# Approximate Hamiltonian for baryons in heavy-flavor QCD

## Abstract

Aiming at relativistic description of gluons in hadrons, the renormalization group procedure for effective particles (RGPEP) is applied to baryons in QCD of heavy quarks. The baryon eigenvalue problem is posed using the Fock-space Hamiltonian operator obtained by solving the RGPEP equations up to second order in powers of the coupling constant. The eigenstate components that contain three quarks and two or more gluons are heuristically removed at the price of inserting a gluon-mass term in the component with one gluon. The resulting problem is reduced to the equivalent one for the component of three quarks and no gluons. Each of the three quark–quark interaction terms thus obtained consists of a spin-dependent Coulomb term and a spin-independent harmonic oscillator term. Quark masses are chosen to fit the lightest spin-one quarkonia masses most accurately. The resulting estimates for bbb and ccc states match estimates obtained in lattice QCD and in quark models. Masses of ccb and bbc states are also estimated. The corresponding wave functions are invariant with respect to boosts. In the ccb states, charm quarks tend to form diquarks. The accuracy of our approximate Hamiltonian can be estimated through comparison by including components with two gluons within the same method.

## 1 Introduction

Theoretically precise and phenomenologically accurate description of triply-heavy baryons as quantum states of quarks and gluons requires the formulation of QCD that satisfies several conditions. It ought to include a construction of the theory ground state, or vacuum, whose excitations are the quanta of quark and gluon fields. Since the canonical QCD Hamiltonian involves singularities and requires regularization, the theoretical formulation should include a mathematically precise renormalization procedure with clear physical interpretation, as a foundation of its predictability. The condition that individually quarks and gluons are not observed, implies that the formulation should allow for inclusion of confinement. The fact that heavy baryons may participate in processes whose description involves motion with speeds close to the speed of light, forces the formulation to be relativistic. In particular, it must guarantee description of baryons that have energies very much larger than their masses. Finally, knowing technical complexity of QCD and realizing that exact solutions are unlikely, it is necessary to demand that the formulation includes an outline of a process of successive approximations that stand a chance of systematically improving precision and accuracy of approximate solutions for observables. This article concerns a pilot application of an approach to heavy-quark QCD that in principle satisfies these requirements.

Quark model represented baryons as bound states of three quarks, e.g. see [1, 2]. In QCD, baryons are instead superpositions of states of quanta of quark and gluon fields. A priori, the number of quanta varies from three to infinity, across an infinite set of components. These quanta may have momenta ranging from zero to infinity. Their interactions diverge with their momenta. This article contributes to a development of a Hamiltonian approach to QCD that appears capable of filling the gap between the complex quantum-field picture and simple quark-model picture for hadrons [3]. Most succinctly, we illustrate a new method for solving the bound-state problem in canonical quantum field theories with asymptotic freedom in terms of its first application to the case of baryons in heavy-flavor QCD. Our method involves three consecutive steps: we solve our renormalization group equation for the front form (FF) Hamiltonian of the theory using the concept of effective particles in the Fock space; we reduce the resulting heavy-baryon eigenvalue problem for low-mass eigenstates to the eigenvalue problem solely for their Fock component of three effective quarks, using a gluon mass ansatz to account for the Fock components with more effective gluons than one; and we draw a qualitative sketch of the estimated low-mass heavy baryons spectrum that follows from the dominant mechanism of binding, while spin and other relatively small corrections to the effects of dominant interactions require future more elaborate calculations of higher-order using the same method. The theoretical challenge our method thus addresses is how to represent states of heavy baryons in terms of the Fock-space wave-functions for quarks and gluons that are invariant with respect to Lorentz boosts. The new results that our pilot study yields for heavy baryons, including the approximate analytic formulae for their mass eigenvalues and corresponding boost-invariant wave functions that can be used in phenomenology of their production and detection, are thus derived in full detail from the heavy-flavor QCD supplied with our gluon mass ansatz. However, our pilot study of the low-mass triply heavy baryons involves severe simplifications. Similar simplifications are made in other approaches but without using the concept of effective particles that we introduce. In our approach, all the simplifications we make in the pilot study can be systematically removed within the same method while increasing its precision, as will be explained later on, but we do not address the question if the RGPEP may be used to derive the pNRQCD, which would require comparison of the dimensional regularization renormalization group equations with equations of the RGPEP, see the pertinent footnote on p. 456 in Ref. [4].

We limit the theory to quarks that have masses much greater than \(\varLambda _{QCD}\), excluding the top quark, and we consider the weak coupling limit [5]. Creation of quark–antiquark pairs is neglected. Components with more than one gluon are eliminated, by assuming that their dominant effect in the component with one gluon is that the gluon has a mass, allowed to be a function of the gluon kinematic relative momentum with respect to the quarks. We use second-order perturbation theory to derive the resulting effective Hamiltonian for baryons that only acts in the component with three quarks. We compare the quark-quark interaction terms in this Hamiltonian to similar terms in the Hamiltonian that only acts in the quark–anti-quark component in quarkonia, previously derived using the same method [3]. Masses of heavy baryons are estimated by solving the resulting eigenvalue equation in the nonrelativistic limit. The parameters involved (the running coupling and the quark masses) are chosen using heavy quarkonia experimental data. In this way, our estimates for baryon masses contain no new parameters. More precisely, the coupling constant is extrapolated from a formally infinitesimal value of weak-coupling limit to the value implied at the quark-mass scale by the known coupling constant at the scale of *Z*-boson mass. Quark masses are adjusted to the known spectra of heavy quarkonia. The scale parameter for hadrons built from different flavors is fixed by a linear interpolation between its one-flavor values. This will be explained in detail later.

