ().

The field operator for the constituent quark of hadron with finite momentum is obtained by Lorentz boosting the field operator of the constituent quark of hadron at rest, which requires the time dependence of the quark field operators. As in the bag model, the time dependence is given by assuming the constituent quarks to be occupying fixed energy levels spm781 (); spm782 (), so that for the -th quark of a hadron of mass at rest, we have

 Qi(x)=Qi(x)exp(−iλimHt), (6)

where is the fraction of the energy (mass) of the hadron carried by the quark, with . For a hadron in motion with four momentum p, the field operators for quark annihilation and antiquark creation, for t=0, are obtained by Lorentz boosting the field operator of the hadron at rest, and are given as spmdiffscat ()

 Q(p)(x,0)=(2π)−3/2∫d\bf kS(L(p))U(k)QI(k+λp)exp(i(k+λp)⋅x) (7)

and,

 ~Q(p)(x,0)=(2π)−3/2∫d\bf kS(L(p))V(−k)~QI(−k+λp)exp(−i(−k+λp)⋅x). (8)

In the above, is the fraction of the energy of the hadron at rest, carried by the constituent quark (antiquark). In equations (7) and (8), is the Lorentz transformation matrix, which yields the hadron at finite four-momentum from the hadron at rest, and is given as spm782 ()

 Lμ0=L0μ=pμmH;Lij=δij+pipjmH(p0+mH), (9)

where, and , and the Lorentz boosting factor is given as

 S(L(p))=[(p0+mH)2mH]1/2+[12mH(p0+mH)]1/2→α⋅→p, (10)

where, , are the Dirac matrices.

## Iii Partial Decay widths of the bottomonium states to B¯B pair in the composite model of the hadrons

The partial decay widths of the bottomonium states, , , and to in the hadronic matter are studied in the present investigation. The medium modifications of these decay widths calculated in the present work arise due to the medium modifications of the decaying bottmonium state and the outgoing and mesons in the hadronic medium. In vacuum, the masses of the bottomonium states, , , , , and the open bottom mesons, () are given as 9460.3 MeV, 10023.26 MeV, 10355.2 MeV, 10579.4 MeV, and 5279 MeV respectively. Hence, in vacuum, the lowest state, which can decay to is . However, the masses of the states as well as and mesons are modified in the hadronic medium, due to which the partial decay widths of the bottomonium states to pair are modified in the medium. In the hadronic matter, the modification of the meson mass turns out to be different from the medium modification of the meson mass, due to their difference in the interactions with the hadronic matter. The modifications of the masses of the open bottom mesons arise due to the interactions with the nucleons, hyperons as well as scalar mesons in the strange hadronic matter dpambmeson (). These in-medium masses have been calculated within an effective hadronic model, where the chiral SU(3) model has been generalized to SU(5) to derive the interactions of the and mesons with the light hadrons dpambmeson (). The bottomonium states are, on the other hand, modified due to their interactions with the gluon condensates in the hadronic medium. The in-medium masses of these states ( , , and ) have been calculated within the same effective hadronic model, where the effect of scale symmetry breaking of QCD through the scalar gluon condensates are simulated by a scalar dilaton field within the hadronic model amdpbottomonium (). For the case of the -state at rest decaying to , the magnitude of , is given by

 (11)

In the above, the medium modifications of the masses of the bottomonium state and the and mesons are considered, so as to calculate the decay width of in the strange hadronic medium.

The explicit construct for the state for the bottomonium state with spin projection at rest as

 |ΥNlm(→0)⟩=∫dk1biI(k1)†aNlm(Υ,k1)~bIi(−k1)|vac⟩, (12)

where, is the color index of the quark and antiquark operators. In the present investigation, we shall assume the harmonic oscillator wave functions for the bottomonium states and shall study the medium modifications of the decay widths of the bottomonium states, , , and , arising from the mass modifications of the bottomonium states as well as of the and mesons.

