An approach to calculation of mass spectra and two-photon decays of mesons in the framework of Bethe-Salpeter Equation
Department of Physics, Chandigarh University, Mohali-140413,
Department of Physics, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia
In this work we calculate the mass spectrum of charmonium for states of , , as well as for states of , and states of along with the two-photon decay widths of ground and first excited state of quarkonia for the process, in the framework of a QCD motivated Bethe-Salpeter Equation. In this BSE framework, the coupled Salpeter equations are first shown to decouple for the confining part of interaction, under heavy-quark approximation, and analyically solved, and later the one-gluon-exchange interaction is perturbatively incorporated leading to mass spectral equations for various quarkonia. The analytic forms of wave functions obtained are used for calculation of two-photon decay widths of . Our results are in reasonable agreement with data (where ever available) and other models.
Key words: Bethe-Salpeter equation, Covariant Instantaneous
Ansatz, Mass spectral equation, charmonium, bottomonium
PACS: 12.39.-x, 11.10.St , 21.30.Fe , 12.40.Yx , 13.20.-v
An important role in applications of Quantum Chromodynamics (QCD) to hadronic physics is played by charmonium () and bottomonium (), which are built up of a heavy quark and heavy anti-quark. By definition, heavy quark has a mass , which is large in comparison to the typical hadronic scale, . Quarkonia are characterized by at least three widely separated scales : the hard scale (the mass of heavy quarks), the soft scale (relative momentum , and the ultra soft scale (typical kinetic energy, of heavy quark and anti-quark). The appearance of all these scales in the dynamics of heavy quarkonium makes its quantitative study extremely difficult. Quarkonium systems are crucially important to improve our understanding of QCD. They probe all the energy regions of QCD from hard region, where perturbative QCD dominates, to the low energy region, where non-perturbative effects dominate. Quarkonium states are thus a unique laboratory where our understanding of non-perturbative QCD may be tested.
There has a been a renewed interest in recent years in spectroscopy of these heavy hadrons in charm and beauty sectors, which was primarily due to experimental facilities the world over such as BABAR, Belle, CLEO, DELPHI, BES etc. [2, 3, 4, 5, 6], which have been providing accurate data on , and hadrons with respect to their masses and decays. In the process many new states have been discovered such as , . The data strongly suggests that among new resonances may be exotic four-quark states, or hybrid states with gluonic degrees of freedom in addition to pair, or loosely bound states of heavy hadrons, i.e. charmonium molecules. Further, there are also open questions about the quantum number assignments of some of these states such as (as to whether it is or [7, 8]). Thus charmonium offers us intriguing puzzles.
However, since the mass spectrum and the decays of all these bound states of heavy quarks can be tested experimentally, theoretical studies on them may throw valuable insight about the heavy quark dynamics and lead to a deeper understanding of QCD further. Studies on mass spectrum of these hadrons is particularly important, since it throws light on the potential, since the long range confinement potential can not be derived from QCD alone. Further, though these states appear to be simple, however, their production mechanism is still not properly understood. These mesons are involved in a number of reactions which are of great importance for study of Cabibbo-Kobayashi-Maskawa (CKM) matrix and CP violation. In this paper we also study the two-photon decays of scalar quarkonia, . These decays are sensitive probes of quarkonium wave functions.
The non-perturbative approaches, such as Effective field theory , Lattice QCD [10, 11, 12], Chiral perturbation theory , QCD sum rules [14, 15], N.R.QCD , Bethe-Salpeter equation (BSE) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], and potential models [27, 28, 29] deal have been employed to study heavy quarkonia. Recent progress in understanding of non-relativistic field theories make it possible to go beyond phenomenological models, and for the first time face the possibility of providing unified description of all aspects of heavy quarkonium physics. This allows us to use quarkonium as a bench mark for understanding of QCD, and for precise determination of Standard Model parameters (e.g. heavy quark masses, QCD coupling constant, ), and for new physics searches.
All this opens up new challenges in theoretical understanding of heavy hadrons and also provide an important tool for exploring the structure of these simplest bound states in QCD and for studying the non-perturbative (long distance) behavior of strong interactions.
In present work, we do the full mass spectral problem of charmonium for states of , , as well as for states of , and along with the two-photon decay widths of ground and first excited state of scalar quarkonia, for the process, in the framework of a QCD motivated Bethe-Salpeter Equation under Covariant Instantaneous Ansatz (CIA), employing the full BS kernel comprising both, the long-range confinement, and the short range one-gluon-exchange (coulomb) interactions.
