Charmed-Baryon Spectroscopyfrom Lattice QCD with N_{f}=2+1+1 Flavors

Charmed-Baryon Spectroscopy
from Lattice QCD with Flavors

Raúl A. Briceñ    Huey-Wen Lin Department of Physics, University of Washington
Box 351560, Seattle, WA 98195, USA
   Daniel R. Bolton Department of Physics, Baylor University
One Bear Pl #97316, Waco, TX 76798, USA

We present the results of a calculation of the positive-parity ground-state charmed-baryon spectrum using flavors of dynamical quarks. The calculation uses a relativistic heavy-quark action for the valence charm quark, clover-Wilson fermions for the valence light and strange quarks, and HISQ sea quarks. The spectrum is calculated with a lightest pion mass around  MeV, and three lattice spacings ( fm,  fm, and  fm) are used to extrapolate to the continuum. The light-quark mass extrapolation is performed using heavy-hadron chiral perturbation theory up to and at next-to-leading order in the heavy-quark mass. For the well-measured charmed baryons, our results show consistency with the experimental values. For the controversial , we obtain the isospin-averaged value  MeV (the three uncertainties are statistics, fitting-window systematic, and systematics from other lattice artifacts, such as lattice scale setting and pion-mass determination), which shows a deviation from the experimental value. We predict the yet-to-be-discovered doubly and triply charmed baryons , , and to have masses  MeV,  MeV,  MeV and  MeV, respectively.


I Introduction

In recent years, interest in charmed-baryon spectroscopy has resurfaced. This excitement has been partly triggered by the first observation of a candidate doubly charmed baryon by SELEX SELEX1 (), as well as its isospin partner  SELEX2 (). The SELEX Collaboration later confirmed their observation of  SELEX_confirm (), but the BABAR BABAR1 (), BELLE BELLE1 (), and FOCUS FOCUS () experiments have seen no evidence for either state of the isospin doublet . The SELEX evidence for this doublet implies unprecedented dynamics, the most surprising of which is the -MeV mass difference between the two states. All other previously observed isospin partners have mass differences one order of magnitude smaller. The ground-state doubly charmed baryon has been previously studied theoretically via various methods, including: the nonrelativistic quark model QM (), the relativistic three-quark model RTQM (), the relativistic quark model RQM (), QCD sum rules QCDsum (), heavy-quark effective theory HQET (), the Feynman-Hellmann theorem FHT (), and lattice quantum chromodynamics (LQCD) latt1 (); latt15 (); latt16 (); latt2 (); latt25 (); latt3 (); latt4 (). Overall, theoretical predictions for this state suggest the mass to be  MeV higher than that observed by SELEX.

There remain many undiscovered doubly and triply charmed baryon states. The recently upgraded Beijing Electron-Positron Collider (BEPCII) detector, the Beijing Spectrometer (BES-III), the LHC, and the future GSI project, the antiProton ANnihilation at DArmstadt (PANDA) experiment, will help further disentangle the heavy-baryon spectrum and resolve puzzles like the one mentioned above. LQCD calculations serve as direct first-principles theoretical input for these experiments.

Currently, LQCD provides the best option for performing reliable calculations of low-energy QCD observables. LQCD is a numerical calculation of QCD, which is necessarily performed in a finite discretized and Euclidean spacetime volume. These approximations introduce an infrared cut-off (the spatial extent ) and an ultraviolet cutoff (the lattice spacing ). The latter of these artifacts has been a source of large systematic errors in the heavy-quark sector of QCD. For heavy-quark masses satisfying , it is natural to control the discretization errors using nonrelativistic QCD (NRQCD) NRQCD (). NRQCD has proven particularly useful when studying physics regarding the bottom quark, but for lattice spacings  fm the charm-quark mass is too small to make the NRQCD approximation justifiable. Alternatively, one can implement relativistic heavy-quark actions RHQ0 (); RHQ1 (); RHQ2 (); RHQ3 (); RHQ4 (), where all corrections are systematically removed.

Several groups have performed lattice charmed-baryon calculations using the quenched approximation latt3 (); latt4 (); latt5 (); latt6 (). Although these have served as benchmark calculations of the charmed-baryon sector, the quenched approximation is a large source of systematic error that is difficult to estimate. Three previous groups have studied the charmed-baryon spectrum using dynamical quarks latt1 (); latt15 (); latt16 (); latt2 (); latt25 (); latt8 (); Alexandrou:2012xk ().

