Masses of Open-Flavour Heavy-Light Hybrids from QCD Sum-Rules

We use QCD Laplace sum-rules to predict masses of open-flavour heavy-light hybrids where one of the hybrid's constituent quarks is a charm or bottom and the other is an up, down, or strange. We compute leading-order, diagonal correlation functions of several hybrid interpolating currents, taking into account QCD condensates up to dimension-six, and extract hybrid mass predictions for all $J^P\in\{0^{\pm},\,1^{\pm}\}$, as well as explore possible mixing effects with conventional quark-antiquark mesons. Within theoretical uncertainties, our results are consistent with a degeneracy between the heavy-nonstrange and heavy-strange hybrids in all $J^P$ channels. We find a similar mass hierarchy of $1^+$, $1^{-}$, and $0^+$ states (a $1^{+}$ state lighter than essentially degenerate $1^{-}$ and $0^{+}$ states) in both the charm and bottom sectors, and discuss an interpretation for the $0^-$ states. If conventional meson mixing is present the effect is an increase in the hybrid mass prediction, and we estimate an upper bound on this effect.


Introduction
Hybrids are hypothesized, beyond-the-quark-model hadrons that exhibit explicit quark, antiquark, and gluonic degrees of freedom. They are colour singlets and so should be permissible within quantum chromodynamics (QCD); thus, the question of their existence provides us with a key test of our characterization of confinement. Despite nearly four decades of searching, hybrids have not yet been conclusively identified in experiment. There are, however, a number of noteworthy candidates. For example, the Particle Data Group (PDG) [1] lists a pair of tentative resonances, the π 1 (1400) and the π 1 (1600), both with exotic J P C = 1 −+ , a combination inaccessible to conventional quark-antiquark mesons [2,3]. There are several non-exotic hybrid prospects as well.
To date, closed-flavour (hidden-flavour or quarkonium) hybrids have received more attention than open-flavour hybrids likely because most promising hybrid candidates are closed. Furthermore, closed-flavour hybrids allow for exotic J P C quantum numbers; open-flavour hybrids, on the other hand, are not eigenstates of C-parity, and so are characterized by non-exotic J P quantum numbers. However, the recent observation of the fully-open-flavour X(5568) containing a heavy (bottom) quark [34] may be a precursor to additional open-flavour discoveries that do not have a simple quark-model explanation (e.g., the X(5568) has been studied as abdsu tetraquark [35]). Hence, computing masses of open hybrids containing heavy quarks is timely and of phenomenological relevance.
Ground states masses of bottom-charm hybrids were recently computed using QCD sum-rules in [36]; therefore, we focus on a QCD sum-rules analysis of open-flavour heavy-light hybrids i.e., hybrids containing one heavy quark (charm or bottom) and one light quark (up, down, or strange).
The seminal application of QCD Laplace sum-rules to open-flavour hybrids was performed by Govaerts, Reinders, and Weyers [37] (hereafter referred to as GRW). Therein, they considered four distinct currents covering J ∈ {0, 1} in an effort to compute a comprehensive collection of hybrid masses. Their QCD correlator calculations took into account perturbation theory as well as mass-dimension-three (i.e., 3d) quark and 4d gluon condensate contributions. Precisely half of the analyses stabilized and yielded viable mass predictions. However, for all heavy-light hybrids, the ground state hybrid mass was uncomfortably close to the continuum threshold (with a typical separation of roughly 10 MeV), so that even a modest hadron width would result in the resonance essentially merging with the continuum [37].