For (N=1,2,3,4) spmddbar80 (),

 aNSm(Υ,k1)=σmuNS(k1), (13)

where,

 u1S(\bf k1)=1√6(R2Υ(1S)π)3/4exp[−12R2Υ(1S)k12] (14)
 u2S(\bf k1)=1√6√32(R2Υ(2S)π)3/4(23R2Υ(2S)\bf k21−1)exp[−12R2Υ(2S)% \bf k21]. (15)
 u3S(\bf k1)=1√6√158(R2Υ(3S)π)3/4(1−43R2Υ(3S)\bf k21+415R4Υ(3S)\bf k41)exp[−12R2Υ(3S)\bf k21]. (16)
 u4S(\bf k1) = −1√6√354(R2Υ(4S)π)3/4(1−2R2Υ(4S)\bf k% 21+45R4Υ(4S)\bf k41−8105R6Υ(4S)\bf k61) (17) × exp[−12R2Υ(4S)\bf k21].

In the above, the factor refers to normalization factor arising from degeneracy factors due to color (3) and spin (2) of the quarks and antiquarks.

The and states, with finite momenta, are explicitly given as

 |B0(p′)⟩=∫di2I(k3+λ1p′)†uB(k3)~bIi2(−k3+λ2p′)d\bf k3. (18)

and

 |¯B0(p)⟩=∫bi1I(k2+λ2p)†uB(k2)~dIi1(−k2+λ1p)d\bf k2 (19)

In the above,

 uB(k)=1√6(RB2π)3/4exp(−RB2k22). (20)

To calculate the partial decay widths of the decay process , we need to know the values of and , the fractions of energy of the hadron carried by its constituent quark and antiquark. These are calculated by assuming that the binding energy of the hadron as shared by the quark/antquark are inversely proportional to the quark/antiquark masses spm782 (); chmdwamspmwg (). The energies of and in meson are then given as spm782 (),

 ω1=Md+MbMb+Md⋅(mB−Mb−Md) (21)

and,

 ω2=Mb+MdMb+Md⋅(mB−Mb−Md), (22)

with

 λi=ωimB. (23)

The motivation for the assumption that the contributions from the quark (antiquark) to the binding energy of the hadron to be inversely proportional to the mass of the quark (antiquark) as in equations (21) and (22) is as follows. In fact, in general, the contributions to the binding energy of the bound state composed of particles of 1 and 2, with masses and , are assumed to be given as , , multiplied by the binding energy of the bound state, where, is the reduced mass of the system, calculated from . In other words, the contributions from the particles to binding energy are inversely proportional to their masses, and the total binding energy is the sum of the individual contributions, i.e., , as it should be. The reason for making this assumption comes from the example of hydrogen atom, which is the bound state of the proton and the electron. As the mass of proton is much larger as compared to the mass of the electron, the binding energy contribution from the electron is of hydrogen atom, and the contribution from the proton is , which is negligible as compared to the total binding energy of hydrogen atom, since . With this assumption, the binding energies of the heavy-light mesons, e.g., and mesons chmdwamspmwg (), as well as for and mesons, mostly arise from the contribution from the light quark (antiquark).

We next evaluate the matrix element of the quark-antiquark pair creation part, , of the Dirac Hamiltonian density, between the initial and the final states for the reaction . The Dirac Hamiltonian density is given as

 H=ψ(x)†(−i→α⋅→▽+βmQ)ψ(x), (24)

where and are the Dirac matrices, with defined following equation (10) and . In the above, is the field operator for the constituent quark, with mass , which is given by equation (1) for t=0. The relevant part of the quark pair creation term for the decay process is through the creation. From equations (7) and (8) we can write down , and then integrate over to obtain the expression

 ∫Hd†~d(x,t=0)dx (25) = ∫d\bf kd\bf k′diI(k+λ1p′)†U(k)†S(L(p′))†δ(−k′+λ1p+k+λ1p′) × (\boldmathα⋅(k+λ1p′)+βMd)S(L(p))V(−k′)~diI(−k′+λ1p),

where, and in the above equation, correspond to the hadrons with finite momenta, and , i.e., and mesons, and is the constituent mass of the quark. has already been defined in equation (10), and, and are given by equation (3).