The concept of effective-gluon mass that we use is explained in Sect. 2, followed by a brief outline of our method in Sect. 3. The method is called the renormalization group procedure for effective particles (RGPEP). Our concept of the gluon mass differs from the concepts discussed in the literature, see, e.g., Refs. [6, 7, 8]. The second-order baryon eigenvalue problem is outlined in Sect. 4. Details of the effective quark–quark interaction terms in *ccc* and *bbb* baryons, implied by the gluon mass, are described in Sect. 5, including a comparison with the case of heavy quarkonia. Section 6 extends the calculation to the *ccb* and *bbc* baryons. The resulting estimates for baryon masses are described in Sect. 7. Comments concerning the RGPEP calculation of effective Hamiltonians in QCD in orders higher than second and beyond perturbation theory conclude the paper in Sect. 8. Details of our fit to the spectra of quarkonia are described in “Appendix A”. Values of the RGPEP scale parameter we use are listed in “Appendix B”. “Appendix C” discusses dependence of harmonic oscillator frequencies on the scale parameter. “Appendix D” provides a detailed description of the baryon wave functions that are used in our estimates and “Appendix E” presents explicit formulas for the associated heavy-baryon masses.

## 2 Assumption of gluon mass

*s*-wave \(c \bar{c}\) or \(b \bar{b}\) mass splittings are on the order of half GeV and they do not decrease with the excitation number as they would if the interaction was purely of the Coulomb type, like in QED with massless photons. The mass splittings in the heavy baryon spectrum are also not expected to rapidly decrease with the excitation number. To describe physical splittings, excitations of the gluon field must involve considerable energy. This requirement can be addressed using the concept of a gluon mass [3].

*t*is the renormalization group parameter. It is useful to think about it as \(t=s^4\), where

*s*has an interpretation of the size of effective particles. Note that the effective-particle size

*s*, as a parameter of renormalization group procedure in quantum field theory, is not mathematically related to the phenomenological size-parameters for gluons, such as, for example, in Ref. [12]. Canonical gluons are considered point-like. Instead of using canonical gluons, the goal is to represent a heavy baryon by a superposition

*s*. We introduce the gluon mass in the effective QCD eigenvalue problem for heavy baryons within the same computational scheme that we previously applied to heavy quarkonia [3].

^{1}

Our leading principle is that the gluon mass \(\mu _t\) is the minimal price we have to pay for limiting the expansion in Eq. (2) to the first two terms. Such limitation makes sense because interactions in the Hamiltonian \(H_t\) contain vertex form factors. The form factors are obtained by solving the RGPEP Eq. (16), with the initial condition provided by the canonical QCD Hamiltonian with regularization and counterterms. These form factors cause that interactions cannot change invariant masses of component states by amounts exceeding 1 / *s*. Therefore, the effective gluons of size *s* cannot be as copiously produced as the point-like gluons can in the canonical representation of QCD. It is plausible that inclusion of a few effective components is sufficient to accurately describe a heavy-baryon solution. Regarding attempts of relating our effective gluon quanta to gluon field degrees of freedom in other approaches, it would be of general interest to find out if lattice studies, such as in Refs. [13, 14], can introduce interpolating operators that are capable of identifying properties of the same degrees of freedom.

Although the number of gluons \(G_t\) that need to be included in the effective representation of a low-mass solution to the QCD eigenvalue problem is expected to be limited, direct inspection shows that inclusion of even a few components still leads to a mathematically difficult equations for their coupled-channel dynamics. The results presented in this paper follow from the effective eigenvalue problem that is obtained by using the hypothesis that the contribution of all components other than \(|3Q_t\rangle \) and \(|3 Q_t \, G_t \rangle \) may be approximated by inclusion of a gluon mass, \(\mu _t\) for the gluon in component \(|3Q_t \, G_t \rangle \).

The mass assumption is falsifiable by extending the calculation to explicitly include more components and relegating the gluon mass ansatz to states with more gluons than one. The purpose would be to verify if the finite value of the RGPEP parameter *s* on the order of quark Compton wave-length is sufficient to prevent the spill of probability to states with many gluons, especially when the coupling constant is small. The latter situation is expected to occur for the quark masses that are much greater than \(\varLambda _{QCD}\). This is precisely the reason for us to study the dynamics of gluons using the RGPEP first in the context of heavy-flavor QCD. In order to simplify the problem and thus increase a chance of understanding the dynamics of effective gluons whose masses are likely to be much larger than \(\varLambda _{QCD}\), we exclude from the theory the quarks that have masses much smaller than \(\varLambda _{QCD}\). If the latter were included in the theory, they could appear in large numbers in the effective Fock-space basis and complicate the dynamics, as massless gluons do.

## 3 RGPEP for hadrons

The RGPEP provides equations for calculating the renormalized Hamiltonian \(H_t\) from the canonical one that includes regularization and counterterms. It is also used to calculate the counterterms. Eigenstates of \(H_t\) define hadrons in terms of effective particle basis in the Fock space. We first consider QCD of only one flavor of heavy quarks, useful in discussing dynamics in baryons made of quarks of one flavor. The case of baryons made of two types of heavy quarks is discussed in Sect. 6. We calculate \(H_t\) using expansion in powers of a formally infinitesimal coupling constant, up to terms of second-order. Results of our second-order calculations are later compared with results obtained in quark models and in lattice approach to QCD.

### 3.1 Canonical Hamiltonian

*c*stand for their spins and colors, respectively. We omit the hats and normal ordering symbols in further formulas. In second-order calculation, only two interaction terms count, the quark–gluon interaction,

### 3.2 Regularization

*free*, or kinetic term is

### 3.3 Renormalized Hamiltonian

*initial Hamiltonian*, since it provides an initial condition for solving the RGPEP equation. Its solution defines a family of renormalized Hamiltonians \(H_t\), which are written in terms of the operators \(q_t\) that create or annihilate particles of size \(s>0\), \(t=s^4\). The latter operators are defined by means of a unitary transformation \({\mathcal {U}}_t\),

*t*. \({\mathcal {G}}_t\) is called a generator of the RGPEP. It is set to

*s*are exponentially suppressed [11], cf. Eq. (20).