In this article, we extend the work of GRW by including both 5d mixed and 6d gluon condensate contributions in our correlator calculations. As noted in GRW, for openflavour heavy-light hybrids, condensates involving light quarks could be enhanced by a heavy quark mass allowing for the possibility of a numerically significant contribution to the sum-rules. By this reasoning, the 5d mixed condensate should also be included. As for the 6d gluon condensate, recent sum-rules analyses of closed-flavour heavy hybrids [9,26,27,28] have demonstrated that it is important and can have a stabilizing effect on what were, in the pioneering work [38,39], unstable analyses. This paper is organized as follows: in Section 2, we define the currents that we use to probe open-flavour heavy-light hybrids and compute corresponding correlation functions; in Section 3, we generate QCD sum-rules for each of the correlators; in Section 4, we present our analysis methodology as well as our mass predictions for those channels which stabilized; in Section 5, we discuss our results and compare them to GRW and to contemporary predictions made using lattice QCD.

Currents and Correlators
Following GRW, we define open-flavour heavy-light hybrid interpolating currents where g s is the strong coupling and λ a are the Gell-Mann matrices. The field Q represents a heavy charm or bottom quark with mass M Q whereas q represents a light up, down, or strange quark with mass m q . The Dirac matrix Γ ρ satisfies and the tensor G a µρ satisfies where G a µρ is the gluon field strength and is its dual defined using the totally antisymmetric Levi-Civita symbol µρνσ . For each of the four currents defined through (1)-(3), we consider a diagonal correlation function where Π (0) probes spin-0 states and Π (1) probes spin-1 states. Each of Π (0) and Π (1) couples to a particular parity value, and, in the case of closed-flavour hybrids, also to a particular C-parity value; however, as noted in Section 1, open-flavour hybrids are not C-parity eigenstates. Regardless, we will refer to Π (0) and Π (1) using the J P C assignments they would have if we were investigating closed-rather than open-flavour hybrids. But, to stress that the C-value cannot be taken literally, we will enclose it in brackets (a notation employed in [36,40]). In Table 1, we provide a breakdown of which currents couple to which J P (C) combinations.
We calculate the correlators (5) within the operator product expansion (OPE) in which perturbation theory is supplemented by a collection of non-perturbative terms, each of which is the product of a perturbatively computed Wilson coefficient and a non-zero vacuum expectation value (VEV) corresponding to a QCD condensate. We include condensates up to 6d: respectively referred to as the 3d quark condensate, the 4d gluon condensate, the 5d mixed condensate, and the 6d gluon condensate. Superscripts on light quark fields are colour indices whereas subscripts are Dirac indices, and σ µν = i 2 [γ µ , γ ν ]. The Wilson coefficients (including perturbation theory) are computed to leading-order (LO) in g s using coordinate-space fixed-point gauge techniques (see [41,42], for example). Note that LO contributions to (5) associated with 6d quark condensates are O(g 4 s ); our calculation is actually O(g 3 s ), and so 6d quark condensates have been excluded from (7)- (10). (In Ref. [27] the numerical effect of the 6d quark condensates has been shown to be small compared to the 6d gluon condensate). Light quark mass effects are included in perturbation theory through a next-to-leading-order light quark mass expansion, and at leading-order in all other OPE terms. The contributing Feynman diagrams are depicted in Figure 1 1 where we follow as closely as possible the labeling scheme of [20]. (Note that there is no Diagram IV in Figure 1 because, in [20], Diagram IV corresponds to an OPE contribution stemming from 6d quark condensates that is absent in the open-flavour heavy-light systems.) The MS-scheme with the D = 4 + 2 convention is used, and µ is the corresponding renormalization scale. We use the program TARCER [44], which implements the recurrence algorithm of [45,46], to express each diagram in terms of a small collection of master integrals, all of which are wellknown. Following [47], we employ a dimensionally regularized γ 5 that satisfies γ 2 5 = 1 and {γ 5 , γ µ } = 0. Note that the imaginary parts of Diagrams I-III were actually first computed between [38] and GRW; for these three diagrams, we were able to successfully bench-mark our results against that original work. 1 All Feynman diagrams are drawn using JaxoDraw [43]. Table 1: The J P (C) combinations