Na et al. latt2 (); latt25 () performed a rather extensive calculation of charm- and bottom-baryon masses at three different lattice spacings ( fm,  fm, and  fm). They used chiral perturbation theory (PT)-inspired polynomial extrapolations of the light-quark masses but refrained from performing a continuum extrapolation of their results. From their results for the doubly charmed baryons, one could infer a  MeV systematic error associated with discretization effects.

Liu et al. latt1 (); latt15 (); latt16 () did a rather nice exploratory calculation over four different pion masses and performed what is probably the best (to this day) chiral extrapolation of the charmed-baryon spectrum using a relativistic heavy quark action for the charm quark. There are a few places where this calculation could be further improved. First, the lightest pion used in their calculation was about  MeV; with advances in technology, we can get closer to the physical point. For baryons with no light degrees of freedom, this is a minor issue, but for isodoublet doubly charmed baryons the light-quark mass dependence is nontrivial. Second, they performed all calculations at a single coarse lattice spacing,  fm, which lies near the upper limit of reliable spacings for studying charm physics. In their work, they used power-counting arguments to give estimates of the discretization effects. In particular, in the doubly charmed sector, they assigned a rather conservative systematic uncertainty associated with discretization effects,  MeV. This is by far their largest uncertainty across all states; for example, their result for the lightest doubly charmed baryon is . Lastly, they restricted themselves to studying the sector. The and charmed baryons are related by heavy-quark symmetries, which results in their chiral extrapolations being coupled. This is particularly relevant when performing a PT-motivated extrapolation of the doublet to the physical point.

The European Twisted-Mass (ETM) Collaboration recently presented results for , , , , , and , using dynamical sea quarks with a lightest pion mass of about  MeV and a relativistic action for the valence charm quark Alexandrou:2012xk (). They used PT-inspired polynomials for the light-quark mass extrapolation, neglecting corrections and chiral-log contributions. Despite having performed calculations at three lattice spacings  fm, they refrained from extrapolating their results to the continuum. Instead, they estimated a discretization error that is incorporated into their systematics. Although historically, the use of dynamical sea quarks was a reasonable approximation, this (like full quenching) introduces a source of systematic error that can only be quantified when results are directly compared to or calculations.

In order to confidently deal with systematics due to discretization effects, it is necessary to perform calculations with highly improved actions, relativistic heavy-quark actions, and multiple lattice spacings in order to extrapolate to the continuum. With these goals in mind, we evaluated the positive-parity ground-state charm-baryon spectrum using two pion masses (with a lightest around 220 MeV) and three lattice spacings ( fm,  fm, and  fm). In this work, we made three extensions to our previous preliminary calculation Briceno (). Firstly, we used an ensemble at the super-fine  fm lattice spacing in order to further constrain the continuum extrapolation. Secondly, when extrapolating the charmed-baryon masses to the physical , we used heavy-hadron PT (HHPT) hhchipt1 (); hhchipt2 (); hhchipt3 (); hhchipt4 () at next-to-leading order (NLO) in and in the heavy-quark mass expansion, while in our previous work we had restricted ourselves to the LO dependence. In order to do this, we extended previous HHPT results hhchipt6 (); hhchipt7 () to include corrections. Thirdly, we quantified systematics associated with finite-volume effects, scale setting, the determination of , corrections to the expressions used to extrapolate to the physical point, and the strange-mass tuning.

This paper is structured as follows. In Sec. II, we outline the formulation of the lattice calculation, including the actions used for the sea, valence light, and valence charm quarks, as well our procedure for setting the scale independently, and the construction of our correlation functions. In Sec. III, we present the tuning of the charm-quark action and show the results for the charmonium spectrum. In this section, we present the results for the splitting, which is shown to have rather large lattice-spacing dependence, but the result presented is in agreement with experiment when extrapolated to the continuum. Section IV outlines our analysis of the charmed-baryon spectrum and includes a detailed discussion of the HHPT expressions for the masses. In this section, the dependence of the charmed-baryon sector is discussed, as well the systematics mentioned at the end of the previous paragraph. Finally, in Sec. V we give a summary of our results and a comparison of the yet-to-be-discovered masses across different models.

