Noname manuscript No. (will be inserted by the editor) A first estimate of triply heavy baryon masses from the pNRQCD perturbative static potential Felipe J. Llanes-Estrada, Olga I. Pavlova, Richard Williams Received: date / Revised version: date Abstract Within pNRQCD we compute the masses some of the values obtained in section 7 below. Like its of spin-averaged triply heavy baryons using the now- meson (quarkonium) counterpart [4,5], we expect this available NNLO pNRQCD potentials and three-body triply heavy baryon to attract much interest. variational approach. We focus in particular on the role For heavy quark systems, the development of poten- of the purely three-body interaction in perturbation tial Non-Relativistic Quantum Chromodynamics (pN- theory. This we find to be reasonably small and of the RQCD) as an effective theory of QCD has allowed a order 25 MeV more systematic treatment of the theoretical uncertain- Our prediction for the Ωccc baryon mass is 4900(250) ties involved in spectroscopic predictions by expand- in keeping with other approaches. We propose to search ing in powers of 1/m [6,7,8]. For those in their ground for this hitherto unobserved state at B factories by ex- state, pNRQCD can additionally be organized in stan- amining the end point of the recoil spectrum against dard perturbation theory as a power expansion in αs [6, triple charm. 9]. While the theory itself has limitations due to the finiteness of the quark masses, the two-body static po- PACS 14.20.Mr 14.20.Lq 12.38.Bx · · tential (quickly reviewed in section 2) has shown to be a good starting point for many meson investigations. With the three-body potential in NNLO perturba- 1 Introduction tion theory now at hand [10] (we will give it in sec- Amongst the staples of hadron physics is Baryon spec- tion 3), it is timely to perform an exploratory study of troscopy. Here, quark model computations of the light the ground-state triply heavy quark spectrum. This we baryon spectrum [1,2,3] find only mild success beyond present in section 8. The necessary QCD parameters αs, ground-states in various channels due to the plethora mc, mb are fixed by describing several common meson of open thresholds and couplings between channels. A spectroscopy observables as explained in section 5. arXiv:1111.7087v1 [hep-ph] 30 Nov 2011 much cleaner system can be provided by baryons com- Finally, we comment on the feasibility of detecting posed of three heavy quarks (i.e. the combinations ccc, the Ωccc in section 10. Some numerical methods are bbb, ccb and bbc). Since a considerable number of states relegated to the appendix. are supposed to be below any strong decay thresholds, one can straightforwardly apply few-body reasoning, and quark model techniques can handle bound states better. 2 Static quarkonium potential in pNRQCD Indeed, a review of the literature reveals many stud- ies that have computed or constrained heavy baryon The static two-body potential for bound states of a quark and anti-quark is well known to NNLO (and be- masses, particularly the ground state Ωccc. We collect yond) [11,12,13] as F. J. Llanes-Estrada, O.I. Pavlova and R. Williams Departamento de F´ısica Te´orica I, Universidad Complutense, (0) (0) (0) (0) 28040 Madrid, Spain V = V + V + V + . (1) LO NLO NNLO ··· 2 Llanes-Estrada et al. The leading order potential is just the color Coulomb with potential MS −2 C1 (x)= a1 + β0 log(x) , (0) 4 αs(r ) VLO = (2) MS 2 2 −3 r C2 (x)= a2 + β0 log (x) + (β1 +2β0a1) log(x) . whereas the NLO and NNLO are, respectively, This obviously simplifies if the renormalization scale is −2 (0) (0) αs(r ) chosen as q2 itself V = V (a +2γ β ) , (3) NLO LO × 4π × 1 E 0 α2(r−2) 2 (0) (0) s 2 4 (4π)αMS (q ) VNNLO = VLO V (q )= (10) × (4π)2 q2 (4) − 3 2 2 π 2 2 α (q2) α (q2) γE(4a1β0 +2β1) + ( +4γE)β0 + a2 . 