We do understand that in quarkonia, the constituents are close enough to each other to warrant a more accurate treatment of the coulomb term. Though for systems, the coulomb term will be extremely dominant in comparison to confining term, and should not be treated perturbatively. However, seeing our mass spectral results for systems, it may not be so unreasonable to treat the coulomb term perturbatively for systems. This is specially so for orbital excitations of these states, where the centrifugal effects  ensure that the separation is large enough to feel the effect of confining term more strongly than the coulomb term. We further wish to state that some of earlier works [34, 35, 36] have treated the OGE(coulomb) term perturbatively for charmed mesons and baryons, while some works [37, 38] did not take into account the importance of coulomb term for heavy quarkonium systems.
Thus, in the present paper, the coupled Salpeter equations for scalar (), and axial vector () quarkonia are first shown to decouple for the confining part of interaction, under heavy-quark approximation, and the analytic forms of mass spectral equations are worked out, which are then solved in approximate harmonic oscillator basis to obtain the unperturbed wave functions for various states of these quarkonia. We then incorporate the one-gluon-exchange perturbatively into the unperturbed spectral equation, and obtain the full spectrum. The wave functions of scalar () quarkonia, are then used to calculate their two-photon decay widths. We further extend the mass spectral calculations of pseudoscalar (), and vector () quarkonia in  with the perturbative inclusion of one-gluon-exchange effects. The approximations used in this analytic treatment of the confining interaction are shown to be fully under control. This work is an improvement on our earlier work  on mass spectral problem for pseudoscalar and vector states of quarkonia on lines of some of the earlier works [31, 32, 33], where we used only the confining interaction.
A quarkonium state is classified by quantum numbers, , where , parity, , and charge conjugation, . With this classification, while the lowest states, are present in (pseudoscalar), the lowest state (), and the second orbitally excited state are present in (vector) quarkonia. However, the first orbitally excited () states are present in (scalar) and (axial-vector) quarkonia. The same holds true for their radial excitations.
This paper is organized as follows: Section 2, deals with the mass spectral calculations of ground and excited states of quarkonia. Section 3 deals with the mass spectral calculation of ground and excited states of , and quarkonia, while Section 4 deals with the mass spectral calculations of ground and excited states of quarkonia. Section 5 deals with the calculation of two-photon decay widths of . Section 6 deals with numerical results and discussions.
2 Mass spectra of scalar quarkonia
In the center of mass frame, where , we can write the general decomposition of the instantaneous BS wave function for scalar mesons , of dimensionality as
where it can be shown by use of a power counting rule proposed in [23, 25] that the Dirac structures associated with the amplitudes and are leading, and will contribute maximum to calculation of any scalar meson observable, while those associated with , and are sub-leading. We now use the two constraint equations in the 3D Salpeter equations, Eq.(16) of , to reduce the four scalar functions into two independent scalar wave functions. For equal mass system, these two equations are reduced to
Applying the above constraint conditions in Eq.(2) to wave function in Eq.(1), we rewrite the relativistic wave function of the state in the form
Here, it is to be noted that by use of the above constraint equations, we have reexpressed in terms of . Thus, the Instantaneous BS wave function of state is determined by only two independent functions, and . Putting the wave function in Eq.(3) above, along with the projection operators defined in Eq.(13) of , into the first two Salpeter equations, Eq.(16) of , and by taking the trace on both sides, we obtain the equations:
Solution of these equations needs information about the BS kernel, [25, 30], which is taken to be one-gluon exchange like as regards the colour () and spin () dependence, and has a scalar part , written as,
Thus, the scalar part of the kernel involves both the OGE term , arising from the one-gluon exchange, as well as the confining term, . We first ignore the term, and work only with the confining part, in Eq.(5), and write Eqs.(4) as,
where the spin dependence of the interaction is contained in the factor, . The scalar part of the confining potential (that involves the colour factor ), is taken to be  , where , and . Here is the part of without the delta function. To handle these equations, we first integrate these equations over , performing the delta function integration that arises due to presence of in the integrand, and get two coupled algebraic equations with on RHS. To decouple them, we first add them. Then we subtract the second equation from the first equation, and get two algebraic equations which are still coupled. Then from one of the two equations so obtained, we eliminate in terms of , and plug this expression for in the second equation of the coupled set so obtained, to get a decoupled equation in . Similarly, we eliminate from the second equation of the set of coupled algebraic equations in terms of , and plug it into the first equation to get