Properties of bottomonium in a relativistic quark model

# Properties of bottomonium in a relativistic quark model

Manjunath Bhat Antony Prakash Monteiro K. B. Vijaya Kumar P. G. Department of Physics, St Philomena college Darbe, Puttur 574 202, India Department of Physics, Mangalore University, Mangalagangothri P.O., Mangalore - 574199, INDIA
###### Abstract

The mass spectrum of states has been obtained using the phenomenological relativistic quark model (RQM). The Hamiltonian used in the investigation has confinement potential and confined one gluon exchange potential (COGEP). In the frame work of RQM a study of M1 and E1 radiative decays of states have been made. The weak decay widths in the spectator quark approximation have been estimated. An overall agreement is obtained with the experimental masses and decay widths.

###### keywords:
relativistic quark model (RQM); radiative decay; confined one gluon exchange potential (COGEP); states
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## 1 Introducing Bottomonium states

The bound state of a bottom quark b and its anti quark known as bottomonium, play an important role in the study of the strong interactions. The spectrum of bottomonium is known very well from the experiment and there are many different approaches to calculate it theoretically. Phenomenological potential models provide an accurate way of doing this. Bottomonium was first observed in the spectrum of pairs produced in 400 GeV protonnucleus collisions at Fermilab. The states were dicovered as spin triplet states , and by E288 collaboration at Fermilab HS77 (); WR77 ().

The spin-singlet state was discovered by the Babar Collaboration and they measured the mass of to be MeVAB08 (). In an another measurement the BaBar found the mass of to be MeV. The CLEO collaboration measured the mass of to be MeVBG10 (). The more precise measurement of mass of state is done by the Belle collaboration which obtained a value of MeVMR12 (). The was successfully observed by the CLEO collaboration in decays at a mass of MeVSD12 (). The Belle collaboration has reported a signal for using the at a mass of MeVMR12 ().

The BaBar collaboration first reported the evidence for the spin singlet P wave state in the transition LJ12 (). Later the Belle collaboration found through transition AI12 (). The measured masses of and were MeV and MeV respectively. The two triplet P-wave states and with J=0,1,2 were discovered in radiative decays of the and respectively KH83 (); EG82 (); CK83 (); FH83 (). The states were produced in the proton - proton collisions at the LHC at TeV and recorded by the ATLAS detectorGA12 (). In addition to this, a new state centred at a mass of MeV has been observed in both the and decay modes. This state was confirmed by the D0 collaboration which observed the state in the final state with mass of MeVVM12 (). The LHCb collaboration has measured the mass of the to be MeVAR14 ().

The spin triplet D-wave state was observed through the decay chain GB04 (); PD10 (). The was discovered in the final state with mass MeVPD10 (). The radial excitations of vector bottomonium family , and were also observedBD85 (); LD85 (). The two states and were observed by the Belle collaboration in the mass spectra of (n=1,2,3) and (n=1,2) pairsAB12 (). These states do not fall into the usual scheme of mesons since they posses electrical charge. These states can be identified as molecular states, since these are very near to the and thresholds.

QCD motivated potential models have played an important role in the understanding of quarkonium spectroscopy. These models have successfully predicted the bottomonium spectrum but over estimated the decay rates. In the present work we study bottomonium spectrum in detail using relativistic quark model.

The paper is organized in 4 sections. In sec. 2 we briefly review the theoretical background for relativistic model and the relativistic description of radiative decay widths. In sec. 3 we discuss the results and the conclusions are drawn in sec. 4 with a comparison to other models.

## 2 Phenomenological Understanding

### 2.1 The Relativistic Harmonic Model

Describing a bottomonium bound state problem, incorporating the elements of non perturbative low energy dynamics is a challenging problem. The two main approaches in this line have been effective field theoriesHG90 (); M97 (); NB05 () and lattice gauge theories. Though the calculations away from threshold have been proven successful, the threshold region remain still tricky. And hence a relativistic potential model would be a befitting one for our study of bottomonium states. Essentially in all phenomenological relativistic QCD base quark models, the Hamiltonian for the quark system consists of a two body confinement potential and Coulomb like potential which takes care of relativistic effects. The Hamiltonian in our model has the confinement potential and a two body confined one gluon exchange potential(COGEP) PC92 (); SB91 (); KB93 (); KB97 ().
The RHM equation is SB83 (); VH04 (); VB09 ():

 (α⋅p+γ0M+12(1+γ0)A2r2)ψ=Eψ (1)

The Dirac wave function is written as ,where is the large componet and is the small component for positive energy state solution in the non-relativistic limit. Using standard matrix form of and ,the Dirac equation is expressed in terms of the large and small components.