Ii Lattice Formulation

ii.1 Light-Quark Action

In this work, we used gauge configurations that were generated by the MILC Collaboration with the highly improved staggered quark (HISQ) MILC (); MILC2 (); HPQCD (); HPQCDUKQCD (); meson4 () action for the sea quarks. The implementation of the HISQ action, first proposed by the HPQCD/UKQCD Collaboration HPQCD (); HPQCDUKQCD (); meson4 (), has been shown to further reduce lattice artifacts as compared to the asqtad action MILC (). Staggered actions reduce the number of doublers to four “tastes”, which are reduced to the desired number of true flavors by taking the fourth-root of the fermionic determinant. As a result, staggered actions have two sources of discretization errors. The first is due to the discretization of the derivative, while the second is associated with taste-exchange interactions in quark-quark scattering. It has been shown that the latter type of errors are suppressed at level when the HISQ action is used for light quarks at lattice spacings of 0.1 fm or less meson4 (). Furthermore, its suppression of errors makes the HISQ action a desirable candidate for studying charm physics on the lattice meson4 (). Lastly, despite the HISQ action being significantly more computationally expensive than the asqtad action hisqvasqtad (), it is still more economical than a non-staggered action. This has allowed the MILC Collaboration to recently generate multiple HISQ ensembles, with a range of lattice spacings  fm and three light-quark (up, down) masses corresponding to  MeV. This variety of ensembles allows for clean extrapolations to the physical pion mass and the continuum limit.

Hypercubic blocking HYP () is implemented on the gauge configurations in order to further reduce the ultraviolet noise from the gauge field. For the valence light (up, down and strange) fermions a tree-level tadpole-improved clover-Wilson action is used 222The light clover propagators were generated and provided by the PNDME Collaboration Lin:2011zz (); Gupta:2012rf (); Bhattacharya:2012wk ()., since the construction of baryon operators with staggered fermions is rather complicated. However, for the coarser and lighter pion mass ensembles (such as 140-MeV pion mass at  fm), one runs into the problem of exceptional configurations where the clover-Dirac operator has near-zero modes exceptional (). Thus, in this work, we were limited to heavier light-quark masses which correspond to  MeV with lattice spacings of around 0.06, 0.09 and 0.12 fm.

Because the actions used for the sea and valence quarks differ, the calculation presented here is partially quenched and violates unitarity. In order to restore unitarity, it is necessary to match the valence- and sea-quark masses, as well as to extrapolate the results to the continuum. Due to the four-fold degeneracy of the staggered action, in the continuum limit it has an chiral symmetry. In this limit, each pion obtains 15 degenerate partners. A finite lattice spacing breaks this symmetry and lifts the degeneracy Orginos:1999cr (); *Orginos:1998ue; *Toussaint:1998sa; *Lagae:1998pe; *Lepage:1998vj. Therefore, there is an ambiguity when tuning the valence-quark mass to the sea-quark mass. We chose to simultaneously tune the light- and strange-quark masses to assure that the valence pion and kaon masses match those of the lightest Goldstone Kogut-Susskind sea pion and kaon masses, as shown in Table 1. The Goldstone Kogut-Susskind sea pion is the lightest pion, the only one that becomes massless in the chiral limit for a nonzero lattice spacing. Ideally, one would want to perform all calculations at a range of light, strange and charm masses and simultaneously extrapolate all masses to their physical values. Due to limited in computational resources, we performed calculations at a single strange quark mass, but as will be discussed in Sec. II.4 our determination of at the continuum and physical is in agreement with experiment. This gives us confidence that the strange-quark mass is tuned properly.

  A1 0.18931(10) 0.32375(12) 0.18850(79)(55) 0.32358(58)(67) 504 2016
A2 0.13407(6) 0.30806(9) 0.13584(79)(59) 0.30894(52)(60) 477 1908
B1 0.14066(13) 0.24085(14) 0.14050(40)(28) 0.24032(39)(23) 391 1564
B2 0.09845(9) 0.22670(12) 0.09950(53)(23) 0.22464(27)(35) 432 1568
C1 6.72 0.09444(9) 0.16204(11) 0.09444(38)(9) 0.16086(29)(68) 330 1320
Table 1: Details of the configurations and propagators used in this work. The subscript “sea” labels the lightest sea pseudoscalar masses from the HISQ action MILC (); MILC2 (), while the subscript “val” labels the valence masses. The sea hadron masses have a single uncertainty due to the statistics, while the valence masses include statistical and systematic uncertainty due to fitting-window selection as defined in Sec. II.2. Additionally, listed are the spatial () and temporal extents () in lattice units, the value of , the number of configurations, and the number of measurements performed for each ensemble.

ii.2 Correlation Functions and Fitting Method

Before discussing the tuning of the charm-quark action, let us explain how we constructed our correlation functions and extracted hadronic masses. For a given interpolating hadron operator, , we construct the two-point correlation functions


where the superscripts and label the smearing type of the annihilation and creation operator, respectively, labels source location, and the sink location. In order to reduce statistical noise, the two-point functions are averaged over four source locations for each gauge configuration.

