Scaling of P-wave excitation energies in heavy-quark systems

A simple regularity in anticipating P-wave excitation energies of states with heavy quarks is noted. It can apply to systems such as the negative-parity $\Sigma_c$, $\Sigma_b$, and $\Omega_c$, $\bar Q Q$ quarkonia, and the bottom-charmed meson $B_c$. When one subtracts a term accounting for phenomenological energies of heavy quarks binding with one another in S-waves, the residual excitation energies display an approximately linear behavior in the reduced mass of constituents, all the way from the $\Lambda$ to the $\Upsilon$.

was taken of quark masses, hyperfine interactions, and S-wave binding terms B(q 1 q 2 ) involving any quark pairs where one quark is heavier than u, d and the other heavier than s. These binding terms are obtained phenomenologically by comparing masses of hadrons containing a single heavy quark (e.g., q 1 or q 2 ) with ones containing two heavy quarks (e.g., q 1q2 ). We find that when these binding terms are taken into account in calculating S-P mass differences, the residual energy differences ∆E R depend approximately linearly on the reduced mass µ 12 = m 1 m 2 /(m 1 + m 2 ) of the pair. This behavior extends from the Λ = uds baryon all the way up to the Υ(1S) and their respective P-wave excitations. We lay out the tools for our estimates in Sec. II, describing assumed quark masses and binding terms. The ground rules for quoting S-P splittings are also given. We quote the observed S-P splittings for a number of pairs in Sec. III. The effects of binding terms, if any, are considered in Sec. IV, giving rise to residual energy differences ∆E R which are plotted as functions of reduced mass. An approximately linear dependence is seen. In baryonic cases the problem is reduced to a two-body one by assuming one quark is excited with respect to two others which remain in a relative S-wave.
The linear dependence of ∆E on reduced mass is used in Sec. V to predict several quantities which were only crudely estimated before. These include P-wave excitation energies for Σ c and Σ b states [10] and for Ω c states [3]. Predictions for Ξ b and B c are also given and compared with others in the literature. Section VI is devoted to a discussion of the possible source of the observed regularity, and a brief conclusion.

II Tools
We use separate constituent-quark masses for mesons and baryons [16]. They are summarized in Table I. Analysis of S-wave mesons and baryons makes use of binding terms B(q 1 q 2 ), also from Ref. [16], summarized in Table II. These terms were calculated by comparing the masses of spin-averaged S-wave bound states (e.g., for charmonium) with the sum of their constituentquark masses as determined from hadrons containing a single heavy quark (e.g., Λ c ).
Ω c = css, where we consider the spin-1 ss diquark to be excited by one unit of orbital angular momentum with respect to the charmed quark [3]. We take the masses listed in Table III to calculate the spin-averaged S-P splittings shown. The masses of excited states are calculated using averagesM P weighted by 2J + 1 factors, where J is the spin of the resonance. Small uncertainties in masses are not quoted.

B Mesons
We consider only those systems for which the spin-averaged ground state and P-wave masses can be calculated. They are cs ("D s "), cc, and bb. For c(ū,d) not all candidates for the 1P level are firmly established, while for b(ū,d) a spin-zero meson and one of two predicted spin-1 mesons are still missing (see Sec. V B). For bc ("B c ") no P-wave states have been seen, but their masses have been predicted (see Sec. V F). The relevant masses are shown in Table IV. Spin averaged masses arē

IV Residual energy differences ∆E R
We now calculate residual energy differences ∆E R ≡ ∆E P −S − B for the above systems, where B denotes the sum of B(q 1 q 2 ) over all relevant heavy quarks q 1 and q 2 (cf. Table  II). The results are shown in Table V. Here [q 1 q 2 ] denotes a spinless color-antitriplet diquark, while (ss) denotes a spin-1 color-antitriplet diquark. We quote isospin-averaged masses where appropriate, letting q stand for u or d.
Whereas the quantities ∆E P −S are not monotonic functions of the reduced mass µ 12 , when the binding energies B are subtracted from them, the residual energies ∆E R are crudely arranged along a straight line, as shown in Fig. 1. A linear fit to the eight experimentally known values in Table V gives the result The root-mean-square deviation of the data from this fit is 18.7 MeV. We discuss some consequences of this regularity, if it is to be taken seriously, in the next Section. (2). Filled diamonds denote data used in the fit. Theoretical predictions for Ξ b (subsection E) and B c (subsection F) systems are plotted as hollow diamonds and hollow squares.

V Consequences and Predictions
A Σ c and Σ b baryons In Ref. [10] a linear extrapolation of excitation energy was used to estimate the S-P wave splittings for Σ c and Σ b baryons. The present discussion gives support to that assumption. The parameters of the present linear fit give slightly different values of ∆E R , as shown in Table VI. For the states in this table, there are no B terms, so ∆E P −S = ∆E R .