*i*. Moreover, the gluon momentum is

*z*.

The differences between Eq. (24) and the corresponding equations for quarkonia [3], are: quark current \(j_{22'}\) instead of anti-quark current \(\bar{j}_{2'2} = \bar{v}_{2'}\gamma ^\mu v_2\), transposition of matrix \(t^4_{22'}\), overall sign difference, color factor \(-2/3\) instead of 4 / 3 (when acting on a color singlet state) and symmetrization factor 1 / 2 that is absent in quarkonia.

### 3.4 Bound-state eigenvalue problem

*t*. This holds for \(\lambda = 1/s\) much larger than the scale of \(\varLambda _\mathrm{QCD}\) in the RGPEP scheme. We formally consider

### 3.5 Gluon mass ansatz in the effective eigenvalue problem

*R*succinctly described in [25]. Up to terms order \(g^2\) and using notation

*r*and

*l*for right and left states in the matrix elements, one obtains

## 4 3Q eigenvalue problem

### 4.1 Small-*x* dynamics

The interaction kernel \(U_{t\,\mathrm eff}(12;1'2')\) in Eq. (39) can be written in terms of relative momentum variables \(x_{1/12}\), \(\kappa _{1/12}^\perp \) and \(x_{1'/12}\), \(\kappa _{1'/12}^\perp \) (that is momenta relative to pair 12). The resulting expression has the same structure as in our quarkonium analysis [3], with the color factor 2 / 3 instead of 4 / 3. The gluon mass ansatz in baryons can be different from the one in quarkonia, since it is a function of gluon momentum relative to the three-quark subsystem instead of quark–antiquark subsystem. The small-*x* singular factors in the interaction do not produce divergences for the same reason as in quarkonia. The gluon-exchange integral is finite because we assume that the gluon mass ansatz vanishes when \(x_5\rightarrow 0\). An example of required behavior in quarkonia is \(\mu ^2 \sim x_5^{\delta _\mu } \kappa _5^2\). In the notation used for baryons, the same behavior is described by \(\mu ^2 \sim x_{4/12}^{\delta _\mu } \kappa _{4/12}^2\) when \(x_{4/12}\rightarrow 0\). Let us assume that \(\mu ^2 \sim x_4^{\delta _\mu } \kappa _4^2\) in the limit \(x_4\rightarrow 0\). When \(x_4\) goes to zero, so does \(x_{4/12} = x_4/(x_1+x_2)\), and \(\kappa _{4/12}^\perp \approx \kappa _4^\perp \). Therefore, \(\mu ^2 \sim x_4^{\delta _\mu } \kappa _4^2\) implies \(\mu ^2 \sim x_{4/12}^{\delta _\mu } \kappa _{4/12}^2\). The same reasoning applies to every pair of the quarks that exchange a gluon. Similarly, in the quark self-interaction terms, the integration variables are \(x_{4/i}\) and \(\kappa _{4/i}^\perp \). In the small-\(x_4\) limit, \(\kappa _{4/i}^\perp \approx \kappa _4^\perp \) and \(x_{4/i} \approx x_4 \rightarrow 0\). Therefore, mass terms are finite when the gluon mass ansatz vanishes properly when \(x_4\rightarrow 0\).

## 5 Effective interactions in the nonrelativistic limit

*m*, take the NR limit and obtain

*V*and

*W*are similar to the ones in the quarkonium case [3].

*m*, \(\lambda \) and \(\alpha \) are the same for mesons and baryons built from one flavor of heavy quarks. This result is very close to the ratio 5/8 suggested by models that employ the concept of gluon condensate in vacuum [27] or only inside hadrons [22].

The oscillator interaction may appear to be in contradiction with the linear confinement picture in QCD. However, the eigenvalues of the FF Hamiltonian are the baryon masses squared, in distinction from the instant form (IF) Hamiltonian eigenvalues that are the baryon energies, reducing to the baryon masses only for bound states at rest. At large distances between quarks, the quadratic potential in the FF corresponds to the linear potential in the IF of Hamiltonian dynamics [26].

## 6 Two flavors of heavy quarks

Several new elements appear when one of the three quarks, say quark 3, is of different flavor than the other two. Besides smaller particle-exchange symmetry and the fact that \(\beta _1\ne \beta _3\ne 1/3\), a new feature emerges that the NR effective quark masses are modified.

The reason for NR mass modification is that, when we deal with two different flavors of quarks, the constant term that cancels completely in Eq. (59) for the same flavor no longer does so for different flavors. A finite function of *x* and \(\kappa ^\perp \) is left and it multiplies \(\psi (1,2,3)\). This effect is small, but in principle ought to be considered. The correction shifts the minimal invariant mass squared value around which the NR approximation is obtained. Namely, the optimal values of \(\beta _i\) around which one expands are slightly altered, cf. Eqs. (48) to (51). The shifts spoil the rotational symmetry of the second-order Coulomb and harmonic oscillator potentials. For *b* and *c* quarks, the deviation from spherical symmetry appears to be on the order of a few percent. It depends on the gluon mass.

This effect is certainly going to change in calculations of higher order than second, because it depends on the gluon mass ansatz and the ansatz will be replaced by theory. Since this effect is relatively small, we neglect it in what follows. Apart from the neglected effect, the Coulomb interactions between quarks are not altered when flavors differ.

*w*is omitted, one obtains

*E*,

## 7 Sketch of triply heavy baryon spectra

In this paper, we focus on qualitative features of our method, and test its capability to describe baryons. Therefore, we only sketch the spectrum of heavy baryons that follows from QCD of quarks *b* and *c* including our pilot simplifications, the latter being gradually removable increasing the order of weak coupling expansion for \(H_{\mathrm{eff} \, t}\) and number of effective Fock components in the eigenvalue problem of \(H_{\mathrm{eff} \, t}\).