probed through different choices of Γ ρ (2) and G a µρ (3). Diagram XII, a 5d mixed condensate contribution, generates some complications. Focusing on the lower portion of the diagram, we see a heavy quark propagator carrying momentum q multiplied by a divergent, one-loop sub-graph. Correspondingly, Diagram XII contributes to the correlator a non-local divergence proportional to Following [6], this divergence is eliminated by renormalization of the composite operators (1) which induces mixing with either Qγ µ q or Qγ µ γ 5 q. Specifically, for those operators with Γ ρ = γ ρ (recall (2)), this mixing results in whereas, for those with Γ ρ = γ ρ γ 5 , we have where k is an as yet undetermined constant emerging from renormalization. For currents that mix according to (12), the VEV under the integral on the right-hand side of (5) gets modified as follows: (14) with an analogous expression for operators that mix according to (13). The first term on the right-hand side of (14) corresponds to the diagrams of Figure 1 whereas the last two terms give rise to new, renormalization-induced contributions to the OPE. Almost all of these new contributions are sub-leading in g s , however, and so are ignored. The only exceptions are those containing the 5d mixed condensate (9); these give rise to the pair of diagrams depicted in Figure 2. Both of these tree-level diagrams contain a heavy quark propagator with momentum q and are multiplied by a factor of 1 in (14), precisely what is needed to cancel the non-local divergence (11) Summing the diagrams from Figures 1 and 2, and then determining the constant k from (12) or (13) such that all non-local divergences are eliminated, we find for either and where Li 2 (z) is the dilogarithm function defined by The remaining quantities in (15) are listed in Tables 2-7 for the distinct J P (C) combinations under consideration. Also, in Table 8, we give the values determined for the  Table 3: The polynomials and constants of the light quark mass correction to perturbation theory.
renormalization parameter k. Finally, we note that, for the sake of brevity, we have omitted all polynomials in z from (15), some of which are divergent, as such terms correspond to dispersion-relation subtractions and provide no contribution to the QCD sum-rules discussed in the next section.

QCD Laplace Sum-Rules
Viewed as a function of Euclidean momentum Q 2 = −q 2 , each of Π (0) and Π (1) from (6) satisfies a dispersion relation where · · · represents subtractions constants, collectively a third degree polynomial in Q 2 , and t 0 represents the appropriate physical threshold. The quantity Π on the lefthand side of (18) is identified with the OPE result (15) while ImΠ on the right-hand side of (18) is the hadronic spectral function. To eliminate the (generally unknown) subtraction constants and enhance the ground state contribution to the integral, the Borel transformB = lim is applied to (18) weighted by (−Q 2 ) k for k ≥ 0 to yield the k th -order Laplace sum-rule (LSR) [48] R k (τ ) = 1 τB The Borel transform annihilates polynomials in Q 2 which eliminates dispersion-relation subtraction constants and justifies our omission of polynomials (divergent or not) from (15). The exponential kernel on the right-hand side of (21) suppresses contributions from excited resonances and the continuum relative to the ground state.
In a typical QCD sum-rules analysis, the hadronic spectral function is parametrized using a small number of hadronic quantities, predictions for which are then extracted using a fitting procedure. We employ the "single narrow resonance plus continuum" model [48] 1 π where m H is the ground state resonance mass, f H is its coupling strength, θ is a Heaviside step function, s 0 is the continuum threshold and ImΠ OPE is the imaginary part of the QCD expression for Π given in (15). Substituting (22) into (21) gives and, defining continuum-subtracted LSRs by we find, between (23) and (24), the result Finally, using (25), we obtain the central equation of our analysis methodology.
To develop an OPE expression for R k (τ, s 0 ), we exploit a relationship between the Borel transform and the inverse Laplace transformL −1 [48] where c is chosen such that f (Q 2 ) is analytic to the right of the integration contour in the complex Q 2 -plane. Applying definitions (20) and (24) to (15) and using (27), it is straightforward to show that and  where a, b are constants given in Table 9 for each J P (C) combination under investigation. Note that the definite integral in (28) can be evaluated exactly; however, the result is rather long and not particularly illuminating, and so is omitted for brevity.