a decoupled equation entirely in . Thus, we get two identical decoupled equations, one entirely in , and the other that is entirely in, . The calculation up to this point is without any approximation. However, we notice that if we employ the approximation, on RHS of the two algebraic equations 111In principle, we should solve the decoupled algebraic equations numerically. However, this would not give explicit dependence of the mass spectra on the principal quantum number N, and nor will this give the explicit algebraic forms of wave functions that can be employed to do analytic calculations of various transition amplitudes for different processes. Our approach may lead to a little loss of numerical accuracy, but it does lead to a much deeper understanding of the mass spectral problem. The approximations used by us have been shown to be totally under control. The plots of our algebraic forms of wave functions for scalar, pseudoscalar, vector and axial vector quarkonia are very much similar to the corresponding plots of wave functions in  obtained by purely numerical methods, which validates our approach.(this is justified since in the confinement region, the relative momentum between heavy quarks in the bound state can be considered small, since the heavy-quark is expected to move with non-relativistic speeds (see Ref.), and these quarks can be treated as almost on mass shell), these decoupled equations can be expressed as:
We see that we get two identical decoupled equations which resemble the harmonic oscillator equations, but for the term involving on the right side of these equations.
We wish to mention that in recent studies on mass spectra of pseudoscalar and vector quarkonia , it was seen that good agreement with data on masses and various decay constants/decay widths of ground and excited states of , and is obtained for input parameters, , GeV., GeV., and , along with the input quark masses GeV. With these numerical values of input parameters, we try to determine the numerical values of , and associated with the terms involving and respectively, for scalar mesons and in RHS of Eqs.(12), and their percentage ratio in Table 1 below, where it can be seen that and hence the second term on RHS of Eqs.(7) contributes , than the first term on RHS of this equation for , so that it can be dropped.
Thus the RHS of both equations in Eq.(7) has only the term, . Now, putting the spatial part, in Eq.(7), the wave functions , and then satisfy identical 3D BSE for equal mass heavy scalar mesons:
where we have used , with . Thus the solutions of these equations, . With the use of the above equality of amplitudes, and reexpressing in terms of , the complete wave function can be expressed as:
We can reduce this equation into the equation of a simple quantum mechanical -harmonic oscillator with coefficients depending on the hadron mass , and total quantum number . The wave function satisfies the 3D BSE:
Now, with the use of leading Dirac structures, we can to a good approximation (as in case of pseudoscalar and vector mesons ) express . This is due to the fact that , due to ( being the unit matrix) being the most leading Dirac structure in the scalar meson wave function, and the terms with have negligible contributions, in heavy quark limit, and can be dropped. Thus the above equation can be put in the form
which can in turn be expressed as,
where, , and with total energy of the system expressed as,
Putting the expression for the laplacian operator in spherical coordinates, we get
where is the orbital quantum number with the values , corresponding to wave states respectively. This is a 3D harmonic oscillator equation, whose solutions can be found by using power series method. Assuming the form of the solutions of this equation as, , the above equation can be expressed as,
The eigen values of this equation can be obtained using the power series method as,
with . Thus, to each value of will correspond a polynomial of order in , that are obtained as solutions of Eq.(15). The odd parity normalized wave functions thus derived are:
The plots of wave functions for scalar quarkonia, are given in Fig. 1 below.
The mass spectrum of ground and excited states for equal mass heavy scalar () mesons is written as:
where the orbital quantum number, . Now, treating the mass spectral equation, Eq.(12) as the unperturbed equation, with the unperturbed wave functions, for scalar mesons in Eq.(17), we now incorporate the OGE (coulomb) term in this equation.
Then the above mass spectral equation can be written as:
Treating the coulomb term as a perturbation to the unperturbed mass spectral equation, we can write the complete mass spectra for the ground and excited states for equal mass heavy scalar () mesons using first order perturbation theory as:
with , where is the expectation value of between the unperturbed states of given quantum numbers (with ) for scalar mesons, and has been weighted by a factor of to have the coulomb term dimensionally consistent with the harmonic term. Its expectation values for the , and states are:
From the mass spectral equation, one can see that, the mass spectra depends not only on the principal quantum number , but also the orbital quantum number . We are now in a position to calculate the numerical values for mass spectral of heavy equal mass scalar meson with the input parameters of our model. The results of mass spectral predictions of heavy equal mass scalar mesons for both ground and excited states with the above set of parameters is given in table 2.