In order to solve the baryon bound-state problem one needs to fix \(\alpha \), \(m_c\) and \(m_b\). To estimate these quantities in the RGPEP scheme, we use data for heavy quarkonia. The first issue one needs to deal with is the strong dependence of oscillator frequencies on \(\lambda \). If our calculations of \(H_t\) and its eigenvalues were exact, the observables we obtain would be independent of \(\lambda \), which hence could be chosen arbitrarily. Since we solve the RGPEP equation only up to order \(\alpha \) and we introduce a gluon mass ansatz to reduce the eigenvalue problem to the hadron dominant Fock component, the effective dynamics we obtain may provide a reasonable approximation only in a certain window of values of \(\lambda \) (see particularly Fig. 4 in Ref. [28] and Fig. 4 in Ref. [29]).

We discuss the approximate hadron spectra that our method produces from heavy-quark QCD using the assumption that \(\lambda \sim \sqrt{\alpha } \ m_Q\), where \(m_Q\) is a suitable quark mass parameter (we introduce its definition in Sect. 7.1). When \(\alpha \) is sufficiently small, this assumption fulfills constraints of Eq. (32) and, moreover, for quark–antiquark system it ensures that \(\lambda \gg k_B \sim \alpha \,\mu \), where \(k_B\) is the strong Bohr momentum and \(\mu \) is the quark reduced mass, which in turn ensures that RGPEP form factors do not influence significantly the eigenvalues of \(H_{\mathrm{eff}\,t}\). For example, if \(\lambda \) were equal \(\alpha \mu \ll \sqrt{\alpha }\mu \), then the form factor of Eq. (57) for \(K = 0\) and \(K' = \alpha \mu \) would be \(f = e^{-16}\), which is practically zero, no matter how small the coupling constant is, and the form factor would play a significant role in the eigenvalue problem. On the other hand, if \(\lambda = \sqrt{\alpha }\mu \), then, for the same *K* and \(K'\) as before, \(f = e^{-16\alpha ^2}\), which is practically one for sufficiently small \(\alpha \), and the form factor is invisible in the first approximation. Such assumption is well justified in QED, where the Schrödinger equation with simple local Coulomb potential gives very good first approximation to the Hydrogen spectrum. Please note, however, that the form factors are necessary in higher order calculations, because they make interactions, which otherwise would be singular, like spin-spin interactions, finite, and actually small in comparison to the leading binding effects that we describe below.

Note also that keeping \(\lambda \) proportional to the square root of \(\alpha \) secures proportionality of the resulting hadron binding energies to \(\alpha ^2\) [5], which resembles analogous scaling in QED. This scaling is maintained with our oscillator terms because their frequencies emerge proportional to \(\alpha ^2\). Knowing that the observed low-mass quarkonium spectra can be characterized as intermediate between the Coulomb and oscillator spectra [10], we expect that the harmonic oscillator frequencies obtained from QCD may be comparable in size with the strong-interaction Rydberg-like constant \(R = \mu (4\alpha /3)^2/2\).

Finally, a comment is in order regarding our use of perturbation theory for calculating effective Hamiltonian while the coupling constant is to be extrapolated from an infinitesimal to a finite value, Formally, the whole calculation is valid only in the limit of infinitesimal coupling constant (or \(\varLambda _\mathrm{QCD} \rightarrow 0\)), because only then one can consider perturbative terms, e.g., the divergent mass counterterm, as small perturbations. Therefore, we stress that we assume that for the values of \(\alpha \) between about one quarter and one half (depending on the system under consideration) the functional form of effective Hamiltonians as a function of \(\alpha \), does not change significantly [28, 29].

The numbers we obtain are listed including four or even five significant digits only because the data we approximately reproduce [9] provide that many digits. We ignore data error bars and use our analytic expressions. The Coulomb effects are estimated in first-order perturbation theory around the relevant oscillator solutions. Only diagonal matrix elements need to be considered because inclusion of non-diagonal matrix elements produces relatively small effects that do not change the main features of lowest-mass heavy-baryon spectrum that we sketch. For example, the ground states of *ccc* and *bbb* shift by about 31 MeV and 47 MeV respectively when instead of first order perturbation theory one diagonalizes Hamiltonian matrix in the basis of harmonic oscillator eigenstates with excitation energy up to \(4\omega _\text {baryon}\). These corrections are small in comparison with the expected effects of spin dependent interactions, which we neglect (but we do include effects due to the Pauli exclusion principle for fermions). Therefore, given the simplifications we have made in the pilot application of our method to solving heavy-flavor QCD, we provide a sketch of the low-mass hadron spectra obtained from first-order perturbation theory around the oscillator spectrum implied by the assumption that gluons develop a mass.

### 7.1 Adjustment of parameters \(\alpha \), \(m_b\), \(m_c\) and \(\lambda \)

*b*and

*c*, ignoring

*u*,

*d*,

*s*and

*t*. Demand that \(\alpha = 0.1181\) for \(\lambda = M_Z = 91.1876\) GeV, would enforce the RGPEP value of \(\varLambda _\mathrm{QCD} = 371\) MeV and we use this value. The resulting spectra do not change significantly when we change \(n_f\) in the range from 2 to 5.

The quark masses are assumed to be independent of \(\lambda \) because their dependence is not known yet in the RGPEP. Confinement poses a conceptual difficulty concerning the definition of quark mass [9]. Quantitative estimates of quark masses would need the RGPEP calculation to at least fourth order while we consider only second. Formulas for running masses of quarks in other approaches, like in Eq. (9.6) in [9], do not concern mass terms in the FF Hamiltonian \(H_t\). At the current level of crude approximation and not knowing the masses precisely, we assume that \(m_b\) and \(m_c\) can be treated as constants in the range of values of \(\lambda \) that we use in fitting data.