 (A2r2+M)ϕ+(σ⋅p)χ=Eϕ (2)
 (σ⋅p)ϕ−(M+E)χ=0 (3)

Hence, solution of the full Dirac equation is written as

 ∣ψ⟩=N(ϕσ⋅pE+Mϕ) (4)

where is the Dirac matrix, M is a constant mass and is the confinement strength.

with normalization constant

 N=√2(E+M)3E+M (5)

E is the eigenvalue of the single particle Dirac equation with interaction potential given by equation (LABEL:eq:A). We perform a unitary transformation to eliminate the lower component of . With this transformation, satisfies the ’harmonic oscillator’ wave equation

 (p2E+M+A2r2)ϕ=(E−M)ϕ (6)

the eigenvalue of which is given by

 E2N=M2+(2NOSC+1)ΩN (7)

where is the oscillator quantum number and is the energy dependent oscillator size parameter given by

 ΩN=A(EN+M)1/2 (8)

The total energy or the mass of the meson is obtained by adding the individual contributions of the quarks. The spurious centre of mass (CM) is corrected GG57 () by using intrinsic operators for the and terms appearing in the Hamiltonian. This amounts to just subtracting the CM motion zero contribution from the expression.

### 2.2 Confined One Gluon Exchange Potential

NRQM employing OGEP, which takes into consideration the confining effect of quarks on mesonic states, does not shed light on confinement of gluons. Hence one needs to incorporate confining effects of gluons on mesonic states since confined dynamics of gluons plays a decisive role in determining the hadron spectrum and in hadron-hadron interaction. There are various confinement models for the gluons. The effect of the confined gluons on the masses of mesons has been studied using the successful current confinement model (CCM). The confined gluon propagators (CGP) are derived in CCM to obtain the COGEP.

The central part of the COGEP isPC92 (); SB91 (); KB93 (); KB97 ()

 VcentCOGEP(→r)=αsN44λi⋅λj[D0(→r)+1(E+M)2[4πδ3(→r)−c4r2D1(→r)][1−23→σi⋅→σj]] (9)

where , are the color matrices, and are the propagators given by

 D0(→r)=Γ1/24π3/2c(cr)−3/2W1/2;−1/4(c2r2) (10) D1(→r)=Γ1/24π3/2c(cr)−3/2W0;−1/4(c2r2) (11)

Here , W’s are Whittaker functions and (fm) is a constant parameter which gives the range of propagation of gluons and is fitted in the CCM to obtain the glue-ball spectra and r is the distance from the confinement center.

The spin orbit part of COGEP is

 VLS12(→r)=αs4N4(E+M)2λ1⋅λ22r×[[→r×(^P1−^P2)⋅(σ1+σ2)](D′0(→r)+2D′1(→r))+[→r×(^P1+^P2)⋅(σ1−σ2)](D′0(→r)−D′1(→r))] (12)

where , are the momentum of quarks. The spin orbit term has been split into the symmetric and anti symmetric spin orbit terms.

The tensor part of the COGEP is,

 VTEN12(→r)=−αs4N4(E+M)2λ1⋅λ2[D′′1(→r)3−D′1(→r)3r]S12 (13)

where

 S12=[3(σ1⋅^r)(σ2⋅^r)−σ1⋅σ2] (14)

The total potential is then

 V(→r)=VcentCOGEP(→r)+VLS12(→r)+VTEN12(→r) (15)

We then obtain Hamiltonian by adding interaction potential given by equation 15 to the the ’harmonic oscillator’ wave equation 6,

 H=2∑i=1P2iE+M−KCM−∑i

We consider two types of radiative transitions of the meson:

#### 2.3.1 Electric Dipole (E1) Transitions

The partial widths for electric dipole (E1) transitions between states and are given by

 Γ(i→f+γ)=(2J′+1)43Q2bαk30SEif∣∣Eif∣∣2 (17)

where is the energy of the emitted photon,

in relativistic model.

is the fine structure constant. is the charge of the b quark in units of . and are the masses of initial and final mesons. The statistical factor , are the total angular momentum of initial and final mesons, are the orbital angular momentum of initial and final mesons and is the spin of initial meson.

 Eif=3k0∫∞0r3Rnl(r)R′nl(r)dr[k0r2j0(k0r2)−j1(k0r2)] (18)

is the radial overlap integral which has the dimension of length, with being the normalized radial wave functions for the corresponding states.