We now derive the mass spectral equations with the incorporation of the OGE (Coulomb) term for pseudoscalar and vector quarkonia, and obtain their solutions in the next section (the preliminary calculations using only the confining part of interaction were done in ).
3 Mass spectral equation for pseudoscalar (), and vector () quarkonia
For pseudoscalar (P), and vector (V) quarkonia, the general decomposition of instantaneous BS wave function of dimensionality in the center of mass frame is given in Eqs. (18), and (25) respectively in . Putting this wave function in Eqs.(18)(for P-mesons), or Eq.(25) (for V-mesons) into the Salpeter equations leads to two coupled equations in leading amplitudes (, and ) for P-mesons, and (, and ) for V-mesons. Decoupling them in the heavy-quark limit, leads to Eqs.(37) (for both P and V-mesons) of  given as,
with non-perturbative energy eigen functions for , and for states obtained as solutions of above spectral equation in approximate harmonic oscillator basis for pseudoscalar quarkonia (for states ) and vector quarkonia (for states ) as Eq.(41) in , for the perturbative calculation of the short ranged one-gluon-exchange interaction shown later in this section.
The mass spectra of vector charmonium and bottomonium states using the above spectral equation was found to have degenerate , and states . However with the incorporation of the OGE (Coulomb) term the mass spectral equation can be written as:
Now, treating the coulomb term as a perturbation to the unperturbed mass spectral equation, Eq.(22), and treating the wave functions in Eq.(41) of , as the unperturbed wave functions, we can write the complete mass spectra of ground (1S) and excited states for equal mass heavy pseudoscalar () and vector () mesons respectively using the first order degenerate perturbation theory as:
where (has again been weighted by a factor of , as in scalar () quarkonia in previous section) is the matrix element of between unperturbed states in Eq.(41) of  of given quantum numbers , and , (with , and for pseudoscalar mesons, while with for vector mesons). It is to be noted that, connects only the equal parity states with the same values of quantum numbers . The only non-vanishing matrix elements of the perturbation between states with the given quantum numbers and are listed below:
The non-zero values of given above not only lead to the lifting up of the degeneracy between the and levels with the same principal quantum number in vector quarkonia, but also leads to bringing the masses of different states of vector and pseudoscalar quarkonia closer to data, as can be seen from the mass spectral results for pseodoscalar (), and vector () quarkonia, which are compared with the experimental data , and other models for each state (where ever available), as given in Tables 3 and 4 respectively as:
|BSE - CIA||Expt.||Pot. Model||QCD sum rule||Lattice QCD|||
|BSE - CIA||Expt.||Rel. Pot. Model||Pot. Model||BSE||Lattice QCD|
4 Mass spectral equation for axial vector quarkonia
The general form for the relativistic Salpeter wave function of state with can be expressed as in [17, 19]. In the center of mass frame, we can then write the general decomposition of the instantaneous BS wave function for axial vector mesons of dimensionality as:
With use of our power counting rule [23, 24], it can be checked that the Dirac structures associated with amplitudes and are , and are leading, and would contribute maximum to any axial vector meson calculation. Following a similar procedure as in the case of scalar mesons, we can write the Salpeter wave function in terms of only two leading Dirac amplitudes , and . Plugging this wave function together with the projection operators into the first two Salpeter equations in Eq.(16) of , and taking trace on both sides and following the steps as for the scalar meson case, we get the coupled integral equations in the amplitudes , and :
To decouple these equations, we follow a similar procedure as in the scalar meson case, and get two identical decoupled equations, one entirely in , and the other that is entirely in, . In the limit, on RHS, and due to the fact that for axial quarkonia again, these equations can be expressed as:
where, it can be checked that both and satisfy the same equation and hence we can approximately write . Thus,
where . Now, with the use of leading Dirac structures, we can again to a good approximation express . This is due to the fact that . This mass spectral equation has the same form as the mass spectral equation for scalar quarkonia, Eq. (10), except for the value of , which is different from . Thus the above equation can be put in a similar form as Eq.(12) (for scalar case), except that inverse range parameter , where, , and the wave function, . The mass spectral equation for would then exactly resemble Eqs.(18) for case, and thus the unperturbed wave functions for would then have the same algebraic form as in Eqs.(17), but with :