*a*and

*b*are fitted to the spectra of heavy quarkonia. Separate fits for a set of \(b\bar{b}\) states and a set of \(c \bar{c}\) states give us most suitable \(\lambda _{b \bar{b}}\) and \(\lambda _{c \bar{c}}\), and quark masses \(m_b\) and \(m_c\). This set of numbers allows us to fix values of

*a*and

*b*in the linear formula of Eq. (72). With

*a*and

*b*fixed, we test Eq. (72) by comparing our theoretical spectrum of \(B_c\) particles, computed for \(\lambda _{bc}\) given by Eq. (72), with experimental data. The agreement is satisfactory, cf. Sect. 7.2. The adjustment of constants

*a*and

*b*reflects the current lack of knowledge of the values of \(\lambda \) at which one can most accurately approximate different hadron eigenvalue problems using merely their lowest Fock components and gluon mass ansatz. Details of our fits of two quark masses, \(m_c\) and \(m_b\) at most suitable values of \(\lambda _{b \bar{b}}\) and \(\lambda _{c \bar{c}}\), are described in “Appendix A”.

*bbb*and

*ccc*spectra resemble results of other approaches, see below.

Our estimates are quite crude. We ask two questions. One is if the oscillator terms that follow from the assumption of gluon mass are capable of providing a reasonable first approximation to heavy hadrons. Provided that in the case of heavy quarkonia the answer is yes, the other question is what character of the heavy baryons spectrum one expects using the assumption that effective gluons develop a mass. To address these qualitative questions, we ignore the BF spin-dependent terms and we estimate strong-Coulomb effects by evaluating expectation values of the corresponding interaction terms in the oscillator eigenstates. Details of unperturbed baryon wave functions are described in “Appendix D”. Comparison with other approaches, including lattice estimates, suggests that our extremely simple oscillator picture and thus possibly also the gluon mass hypothesis, appear reasonable. Reliable estimates of better accuracy require fourth-order solution to the RGPEP Eq. (16).

### 7.2 Masses of quarkonia

### 7.3 Estimates of masses of heavy baryons

The fit to quarkonia described in Sect. 7.2 establishes optimal values of \(\lambda \) for all baryons. Values of the coupling constant are obtained from Eq. (71). The optimal values we obtain for these parameters are listed in “Appendix B”. The resulting masses of heavy baryons are shown in Fig. 4. Labels of states describe internal orbital motion of quarks, where the first part of a label corresponds to the motion of quark 1 with respect to quark 2 and the second part corresponds to the motion of quark 3 with respect to the pair of quarks 1 and 2. For example, in the state 1*P*1*S*, the pair 12 is in a *p*-wave without radial excitation, while the quark 3 in its motion with respect to the pair 12 is in an *s*-wave state without radial excitation. In 1*S*2*S*, both 1 with respect to 2 and 3 with respect to 12 are in an *s*-wave state but the latter is radially excited. States *A*, *B*, *C* and *D* in *ccc* and *bbb* correspond to the second excitation of harmonic oscillator with excitation energy \(2\omega \) above the ground state (\(\omega \equiv \omega _\text {baryon}\)). These states have spin-momentum wave functions that are symmetrized in a way due for fermions in colorless states. Details of the harmonic oscillator basis wave functions are described in “Appendix D”. Analytical formulas for masses of baryons are given in “Appendix E”.

*bbb*and

*ccc*baryons agree well with model calculations [31, 32, 33, 34, 35, 36, 37] including quark–diquark [38] and hypercentral approximations [39, 40], bag models [41, 42, 43], Regge phenomenology [44, 45], sum rules [46, 47, 48, 49], pNRQCD [50], Dyson–Schwinger approach [51, 52] and lattice studies [53, 54, 55, 56, 57, 58, 59], where comparison is available. As an example of comparison, we note that the ground state of

*ccc*is assigned masses from 4733 to 4796 MeV, by different lattice calculations, with an average of 4768 MeV. Our result is 4797 MeV, differing by 29 MeV, or 0.6 % from the average. For

*bbb*, the average of two lattice results we have identified is 14369 MeV, and our result is 14346 MeV, which is 23 MeV difference, or 0.2 %. These comparisons refer to Table I in Ref. [40] that summarizes results of calculations of masses of \(\varOmega _{ccc}\) and \(\varOmega _{bbb}\) reported in twenty different articles. Ground state of

*bbc*is also very close to the lattice result [53]. In contrast, our

*ccb*differs by about 300 MeV from the lattice. We comment on this feature below. Comparison with lattice calculations reported in Ref. [54] shows that our splittings in

*bbb*differ only by about 10%. In case of

*ccc*[56], the difference of splittings does not exceed 20%. This degree of agreement is surprising in view of the complexity of lattice calculations in comparison with the simplicity of our effective Hamiltonian calculation.

The prominent feature visible in Fig. 4 is the extraordinary magnitude of splittings in the *ccb* baryons. It is a consequence of large oscillator frequency in *cc* subsystem in Eq. (64), due to large ratio of \(\lambda _{ccb}/m_c\), in which the scale parameter \(\lambda _{ccb}\) is large in comparison with \(m_c\) due to \(m_b\). This separation of scales may make precise calculations of masses of excited *ccb* baryons difficult. Since the harmonic excitation is so high, it is likely that components with gluons of mass on the order of 1 GeV have to be included in a nonperturbative way.

The surprising feature that the very crude, first approximation based on the RGPEP, with no free parameters left after adjusting quark masses and scale to \(b \bar{b}\) and \(c \bar{c}\) data, produces in an elementary analytic way similar splittings to the ones resulting from advanced calculations, is further illustrated in Fig. 5. It presents splittings in a second band of harmonic oscillator caused by Coulomb interactions. Splittings \(m_D-m_C\), \(m_C-m_B\), \(m_B-m_A\) are in relation 2:1:5, which is the general result in the first order of perturbation theory for harmonic oscillator perturbed by any potential [60].