1+ a MS + a MS . × 3 × 1 4π 2 4π ! By standard convention γ 0.57721 ... is the Euler- E ≃ Mascheroni constant; for three colors and in terms of the number of quark flavors N below the renormaliza- f 2.1 1/m Potential tion scale, the beta function that determines the run- ning of the coupling is expanded as In QED the 1/m corrections to the static potential were 2N long ago shown to vanish2. β 11 f , (5) 0 ≡ − 3 In perturbative QCD the non-Abelian vertex correc- 38Nf tion produces a 1/m potential that cannot be gauged β 102 , (6) 1 ≡ − 3 away. It can be nominally assigned to the 1/m2 order and the coefficients in the potential that remain in a through a field redefinition [7]. Since a recent study of conformal theory are the meson spectrum [16] finds reasonably large effects for the 1S states, especially in charmonium, we also 31 10Nf consider it here. a1 , (7) ≡ 3 − 9 At leading order, the 1/m potential vanishes. For 4 4343 2 9π NLO and NNLO we employ the convention of [17]. Al- a2 + 36π ≡ 18 − 4 (8) ternatively, one can use the NLO result (see Eq. (11) 1229 52 100 2 below) without [7] the factor (7/9), in order to match +66ζ(3) + ζ(3) Nf + N , − 27 3 81 f a lattice computation. In coordinate space the potential reads where ζ(3) 1.202 ... is Riemann’s zeta function. ≃ The NLO potential can be understood not only from 2 αs(µ) 7 pNRQCD (first, construct an effective theory around V −1 = m − m r2 × 9 the heavy quark limit, then use Coulomb gauge in in- r (11) 3 termediate steps to obtain the gauge-invariant poten- α (µ) 7β0 68 s b + log(e2γE µ2r2) + , tial that is a matching coefficient in pNRQCD) but also − 3πm r2 − 2 6 3 r from the D ( q ) time-like gluon propagator obtained σσ | | in Coulomb gauge by Watson and Reinhardt [14], re- where mr is the reduced (pole) mass of the qq¯ system, ducing it to its simpler Heavy Quark limit (there, sev- b 20.836 for N = 3 (appropriate for charmonium) 2 ≃− f eral terms do not contribute to the Wilson loop poten- and b2 18.943 for Nf = 4 (appropriate for bottomo- 1 ≃− tial) . nium) and b 17.049 for higher scales where N =5 2 ≃− f In momentum space, the potential can be given to are given in [17]. The last term with a logarithm van- order NNLO [12] at an arbitrary renormalization scale ishes if the scale is chosen as the BLM scale defined by 2 µ as Eq. (49) in the appendix. We have performed compu- tations with both this running scale and a fixed scale 4 (4π)α V (q2)= MS (9) (m2 or m2). −3 q2 × c b 2 2 α µ α 2 µ 2 1+ MS CMS + MS CMS For example, in [15] it is shown that the Bethe-Salpeter 4π 1 q2 4π 2 q2 ladder approximation generates Feynman gauge 1/m terms that vanish upon including the crossed-ladder box. No such 1 We thank Jacopo Ghiglieri for this observation terms are present in Coulomb gauge. Triply heavy baryons in pNRQCD 3 If the potential is constructed in momentum space, 2.2 Running of the strong coupling constant the 1/m correction to the central static potential reads The renormalization group equation that determines 2 2 2 αs(µ ) 7 the running of the strong coupling constant to NNLO V −1 = 2π m − m q × 9 r (12) is 3 2 2 2παs(µ ) µ 7β0 68 ∂αs β0 2 β1 3 b2 + log + . = α α . (17) − 3m q − q2 6 3 2 s 2 s r ∂ log q −4π − (4π) Again, judicious choice of the scale µ = q disposes of By keeping the first term on the right hand side, or both the logarithm. terms, this equation can be solved to NLO or NNLO A counterintuitive result is that matrix elements of respectively. the Vm−1 potential can actually be similar or, in ex- In general we employ the Runge-Kutta algorithm to treme cases, even larger for bottom systems than for numerically solve Eq. (17). To NLO the equation is also charm systems, since in a Coulombic system all ener- very simply analytically solvable and provides a handy gies scale with the reduced mass mr. check for the computer programme. Following [18] we To see this, let us restrict ourselves to NLO and introduce a scale Λ as is customary, so that employ the convention of [7] in momentum space 1 αNLO(Q2)= , (18) 2 2 s Q2 2 αs(µ ) b log Λ2 V −1 = 2π , (13) m − m q r with and compute ψ Vm−1 ψ with a hydrogen like wave- β 33 2N h | | i b = 0 = − f . function. Taking the Fourier transform of a 1S state 4π 12π −r/a0 3 −1 2e / 4πa , with Bohr radius a = mrαs yields 0 0 Inverting the equation yields p 3 4 4πa0 ψ(q)= , (14) 2 2 1 (1 + q2a2)2 Λ = Q exp . (19) p 0 −bαNLO(Q2) s and therefore In Table 1 we give for convenience, and as benchmarks, 3 3 3 2 2a0 d kd q αs( k q ) the values obtained by running back to low scales the V −1 = . m − 3 | − | 2 renormalization group equation from the Z-boson pole, h i mrπ k q ((1 + q2a2)(1 + k2a2)) Z | − | 0 0 (15) where the coupling constant is very accurately con- strained by many analyses [19].