Both the baryonic and mesonic correlation functions are calculated with gauge-invariant Gaussian-smeared (S) sources and point (P) sinks. For the mesons, we use the generalized Prony-Matrix (PM) method prony () over the smeared-smeared (SS) and smeared-point (SP) correlation functions. The PM method uses the fact that each choice of smearing parameters corresponds to a particular linear combination of the exponentiated masses () and the corresponding overlap factors (), . By computing correlation functions with two sets of smearing parameters, we can determine the two lowest energy states that have overlap with the interpolating operator used by solving the eigenvalue equation


where . One solution to this equation is given by prony ()


where the window size must be in order to ensure the matrices within the brackets are invertible. For each hadron, is chosen in order to maximize the plateau of the ground state. The statistical uncertainties of the extracted hadron masses are evaluated using the jackknife method.

We test the PM method for a subset of the baryonic masses and compare the results with those extracted from a single-exponential and double-exponential fits to the SP correlation function at large Euclidean time. We find these to be in agreement within our systematics, with the single-exponential having the smaller uncertainty. As a result, we choose to extract all masses from the single-exponential behavior of the SP correlation function.

For all energies extracted, we determine the statistical uncertainty and a systematic associated with choosing a fitting window . In order to estimate the latter, for all fitting windows that fall within we calculate the energy, , and goodness of fit (defined as ), which depends on the number of degrees of freedom and is optimally near 1. From this ensemble of energies, we define the systematic as the standard deviation of the energies weighted by .

ii.3 Charm-Quark Action

Since the charm-quark mass is too light to justifiably implement a nonrelativistic action for the lattice spacings used in our calculation, it is necessary to use a relativistic action. To systematically remove the discretization artifacts (where is the charm-quark mass), we use the following relativistic heavy-quark action for the valence charm quark RHQ1 (); RHQ2 (); RHQ3 (); RHQ4 ():


where is the heavy-quark field at the site , are the Hermitian gamma matrices that satisfy the Euclidean Clifford algebra , is the first-order lattice derivative, and is the Yang-Mills field-strength tensor. The parameters must be tuned to assure terms have been removed. For the coefficients and we use the tree-level tadpole-improved results latt1 (); latt15 (); latt16 (); clover_terms () with the tadpole factor defined as , where is the product of gauge links around the fundamental lattice plaquette .

The coefficients and were simultaneously determined nonperturbatively by requiring the ratio to be equal to its experimental value, 1.83429(56), and {, } to satisfy the correct dispersion relation, . In constructing the charmonium correlation functions, we used the local interpolating operators shown in Table 3. The dispersion relation was matched using and energies at the six lowest momenta: , , , in units of , and their rotational equivalents. In practice, we performed the initial tuning with a subset of 40 gauge configurations (with four sources each). Clearly this procedure does not guarantee correct tuning upon analysis of the full ensemble. Therefore, we used two separate charm-quark masses and extrapolated to the physical charmonium mass. These two points allowed us to interpolate linearly in to the physical charm-quark mass defined by . The valence charm-quark masses used for each ensemble are shown in Table 4. Figure  1 shows examples of the resulting dispersion relations for the and with full statistics after extrapolating to the physical charm mass from one of the ensembles, C1, and they show that the slopes are consistent with 1.

  A1 0.11926(77)(51) 1.0291(56)(37) 0.901 0.872
A2 1.0192(31)(21) 0.900 0.853
B1 0.7562(81)(52) 0.561 0.536
B2 0.7463(52)(25) 0.552 0.522
C1 0.5148(17)(39) 0.319 0.309
Table 2: The lattice spacings and masses cited include the statistical and systematic uncertainties due to the fitting window. The lattice spacings are determined by the chiral extrapolation of the mass to the physical value of for each value of 444Note that we calculate the mass on 200 configurations for  fm, 220-MeV to fix the lattice spacing for ensemble C1.. Additionally listed are the two bare masses of the valence charm quarks used for each ensemble.
Figure 1: A sample dispersion relation for the and . The six points correspond to energies (and uncertainties) for the at the six lowest-momenta states: , , , in units of , and their permutations. The red line is the resulting fit to the data using the relativistic dispersion relation , and the blue band includes the statistical and systematic errors added in quadrature. The energies shown are obtained using the full statistics of the C1 ensemble and have been extrapolated to the physical charm mass. From the fit we obtain the speed of light and its statistical and systematic uncertainties, and .

ii.4 Lattice-Spacing Determination and Discussion of Ratios

Figure 2: PT and continuum extrapolations of the kaon mass. The red line indicates the fit of the data that has been extrapolated to . The blue band includes the statistical and systematic errors added in quadrature.