Interestingly, analogous lattice QCD splittings with spin dependent interactions turned off [54], also appear in the ratios 2:1:5. These results suggest that the RGPEP constituent picture with a gluon mass ansatz may be grasping the physics of lowest-mass heavy baryons. Since experimentally triply heavy baryons are difficult to produce and detect [61], their theoretical understanding using standard techniques is weakly motivated and hence also limited [62, 63]. Therefore, the ease with which our method yields results for heavy baryons in agreement with complex approaches suggests that application of the RGPEP in fourth order and including components with one or more effective gluons in the eigenvalue problem beyond perturbation theory, are worth attempting.

## 8 Conclusion

The effective Hamiltonians we finesse for heavy quarkonia and baryons from QCD of charm and beauty quarks using our gluon mass ansatz, lead to the baryon mass spectra in the ball park of expectations from other approaches to physics of *ccc* and *bbb* systems. In addition, the Hamiltonians suggest that quarks *c* form tight diquarks in *ccb* baryons. Diquarks are less likely in *bbc* baryons. Other approaches do not foresee tight diquarks in *ccb* baryons. This feature may thus distinguish a physically proper approach in future. However, such tight diquarks are hard to excite and mass splittings due their excitation are comparable or even exceed values of the gluon mass one may expect in theory. In that case, the highly excited baryon component with a heavy gluon may be large and our approximation to the three-quark component as dominant may be invalid. Calculations that treat the highly excited baryons as having significant components with one heavy effective gluon may yield smaller masses than our approximation based on the dominance of the three-quark component. If it were the case, the RGPEP approach would still apply, but in the domain of hadron physics in which gluons appear as constituents in competition with quarks for probability of appearance.

Taking into account that the method of RGPEP that we use is invariant under boosts and that it is a priori capable of providing a relativistic theory of hadrons in terms of a limited number of their effective constituents with suitably adjusted size, an extension of the RGPEP calculation to fourth order appears worth undertaking. It is certainly needed for verifying if the gluon mass ansatz we introduced provides an adequate representation of dynamics of gluons in the presence of heavy color sources. Fourth-order Hamiltonian is also needed for control on the spin splittings and rotational symmetry.

The ratio \(\sqrt{8/6}\) of harmonic oscillator frequencies in heavy quarkonia and triply heavy baryons is close to the ratio \(\sqrt{8/5}\) obtained for *u* and *d* constituent quarks in models using the concept of gluon condensate. If this is not accidental, one may hope that the RGPEP formalism shall apply also to light hadrons as built from constituent quarks and massive gluons, the latter nearly decoupled after generating effective interactions for quarks on the way down in \(\lambda \) toward 1/fm [28, 29]. But even for heavy baryons alone, the effective oscillator picture provides simple wave functions that can be used in description of relativistic processes that involve heavy hadrons.

## Footnotes

- 1.
Note that the gluon mass is introduced not in the Lagrangian or canonical QCD Hamiltonian, but as a candidate for an approximation in solving the eigenvalue equation. Thus, the ansatz does not violate the gauge symmetry in the canonical theory.

## Notes

### Acknowledgements

María Gómez-Rocha acknowledges financial support from the European Centre for Theoretical Studies in Nuclear Physics and Related Areas, MINECO FPA2016-75654-C2-1-P and the European Commission under the Marie Skłodowska-Curie Action Cofund 2016 EU project 754446 – Athenea3i. Stan Głazek acknowledges support of ECT* during his visit in February 2017.