From equation (25) we can then evaluate that

 ⟨¯B0(p)|⟨B0(p′)|∫Hd†~d(x,t=0)dx|Υ(NS)m(→0)⟩=δ(p+p′)∫d\bf k1AΥ(NS)m(p,k1), (26)

using the explicit forms of the -states and and states. We obtain the form of , including summing over color,

 AΥ(NS)m(p,k1) = 3u¯B0(k)uB0(k)⋅Tr[am(Υ(NS),k1)U(k)† (27) × S(L(p′))†(\boldmathα⋅~q+βMd)S(L(p))V(−k)],

where, , and .

We shall now simplify . Firstly, since the mesons are completely nonrelativistic, we shall be assuming that and as identity. The integral in the R.H.S. of the equation (26) can be written as,

 ∫d\boldmathk1AΥ(NS)m(p,k1)=3∫d\boldmathk1u¯B0(k)uB0(k)⋅Tr[am(Υ(NS),k1)B(\boldmathk,\boldmath~q)], (28)

where,

 B(\boldmathk,\boldmath~q)=\boldmathσ⋅\boldmath~q−(2(\boldmathk⋅% \boldmath~q)g2+f(\boldmathk))\boldmathσ⋅\boldmathk. (29)

We use the approximate forms of and at small momentum, i.e., , and , for simplifying the integral given by equation (28). After simplification, this integral can be written as

 ∫d\boldmathk1AΥ(NS)m(p,k1)=6cΥ(NS)exp[(aΥ(NS)b2Υ(NS)−RB2λ22)|\boldmathp|2]∫d\boldmathk1TΥ(NS)m(p,k1), (30)

where, , for , are given as,

 TΥ(1S)m(p,k1) = 12Tr[σmB(\boldmathk,\boldmath~q)], TΥ(2S)m(p,k1) = 12Tr[σmB(\boldmathk,\boldmath~q)]⋅(23R2Υ(2S)\bf k21−1), TΥ(3S)m(p,k1) = 12Tr[σmB(\boldmathk,\boldmath~q)]⋅(1−43R2Υ(3S)\bf k21+415R4Υ(3S)\bf k41), TΥ(4S)m(p,k1) = 12Tr[σmB(\boldmathk,\boldmath~q)]⋅(1−2R2Υ(4S)% \bf k21+45R4Υ(4S)\bf k41−8105R6Υ(4S)\bf k61). (31)

In the above, the parameters and are given as

 aΥ(NS)=12R2Υ(NS)+R2B;bΥ(NS)=R2Bλ2/aΥ(NS), (32)

with as the radius of the bottomonium state, , and,

 cΥ(1S) = 16√6⋅⎛⎝R2Υ(1S)π⎞⎠3/4⋅(R2Bπ)3/2, cΥ(2S) = 16√6√32⎛⎝R2Υ(2S)π⎞⎠3/4⋅(R2Bπ)3/2, cΥ(3S) = 16√6√158⎛⎝R2Υ(3S)π⎞⎠3/4⋅(R2Bπ)3/2, cΥ(4S) = 16√6(√354)⎛⎝R2Υ(4S)π⎞⎠3/4⋅(R2Bπ)3/2. (33)

We now change the integration variable to q in equation (28) with the substitution and write

 ∫AΥ(NS)m(p,k1)d\bf k1=6cΥ(NS)exp[(aΥ(NS)b2Υ(NS)−λ22R2B)|p|2]⋅∫exp(−aΥ(NS)\bf q2)TΥ(NS)md\bf q, (34)

where, in the above equation, is the expression given by equation (31), rewritten in terms of . We next proceed to simplify the above integral, by using the fact that the terms odd in q in equation (34) will vanish. Also, using and , where, is an even function of , in the above integrand can be recast into the form