Renormalization-group (RG) improvement [49] dictates that the coupling constant and quark masses in (28) be replaced by their (one-loop, MS) running counterparts. The running coupling is given by where N F is the number of active quark flavors and M X is a reference scale for experimental values of α s . In addition, the running heavy quark mass can be expressed as where M is defined by M (M ) = M , and the running light quark mass can be expressed as in anticipation of using the Ref.
The QCD predictions (15) have isospin symmetry because ūu = d d = nn and the sub-leading effect of nonstrange quark masses is negligible (i.e., we are effectively in the chiral limit for nonstrange systems). In addition to specifying expressions for the running coupling and quark masses, we must also specify the numerical values of the condensates (7)- (10). Because of the form of (15), for qq we consider the product where PDG values are used for the meson masses [1] and the decay constants are [52] f π = 92.2 MeV GeV, f K = 110 MeV.
The quark mass ratios of strange to light and charm to strange quarks are given in [1]; however, in order to consider the RG-invariant product (37) For the purely gluonic condensates (8) and (10), we use values from [54,55]: The 5d mixed condensate can be related to the 3d quark condensate through [19,56,57] gqσGq qq Because we are using (37) to specify the chiral-violating condensates, in the analysis below, the qq effects are subsumed within dimension-four contributions and gqσGq effects within dimension-six contributions. As noted above, we choose the renormalization scale µ to be M c for the charm systems and M b for the bottom systems.

Analysis Methodology and Results
In order to extract stable mass predictions from the QCD sum-rule, we require a suitable range of values for our Borel scale (τ ) within which our analysis can be considered reliable. Within this range, we perform a fitting (i.e., minimization) procedure to obtain an optimized value of the continuum onset (s 0 ) associated with our resulting mass prediction. We determine the bounds of our Borel scale by examining two conditions: the convergence of the OPE, and the pole contribution to the overall mass prediction, mirroring our previous work done in charmonium and bottomonium systems [27]. To enforce OPE convergence and obtain an upper-bound on our Borel window (τ max ), we require that contributions to the dimension-four condensate be less than one-third that of the perturbative contribution, and the dimension-six gluon condensate contribute less than one-third of the dimension-four condensate contributions. (See Figure 3 for an example.) To determine a lower bound for our Borel window, we examine the pole contribution defined as To extract a lower bound for our Borel window (τ min ), we must first provide a reasonable estimate of the continuum s 0 as a seed value for the minimization. To do this, we look for a stability in the hadronic mass prediction as a function of s 0 with variations in τ ( Figure 4). We optimize the initial s 0 and m H predictions by minimizing where we sum over a discretized τ range inside the Borel window. Minimizing (47) results in an optimized values for the continuum s 0 . Once s 0 is found, we may use (46) to determine a lower bound on τ by requiring a pole contribution of at least 10%. Note that this procedure involving (46) should be iterated to ensure that the values of s 0 and τ min are self-consistent.
We present results for the Borel window, continuum, and predicted hybrid mass for open-charm and open-bottom hybrids in Tables 10 through 13. Channels that do not stabilize have been omitted from the tables. The errors presented encapsulate contributions from the heavy quark masses and the condensate values. We do not attempt to estimate errors resulting from the truncation of the OPE series, s 0 , or the variations in τ .