## References

- 1.S. Capstick, W. Roberts, Prog. Particle Nucl. Phys.
**45**, S241 (2000). https://doi.org/10.1016/S0146-6410(00)00109-5 ADSCrossRefGoogle Scholar - 2.S. Capstick, N. Isgur, Phys. Rev. D
**34**, 2809 (1986). https://doi.org/10.1103/PhysRevD.34.2809 ADSCrossRefGoogle Scholar - 3.S.D. Głazek, M. Gómez-Rocha, J. More, K. Serafin, Phys. Lett. B
**773**(9), 172 (2017). https://doi.org/10.1016/j.physletb.2017.08.018 ADSCrossRefGoogle Scholar - 4.G. t Hooft, Nucl. Phys. B
**61**, 455 (1973). https://doi.org/10.1016/0550-3213(73)90376-3 ADSCrossRefGoogle Scholar - 5.K.G. Wilson, T.S. Walhout, A. Harindranath, W.M. Zhang, R.J. Perry, S.D. Glazek, Phys. Rev. D
**49**, 6720 (1994). https://doi.org/10.1103/PhysRevD.49.6720 ADSMathSciNetCrossRefGoogle Scholar - 6.G. Parisi, R. Petronzio, Phys. Lett.
**94B**, 51 (1980). https://doi.org/10.1016/0370-2693(80)90822-9 ADSCrossRefGoogle Scholar - 7.J.M. Cornwall, Phys. Rev. D
**26**, 1453 (1982). https://doi.org/10.1103/PhysRevD.26.1453 ADSCrossRefGoogle Scholar - 8.A.C. Aguilar, D. Binosi, C.T. Figueiredo, J. Papavassiliou, Eur. Phys. J. C
**78**(3), 181 (2018). https://doi.org/10.1140/epjc/s10052-018-5679-2 ADSCrossRefGoogle Scholar - 9.C. Patrignani et al., Chin. Phys. C
**40**(10), 100001 (2016). https://doi.org/10.1088/1674-1137/40/10/100001 ADSCrossRefGoogle Scholar - 10.E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Phys. Rev. D
**17**, 3090 (1978). https://doi.org/10.1103/PhysRevD.17.3090 (erratum: Phys. Rev. D 21, 313, 1980) - 11.S.D. Glazek, Acta Phys. Polon. B
**43**, 1843 (2012). https://doi.org/10.5506/APhysPolB.43.1843 CrossRefGoogle Scholar - 12.C. Semay, B. Silvestre-Brac, Nucl. Phys. A
**618**, 455 (1997). https://doi.org/10.1016/S0375-9474(97)00060-2 ADSCrossRefGoogle Scholar - 13.K.J. Juge, J. Kuti, C. Morningstar, Phys. Rev. Lett.
**90**, 161601 (2003). https://doi.org/10.1103/PhysRevLett.90.161601 ADSCrossRefGoogle Scholar - 14.T.T. Takahashi, H. Suganuma, Phys. Rev. Lett.
**90**, 182001 (2003). https://doi.org/10.1103/PhysRevLett.90.182001 ADSCrossRefGoogle Scholar - 15.P.A.M. Dirac, Rev. Mod. Phys.
**21**, 392 (1949). https://doi.org/10.1103/RevModPhys.21.392 ADSCrossRefGoogle Scholar - 16.G.P. Lepage, S.J. Brodsky, Phys. Rev. D
**22**, 2157 (1980). https://doi.org/10.1103/PhysRevD.22.2157 ADSCrossRefGoogle Scholar - 17.S.J. Brodsky, H.C. Pauli, S.S. Pinsky, Phys. Rep.
**301**, 299 (1998). https://doi.org/10.1016/S0370-1573(97)00089-6 ADSMathSciNetCrossRefGoogle Scholar - 18.S.D. Glazek, K.G. Wilson, Phys. Rev. D
**48**, 5863 (1993). https://doi.org/10.1103/PhysRevD.48.5863 ADSCrossRefGoogle Scholar - 19.A. Casher, L. Susskind, Phys. Rev. D
**9**, 436 (1974). https://doi.org/10.1103/PhysRevD.9.436 ADSCrossRefGoogle Scholar - 20.P. Maris, C.D. Roberts, P.C. Tandy, Phys. Lett. B
**420**, 267 (1998). https://doi.org/10.1016/S0370-2693(97)01535-9 ADSCrossRefGoogle Scholar - 21.S.J. Brodsky, C.D. Roberts, R. Shrock, P.C. Tandy, Phys. Rev. C
**82**, 022201 (2010). https://doi.org/10.1103/PhysRevC.82.022201 ADSCrossRefGoogle Scholar - 22.S.D. Glazek, Acta Phys. Polon. B
**42**, 1933 (2011). https://doi.org/10.5506/APhysPolB.42.1933 CrossRefGoogle Scholar - 23.S.J. Brodsky, C.D. Roberts, R. Shrock, P.C. Tandy, Phys. Rev. C
**85**, 065202 (2012). https://doi.org/10.1103/PhysRevC.85.065202 ADSCrossRefGoogle Scholar - 24.M. Gómez-Rocha, S.D. Głazek, Phys. Rev. D
**92**(6), 065005 (2015). https://doi.org/10.1103/PhysRevD.92.065005 ADSCrossRefGoogle Scholar - 25.K.G. Wilson, Phys. Rev. D
**2**, 1438 (1970). https://doi.org/10.1103/PhysRevD.2.1438 ADSMathSciNetCrossRefGoogle Scholar - 26.A.P. Trawiński, S.D. Głazek, S.J. Brodsky, G.F. de Téramond, H.G. Dosch, Phys. Rev. D
**90**(7), 074017 (2014). https://doi.org/10.1103/PhysRevD.90.074017 ADSCrossRefGoogle Scholar - 27.S.D. Głazek, M. Schaden, Phys. Lett. B
**198**, 42 (1987). https://doi.org/10.1016/0370-2693(87)90154-7 ADSCrossRefGoogle Scholar - 28.S.D. Glazek, K.G. Wilson, Phys. Rev. D
**57**, 3558 (1998). https://doi.org/10.1103/PhysRevD.57.3558 ADSCrossRefGoogle Scholar - 29.S.D. Glazek, J. Mlynik, Phys. Rev. D
**67**, 045001 (2003). https://doi.org/10.1103/PhysRevD.67.045001 ADSMathSciNetCrossRefGoogle Scholar - 30.M. Gómez-Rocha, T. Hilger, A. Krassnigg, Phys. Rev. D
**93**(7), 074010 (2016). https://doi.org/10.1103/PhysRevD.93.074010 ADSCrossRefGoogle Scholar - 31.J.D. Bjorken, AIP Conf. Proc.