 TΥ(NS)m(\bf p,\bf q) ≡ pm[FΥ(NS)0(|\bf p|)+FΥ(NS)1(|\bf p|)|\bf q|2+FΥ(NS)2(|\bf p|)(|\bf q|2)2 (35) + FΥ(NS)3(|\bf p|)(|\bf q|2)3+FΥ(NS)4(|\bf p|)(|\bf q|2)4]

The coefficients, are given as

 FΥ(1S)0 = (λ2−1) − 2g2|\bf p|2(bΥ(1S)−λ2)(34b2Υ(1S)−(1+12λ2)bΥ(1S)+λ2−14λ22), FΥ(1S)1 = g2[−52bΥ(1S)+23+116λ2], FΥ(1S)2 = 0, FΥ(1S)3 = 0, FΥ(1S)4 = 0, (36)
 FΥ(2S)0 = (23R2Υ(2S)b2Υ(2S)|% \bf p|2−1)FΥ(1S)0, FΥ(2S)1 = 23R2Υ(2S)FΥ(1S)0+(23R2Υ(2S)b2Υ(2S)|\bf p|2−1)FΥ(1S)1 − 89R2Υ(2S)bΥ(2S)g2|\bf p% |2[94b2Υ(2S)−bΥ(2S)(2+52λ2)+2λ2+14λ22], FΥ(2S)2 = 23R2Υ(2S)g2[−72bΥ(2S)+23+116λ2], FΥ(2S)3 = 0, FΥ(2S)4 = 0, (37)
 FΥ(3S)0 = (−1+λ2+g2|p|22(bΥ(3S)−λ2)2(3bΥ(3S)+λ2−4)) × (1−43R2Υ(3S)b2Υ(3S)|p|2+415R4Υ(3S)b4Υ(3S)|p|4), FΥ(3S)1 = 43R2Υ(3S)(1−λ2)(1−23b2Υ(3S)R2Υ(3S)|p|2)+g26(3bΥ(3S)−7λ2+4) + g2|p|2R2Υ(3S)9[(−3bΥ(3S)+7λ2−4)b2Υ(3S) + 4(3bΥ(3S)−λ2−2)(bΥ(3S)−λ2)(−2bΥ(3S)+3λ2)] + 2g2|p|4R4Υ(3S)b2Υ(3S)45[(3bΥ(3S)−7λ2+4)b2Υ(3S) + 4(3bΥ(3S)−4λ2)(3bΥ(3S)−λ2−2)(bΥ(3S)−λ2)] FΥ(3S)2 = 415(λ2−1)R4Υ(3S)−29g2R2Υ(3S)(9bΥ(3S)−7λ2+4) + g2R4Υ(3S)|p|215[8b3Υ(3S)−83bΥ(3S)(bΥ(3S)−λ2)(3bΥ(3S)+λ2−4) + 2(bΥ(3S)−λ2)2(3bΥ(3S)+λ2−4)+14b2Υ(3S)(bΥ(3S)−λ2) − 8815b2Υ(3S)(3bΥ(3S)−λ2−2)], FΥ(3S)3 = 2g245R4Υ(3S)(15bΥ(3S)−7λ2+4), FΥ(3S)4 = 0, (38)

and,

 FΥ(4S)0 = 12(bΥ(4S)−1)(bΥ(4S)−λ2)(3bΥ(4S)+λ2−4)g2|p|2 ×(1−2R2Υ(4S)b2Υ(4S)|p|2+45R4Υ(4S)b4Υ(4S)|p|4−8105R6Υ(4S)b6Υ(4S)|p|6) FΥ(4S)1 = g26(9(bΥ(4S)−1)−2(3bΥ(4S)−λ2−2)) + g2|p|2R2Υ(4S)3[(−5bΥ(4S)+3)(3bΥ(4S)+λ2−4