Discussion
For each open-flavour heavy-light hybrid combination under consideration, we performed a LSR analysis of all eight J P (C) combinations defined according to Table 1. As Table 10: QCD sum-rules analysis results for ground state charm-nonstrange hybrids.  Table 11: QCD sum-rules analysis results for ground state charm-strange hybrids.     J P (C) ∈ {0 +(+) , 0 −(−) , 1 +(+) , 1 −(−) } for all heavy-light flavour hybrids. Note that, for all heavy-light quark combinations considered, we did arrive at a unique mass prediction for each J P . GRW found something similar, but, as can be seen from Tables (14) and (15), our updated mass predictions are significantly different from theirs. In all stable channels, the most significant non-perturbative contribution to the LSRs is the 4d gluon condensate term of the OPE. At the corresponding optimized value of s 0 and over the Borel window indicated in Tables 10-13, the 4d gluon condensate term accounts for roughly 10-30% of the area underneath the R 0 (τ, s 0 ) curve. The second most significant contribution comes from the 3d quark condensate term which accounts for roughly 10% of the area while the 5d mixed and 6d gluon condensate contributions each account for 5%. Light quark mass corrections to massless perturbation theory are numerically insignificant leading to isospin invariance of our results.   The dominant contributions to the error in the charm-light systems come from the gluon condensates and 5d mixed condensate; in the bottom systems, the dominant error originates exclusively from the mixed condensate. All channels are relatively insensitive to errors in the quark condensate, the heavy quark masses, and the quark mass ratios.
Within computational uncertainty, the mass spectra of the heavy-nonstrange hybrid systems are predicted to be degenerate with their heavy-strange counterparts. (Compare Tables 10 and 11 as well as Tables 12 and 13. Also, see Figure 6.) This can be attributed to the small size of the light quark mass correction to massless perturbation theory and to the presence of a heavy quark mass factor as opposed to a light quark mass factor in the 3d quark and 5d mixed condensate contributions to the OPE. In addition, the two spin-1 mass predictions are degenerate in the charm systems and are nearly so in the bottom systems. Furthermore, for charm hybrids, there is a large splitting between the spin-1 mass and the (mean) spin-0 mass with the spin-0 states coming in significantly heavier than the spin-1 states. For bottom hybrids, on the other hand, the relative splittings between states is much less pronounced than for charm hybrids.
In GRW, for each heavy-light hybrid channel whose LSR analysis was stable, the authors pointed out that the difference between the square of the predicted resonance mass and the continuum threshold parameter was small, typically a couple of hundred MeV which did not seem to allow for much in the way of resonance width. In our updated analysis, Tables 10-13 shows that even a relatively wide resonance would be well-separated from the continuum.
We can compare our negative parity charm hybrid results to those computed on the lattice in [31]. For J P = 1 − , we are in agreement on a ground state mass of approximately 3.1 GeV (where we have used M ηc = 2.98 GeV from [1] in order to make the comparison). For J P = 0 − , however, the lattice sees a ground state mass nearly degenerate with that of the J P = 1 − in contrast to the large splitting we observe here between these two states.
As noted in Section 1, the open-flavor structure of the hybrid systems in question precludes the possibility of explicitly exotic J P C states. As such, we might expect a degree of mixing with conventional mesonic states. In our previous work on heavy quarkonium hybrids [27], this possibility of mixing was examined through the addition of a conventional meson to the single narrow resonance model (22) such that (25) becomes where the parameters f conv and m conv are the coupling constant and mass of some nearby conventional meson sharing the same J P values. By including these terms, we can form a sum-rule coupled to the conventional state, which can be used to investigate the dependence of the hybrid mass on the coupling to the conventional state by using known values of conventional meson masses to specify m conv . We see in the resulting Figure 7 that increasing the coupling to the conventional state tends to increase the hybrid mass prediction, indicating our results presented here may outline a lower bound on the hybrid mass if mixing with conventional states is substantial.
In summary, we have performed a QCD LSR analysis of spin-0,1, heavy-light open flavour hybrids. In the OPE, we included condensates up to dimension-six as well as leading-order light quark mass corrections to massless perturbation theory. For all flavour combinations, we extracted a single mass prediction for each J P ∈ {0 ± , 1 ± } (see . Our results were isospin-invariant and degenerate under the exchange of light nonstrange and strange quarks. For a fixed spin, we found that mass predictions were (nearly) degenerate under parity flips, and that the spin-0 hybrids were heavier than the spin-1, particularly in the charm channels. Finally, given that open-flavour hybrids cannot take on exotic J P C quantum numbers, mixing with conventional mesons could be important, and our analysis suggests that such mixing would tend to increase the hybrid mass predictions.  Table 10 value are shown.