**132**, 390 (1985). https://doi.org/10.1063/1.35379 ADSCrossRefGoogle Scholar - 32.M. Tsuge, T. Morii, J. Morishita, Mod. Phys. Lett. A
**1**, 131 (1986). https://doi.org/10.1142/S0217732386000191 (erratum: Mod. Phys. Lett. A 2, 283, 1987) - 33.B. Silvestre-Brac, Few Body Syst.
**20**, 1 (1996). https://doi.org/10.1007/s006010050028 ADSCrossRefGoogle Scholar - 34.Y. Jia, JHEP
**10**, 073 (2006). https://doi.org/10.1088/1126-6708/2006/10/073 ADSCrossRefGoogle Scholar - 35.A.P. Martynenko, Phys. Lett. B
**663**, 317 (2008). https://doi.org/10.1016/j.physletb.2008.04.030 ADSCrossRefGoogle Scholar - 36.W. Roberts, M. Pervin, Int. J. Mod. Phys. A
**23**, 2817 (2008). https://doi.org/10.1142/S0217751X08041219 ADSCrossRefGoogle Scholar - 37.J. Vijande, A. Valcarce, H. Garcilazo, Phys. Rev. D
**91**(5), 054011 (2015). https://doi.org/10.1103/PhysRevD.91.054011 ADSCrossRefGoogle Scholar - 38.F. Giannuzzi, Phys. Rev. D
**79**, 094002 (2009). https://doi.org/10.1103/PhysRevD.79.094002 ADSCrossRefGoogle Scholar - 39.Z. Ghalenovi, A.A. Rajabi, Sx Qin, D.H. Rischke, Mod. Phys. Lett. A
**29**, 1450106 (2014). https://doi.org/10.1142/S0217732314501065 ADSCrossRefGoogle Scholar - 40.Z. Shah, A.K. Rai, Eur. Phys. J. A
**53**(10), 195 (2017). https://doi.org/10.1140/epja/i2017-12386-2 ADSCrossRefGoogle Scholar - 41.W. Ponce, Phys. Rev. D
**19**, 2197 (1979). https://doi.org/10.1103/PhysRevD.19.2197 ADSCrossRefGoogle Scholar - 42.P. Hasenfratz, R.R. Horgan, J. Kuti, J.M. Richard, Phys. Lett.
**94B**, 401 (1980). https://doi.org/10.1016/0370-2693(80)90906-5 ADSCrossRefGoogle Scholar - 43.A. Bernotas, V. Simonis, Lith. J. Phys.
**49**, 19 (2009). https://doi.org/10.3952/lithjphys.49110 CrossRefGoogle Scholar - 44.K.W. Wei, B. Chen, X.H. Guo, Phys. Rev. D
**92**(7), 076008 (2015). https://doi.org/10.1103/PhysRevD.92.076008 ADSCrossRefGoogle Scholar - 45.K.W. Wei, B. Chen, N. Liu, Q.Q. Wang, X.H. Guo, Phys. Rev. D
**95**(11), 116005 (2017). https://doi.org/10.1103/PhysRevD.95.116005 ADSCrossRefGoogle Scholar - 46.J.R. Zhang, M.Q. Huang, Phys. Lett. B
**674**, 28 (2009). https://doi.org/10.1016/j.physletb.2009.02.056 ADSCrossRefGoogle Scholar - 47.Z.G. Wang, Commun. Theor. Phys.
**58**, 723 (2012). https://doi.org/10.1088/0253-6102/58/5/17 ADSCrossRefGoogle Scholar - 48.T.M. Aliev, K. Azizi, M. Savci, JHEP
**04**, 042 (2013). https://doi.org/10.1007/JHEP04(2013)042 ADSCrossRefGoogle Scholar - 49.T.M. Aliev, K. Azizi, M. Savcı, J. Phys. G
**41**, 065003 (2014). https://doi.org/10.1088/0954-3899/41/6/065003 ADSCrossRefGoogle Scholar - 50.F.J. Llanes-Estrada, O.I. Pavlova, R. Williams, Eur. Phys. J. C
**72**, 2019 (2012). https://doi.org/10.1140/epjc/s10052-012-2019-9 ADSCrossRefGoogle Scholar - 51.H. Sanchis-Alepuz, R. Alkofer, G. Eichmann, R. Williams, PoS
**QCD-TNT-II**, 041 (2011)Google Scholar - 52.S.X. Qin, C.D. Roberts, S.M. Schmidt, Phys. Rev. D
**97**(11), 114017 (2018). https://doi.org/10.1103/PhysRevD.97.114017 ADSCrossRefGoogle Scholar - 53.Z.S. Brown, W. Detmold, S. Meinel, K. Orginos, Phys. Rev. D
**90**(9), 094507 (2014). https://doi.org/10.1103/PhysRevD.90.094507 ADSCrossRefGoogle Scholar - 54.S. Meinel, Phys. Rev. D
**85**, 114510 (2012). https://doi.org/10.1103/PhysRevD.85.114510 ADSCrossRefGoogle Scholar - 55.R.A. Briceno, H.W. Lin, D.R. Bolton, Phys. Rev. D
**86**, 094504 (2012). https://doi.org/10.1103/PhysRevD.86.094504 ADSCrossRefGoogle Scholar - 56.M. Padmanath, R.G. Edwards, N. Mathur, M. Peardon, Phys. Rev. D
**90**(7), 074504 (2014). https://doi.org/10.1103/PhysRevD.90.074504 ADSCrossRefGoogle Scholar - 57.Y. Namekawa et al., Phys. Rev. D
**87**(9), 094512 (2013). https://doi.org/10.1103/PhysRevD.87.094512 ADSCrossRefGoogle Scholar - 58.C. Alexandrou, V. Drach, K. Jansen, C. Kallidonis, G. Koutsou, Phys. Rev. D
**90**(7), 074501 (2014). https://doi.org/10.1103/PhysRevD.90.074501 ADSCrossRefGoogle Scholar - 59.K.U. Can, G. Erkol, M. Oka, T.T. Takahashi, Phys. Rev. D
**92**(11), 114515 (2015). https://doi.org/10.1103/PhysRevD.92.114515 ADSCrossRefGoogle Scholar - 60.N. Isgur, G. Karl, Phys. Rev. D
**19**, 2653 (1979). https://doi.org/10.1103/PhysRevD.19.2653 (erratum: Phys. Rev. D 23, 817, 1981) - 61.S.L. Olsen, T. Skwarnicki, D. Zieminska, Rev. Mod. Phys.
**90**(1), 015003 (2018). https://doi.org/10.1103/RevModPhys.90.015003 ADSCrossRefGoogle Scholar - 62.N. Brambilla et al., Eur. Phys. J. C
**71**, 1534 (2011). https://doi.org/10.1140/epjc/s10052-010-1534-9 ADSCrossRefGoogle Scholar - 63.N. Brambilla et al., Eur. Phys. J. C
**74**(10), 2981 (2014). https://doi.org/10.1140/epjc/s10052-014-2981-5 CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}