B Charm and bottom mesons
The reduced masses for D and B mesons are displayed in Table VII. They lead to predictions via Eq. (2) of ∆E R , which is equal to ∆E P −S because the binding correction B is zero. In order to compare these predictions with experiment, one must know the masses of all four P-wave states. Our partial information is summarized in Table VIII. The D and B mass eigenstates have j (the vector sum of light-quark spin and orbital angular momentum) equal to 1/2 or 3/2. Those with j = 3/2 (total J = 1, 2) decay predominantly via D waves, are narrow, and are firmly established [18]. Those with j = 1/2 (J = 0, 1) are expected to decay via S waves and are very broad, with consequent mass uncertainty. The j = 1/2 D mesons would satisfy the linear fit if their widths, exceeding 200 MeV, were included as error bars. No candidates for the j = 1/2 B states have been identified. They would have to be considerably lighter than the j = 3/2 states if they were to obey the prediction in Table VII. The outlier nature of D and B states is further discussed in Sec. VI.
The predicted spin-averaged P-wave mass for B s is low enough that the j = 1/2 B s Pwave states are probably below the respective BK and B * K thresholds for the J = 0 and   (2317) and D s1 (2460) (see below), they are expected to be very narrow, decaying only via B s0 → γB * s and B s1 → γB s or γB * s , or with isospinviolating processes involving π 0 emission. The properties of these states have been discussed in Refs. [19,20].

C D s mesons
The observed masses of D s0 (2317) and D s1 (2460) were considerably below predictions of potential models, leading to some initial surprise. The present regularity (Fig. 1) indeed supports the picture of these states as lying below those predictions.
D Ω c baryons The residual energy ∆E R for the five narrow Ω c states observed by LHCb [1] lies right on the linear fit, supporting their assignment as five P-wave states [3][4][5][6][7][8] and disfavoring an alternate assignment (see, e.g., [3]) in which the two highest states are 2S excitations and two lower-mass P-wave states remain to be discovered. Table IX: Values of ∆E P −S and ∆E R for Ξ b states from models compared with predictions of linear fit (2). Masses in MeV. a Spin-averaged ground state mass. b Ref. [9]. c Ref. [21].

E Ξ b baryons
In Table IX we compare a recent prediction [9] for the masses of P-wave excitations of the scalar [sq] quark in Ξ b baryons (Fig. 1, hollow diamond), with an earlier one ( [21], Fig. 1, hollow square), and with the result of the linear fit for reduced mass µ bc = 690.3 MeV. The fit is more consistent with the later prediction.

F B c states
One can obtain a value of∆E P −S for the B c system by interpolating between the nearly equal values for the cc and bb systems, as one might expect if the interquark potential is close to the logarithmic one proposed in Ref. [22]. One thus obtains ∆E P −S = 456 MeV, corresponding to the open diamond in Fig. 1 when a binding term of 341.5 MeV is taken into account. An early potential-model prediction [11]

VI Discussion and conclusions
The binding terms B used to calculate ∆E R represent corrections to the picture of spectra due to constituent-quark masses and hyperfine terms [23], when quarks are heavy enough to experience the short-distance Coulomb-like force of single gluon exchange. In a purely Coulombic potential V (r) = −(4/3)α s /r the energy levels are given by E n = −[(4/3)α s ] 2 µ/(2n 2 ). In the simplest approximation the P-wave excitation energy is given by ∆E P −S = E 2 − E 1 . We have subtracted the S-wave binding energy B from this P-wave excitation energy to obtain the residual energy difference In our convention this S-wave binding energy is positive. In this simple example it is just the minus the ground-state energy, −E 1 . The upshot is that here the residual excitation energy is just the energy eigenvalue of the P-wave: So in this case the slope in Fig. 1 is just −[(4/3)α s ] 2 µ/8. In a more realistic potential with a confining piece the slope will be different and there is likely to be also a constant term. For light quarks (u, d, s) the use of constituent-quark masses means that it is not necessary to subtract a B term; the constituent-quark masses already embody such a term. Nonetheless, the negative slope in the relation between residual energy and reduced mass is generic. It just reflects the fact that the P-wave energy (as opposed to energy splitting) is negative. This is surprising, as relativistic corrections (important even for systems as heavy as bottomonium) do not depend purely on the reduced mass. This is true for quantum electrodynamics, as shown by the comparison between positronium and the hydrogen atom [24]. The linear dependence of residual energy must be the result of compensating effects, not some fundamental relation. What we have done is to construct a phenomenological "bridge" between confinement and short-distance Coulomb-like behavior. This picture then explains why the B and D mesons are outliers. Their radii are of order 1/Λ QCD , rather than 1/(α s µ). The fact that α s runs between µ = 500 MeV and 2500 MeV will make the slope slightly scale dependent.
The potential for learning about P-wave excitations of heavy-quark baryons and mesons makes the present discussion timely. Consequences have been noted for charmed and bottomflavored baryons and mesons. It will be interesting to see if some of these regularities are further supported by experiment.