#### 2.3.2 Magnetic Dipole (M1) Transitions

The M1 partial decay width between S wave states is WJ88 (); NL78 (); CE05 (); WE86 (); BB95 (); NB99 (); BA96 ()

 Γi→fγ=δLL′4αk30Eb(k0)mi(Qbmb+(−1)S+S′Q¯bm¯b)2(2S+1)×(2S′+1)(2J′+1){SLJJ′1S′}2{1121212SS′}2×[∫∞0Rn′L′(r)r2j0(kr/2)RnL(r)dr]2 (19)

where is the overlap integral for unit operator between the coordinate wave functions of the initial and the final meson states, is the spherical Bessel function, is the mass of bottom quark. and are the total angular momentum of initial and final meson states respectively. , , and are the orbital angular momentum ans spin angular momentum of initial and final meson states respectively

## 3 Discussion on Results

### 3.1 Mass Spectra

The quark-antiquark wave functions in terms of oscillator wave functions corresponding to the relative and center of mass coordinates have been expressed here, which are of the form,

 Ψnlm(r,θ,ϕ)=N′(rb)l Ll+12n(r2b2)exp(−r22b2)Ylm(θ,ϕ) (20)

where is the normalising constant given by

 |N′|2=2n!b3π1/22[2(n+l)+1](2n+2l+1)!(n+l)! (21)

are the associated Laguerre polynomials.

The harmonic oscillator wave function allows the separation of the motion of the center of mass and has been widely used to classify the spectra of baryons and mesons FD69 (); RP71 () and extending to nucleon-nucleon interaction is straight forward KS89 (); FA83 (); KB93 (). If the basic states are the harmonic oscillator wave functions, then it is straightforward to evaluate the matrix elements of few body systems such as mesons or baryons. Since the basic states are the products of the harmonic oscillator wave functions they can be chosen in a manner that allows the product wave functions to be expanded as a finite sum of the corresponding products for any other set of Jacobi coordinates. It is advantageous to use the Gaussian form since in the annihilation of quark-anti quark into lepton pairs, the amplitude of the emission or absorption processes depends essentially on the overlap of initial and final hadrons and hence the overlap depends only on the intermediate distance region of the spatial wave functions which can extend up to 0.5 fm. This intermediate region can be described by potentials that are similar in this region and hence harmonic oscillator wave functions are expected to reproduce emission and absorption processes quite well.

The five parameters used in our model are the mass of beauty quark , the harmonic oscillator size parameter , the confinement strength , the CCM parameter and the quark-gluon coupling constant . There are several papers in literature where the size parameter is defined SN86 (); IM92 (). The value of is fixed by minimizing the expectation value of the Hamiltonian for the vector meson. To start with, we construct the Hamiltonian matrix for states in the harmonic oscillator basis. The confinement strength is fixed by the stability condition for variation of mass of the meson against the size parameter .

 ∂∂b⟨ψ|H|ψ⟩=0

The parameter in CCM PC92 (); SB85 (); SB87 () was obtained by fitting the iota (1440 MeV) as a digluon glue ball.To fit , we start with a set of reasonable values and diagonalize the matrix for meson. Then we tune these parameters to obtain an agreement with the experimental value for the mass of meson. In literature we find different sets of values for , which are listed in Table 1.

The values of strong coupling constant in literature are listed in Table 2. The value of strong coupling constant () used is compatible with the perturbative treatment.

The mass spectrum has been obtained by diagonalizing the Hamiltonian in a large basis of matrix. The calculation clearly indicates that masses for both pseudo scalar and vector mesons converge to the experimental values when the diagonalization is carried out in a larger basis. In our earlier work also, we had come to the similar conclusion while investigating light meson spectrumVH04 (); BK05 (). The diagonalization of the Hamiltonian matrix in a larger basis leads to the lowering of the masses and justifies the perturbative technique to calculate the mass spectrum. The calculation clearly indicates that when diagonalization is carried out in a larger basis convergence is achieved both for pseudo-scalar mesons and vector mesons to the respective experimental values.
We use the following set of parameter values.

 mb=4770.0 MeV  ;b=0.35 fm;   αs=0.2 to 0.3;  A2=220 MeV fm−2;  c=1.72 fm−1 (22)

The calculated masses of the states after diagonalization are listed in Table  3. Our calculated mass value for is 9399 MeV which agrees with the experimental value 9398.0 MeVPDG () and for (1S) is 9460.30 . (1S) is heavier than by 61 MeV. This difference is justified by calculating the splitting of the ground state which is given by

 M(3S1)−M(1S0)=32παs|ψ(0)|29m2b (23)

The experimental value of hyperfine mass splitting for the ground state is given by PDG ()

 M(3S1)−M(1S0)=62.3±3.2 MeV (24)

The hyperfine mass splitting calculated in our model is in good agreement with both experimental dataPDG () and lattice QCD result ( MeVSM10 (). The EFTs have predicted a hyperfine mass splitting of MeVBA04 () which is lower than the experimental value.

We predict a mass of 10001.12 MeV for the state which is in good agreement with the experemental value MeV PDG (). The calculated mass of spin triplet state() of is 10024.50 MeV which is found to be in good agreement with the experemntal value MeV. The hyperfine mass splitting for the 2S state MeV is in good agreement with both the experemental one MeVPDG () and with the lattice QCD results MeVSM10 (). The predicted mass values for and states in our model are 10375.96 MeV and 10392.00 MeV respectively. The corresponding hyperfine mass splitting is MeV. We have also claculated masses of higher radially excited states of and its spin triplet state . The mass of spin singlet P wave state calculated in our model is compatible with the experimental one. The masses of spin triplet states are in good agreement with experimental values of masses of these states. Our prediction for masses of higher excited P wave states are 10-20 MeV higher than the experimental values and other model calculations. Some of the higher excited states are 50-100 MeV heavier in our model.

Radiative decays of excited bottomonium states are the powerful tools which can be used to study the internal structure of states and they provide a good test for the predictions from the various models. Radiative transitions between different bottomonium states can be used to discover new bottomonium states experimentally.

The possible decay modes have been listed in Table 4 and the predictions for E1 decay widths are given. Also our predictions have been compared with other theoretical models. Most of the predictions for transitions are in qualitative agreement. However, there are some differences in the predictions due to differences in phase space arising from different mass predictions and also from the wave function effects particularly from relativization procedure. We find our results are compatible with other theoretical model values for most of the channels. Relativistic corrections to the wave functions tend to reduce the E1 transition widths for most of the channels. We find that the E1 transitions are suppressed compared to other E1 transitions and it is interesting to see that the E1 transition rates for are zero in the nonrelativistic caseEG08 (); GR92 (). We find that these E1 transisitions are very sensitive to relativistic effects.

The M1 transitions contribute little to the total decay widths of bottomonium states and are weaker than the E1 transitions of bottomonium states. M1 transitions have been used to observe the spin-singlet statesSG16 (). Allowed M1 transitions correspond to triplet-singlet transitions between S-wave states of the same n quantum number, while hindered M1 transitions are either triplet-singlet or singlet-triplet transitions between S-wave states of different n quantum numbers.

In order to calculate decay rates of hindered transitions we need to include relativistic corrections.There are three main types of corrections: relativistic modification of the non relativistic wave functions, relativistic modification of the electromagnetic transition operator, and finite-size corrections. In addition to these there are additional corrections arising from the quark anomalous magnetic moment. Corrections to the wave function that give contributions to the transition amplitude are of two categories. Firstly from higher order potential corrections, which are distinguished as a) Zero recoil effect and b) recoil effects of the final state meson and secondly from colour octet effects. The colour octet effects are not included in potential model formulation and are not considered so far in radiative transitions.

The M1 transition rates of bottomonium states have been calculated using equation (19). The resulting M1 radiative transition rates of these states are presented in Table 5. In this table we give calculated values for decay rates of M1 radiative transition in comparison with the other relativistic and non relativistic quark models. We see from these results that the relativistic effects play a very important role in determining the bottomonium M1 transition rates. The relativistic effects reduce the decay rates of allowed transitions and increase the rates of hindered transitions. The M1 transition rates calculated in our model agree well with the values predicted by other theoretical models.

## 4 Conclusions

The bottomonium mass spectrum and electromagnetic transitions are investigated by adopting the quark-antiquark potential consisting of the confined one-gluon-exchange and the Lorentz scalar plus vector harmonic oscillator confinement potentials. We perform a nonperturbative calculations with Hamiltonian including spin-independent and dependent potentials.

The bottomonium spectrum predicted by our quark model is in good agreement with the experimental data. We have calculated a large number of electromagnetic decay widths showing that our results are in reasonable agreement with the other model caculations in most of the cases. The hyperfine mass splitting between the singlet and triplet 1S as well as 2S states are consistent with the experimental data and also with lattice QCD results. We have also predicted radially excited states of and their triplet states. The masses of P wave states calculated in our model are in good agreement with the experimental values and other theoretical models. Some of the higher excited states are 50-100 MeV heavier in our model. This may be due to the fact that coupled channel effects are significant in these states.

Radiative decays are the dominant decay modes of the excited states having widths of about a fraction of MeV. In order to understand the spectrum and distinguish exotic states from conventional ones, it is very essential that the masses and the radiative decay widths of states are accurately determined. The calculated M1 transition rates reasonably agree with the other theoretical model predictions as listed in table 5. It is clearly seen in this calculation that the relativistic effects play an important role in determining the M1 radiative transition rates, since the hindered transition rates are zero due to the wave function orthogonality in the NRQM formalism. The inclusion of relativistic effects enhances the non-relativistically-hindered radiative transition rates, predicting them larger than the allowed ones by an order of magnitude. It is a good example for the importance of relativity, even for some properties of heavy mesons.

Acknowledgements

One of the authors (APM) is grateful to BRNS, DAE, India for granting the project and JRF (37(3)/14/21/2014BRNS).

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