As mentioned earlier, it is necessary to evaluate the spectrum at multiple lattice spacings in order to simultaneously restore unitarity and remove discretization errors. With this in mind, we perform the calculation at three lattice spacings,  fm, 0.09 fm and 0.12 fm. For the coarse ( fm) and fine ( fm) lattice spacings, we use two different light-quark masses corresponding to  MeV; for the super-fine ( fm) ensemble we use a single light quark,  MeV. We calculate the mass on 200 configurations for  fm and  MeV to fix the lattice spacing for ensemble C1.

In order to obtain physical masses in the continuum, it is necessary to determine the lattice spacing for the five ensembles used. Currently, the most precise determination of lattice spacings for the MILC ensembles is by the HPQCD Collaboration HPQCD (); however, their determinations of the lattice spacings for the B2 and C1 ensembles remain unpublished. For this reason, we perform our own determination. Due to the small -dependence of (at the few-percent level) we choose to set the scale by extrapolating the value of across all ensembles with the same value of to the physical pion mass. We define the lattice spacing by dividing by the physical mass,  MeV.

In constructing the correlation functions for the , we use as the interpolating operator. The strange-quark annihilation operator is denoted with color index , are the symmetric spin matrices (where is the charge-conjugation matrix), is the positive-parity projection operator, and are the spin-projection operators for spin- particles.

One can determine as a function of via PT, but this expression suffers from rather large expansion parameters (, ) and does not always describe lattice baryon masses well. Alternatively, it has been proposed that the hyperon masses can be extrapolated using a two-flavor chiral perturbation theory su2 (). With a faster convergence than its three-flavor counterpart, the advantages of this approach are clear. The cost is manifested in a larger set of unknown coefficients. Using PT for the hyperons, the mass as a function of up to is su2 ()


where  MeV is the pion decay constant, and the {} are the low-energy coefficients (LECs) of the theory. Because at each lattice spacing we have (at most) two ensembles with two corresponding values of , we are forced to truncate Eq. 5 at in order to retain a reasonable level of precision. This truncation introduces a systematic uncertainty into our calculations which will be accounted for in Sec. IV.2.

Further details of the ensembles, including our determination of the lattice spacing and the mass are listed in Table 4. The values determined by the MILC Collaboration are  fm for the coarse and  fm for the fine. The HPQCD Collaboration performed a rather extensive program in which they determined the lattice spacing for each ensemble using three different quantities: Upsilon - splitting, the decay constant of the meson, and the ratio HPQCD (). We determine a single lattice spacing for each value of and find central values that are consistently below both the MILC and HPQCD central values. This difference in the definition of the lattice spacing should have no impact on continuum-extrapolated results.

Table 4 shows that the lattice spacing for the ensemble C1 is currently determined at the level of precision. For the same reasons discussed above, we choose to determine the physical hadron masses using the ratio. As will be shown, the is determined at the sub- level for all ensembles and particles. Due to the removal of the -terms in Eq. 5, we proceed to truncate all of our chiral fits at the level of accuracy, and estimate a systematic error associated with this approximation (see Sec. IV.2).

Because we are using the strange mass to set the scale, it is important to first test the strange-mass tuning, which we do using the kaon mass. For all the pseudoscalar mesons, we use the standard local operators , where is the annihilation operator for a quark with flavor and color index . As discussed in Ref. kchipt (), when reducing the symmetry of PT from to , kaons can be represented as a matter field that couples to the chiral currents. This treatment of the kaons is referred to as PT. The advantage of PT is that the largest expansion parameter is . Using PT, the kaon mass as a function of is found to be kchipt ()


where is the bare kaon mass, and the -dependence is parametrized by . For the kaon and for all other hadrons studied in this work, the continuum-limit mass is recovered by multiplying the ratio at the physical point by the physical mass,  MeV.

In Table 1 the valence kaon masses are shown for each ensemble. In Fig. 2 we show the values for the kaon mass for each ensemble with the corresponding statistical and systematic uncertainties as a function of , as well as the chiral extrapolation at the continuum. Figure 2 shows that the lattice-spacing dependence of the kaon is rather small, and that the extrapolated value,  MeV (the three uncertainties are statistics, fitting-window systematic, and systematics from scale setting, -corrections to the expressions used to extrapolate to the physical point, finite volume, and strange-mass tuning as discussed in Sec. IV.2), agrees with experiment within our systematics. This confirms our strange-mass tuning as well as our scale determination and extrapolation procedure using the ratio.

Iii Charmonium Spectrum

Figure 3: Diagrams that contribute to the charmonium correlation functions. In this work we evaluate the contribution from connected diagrams (left) and neglect disconnected diagrams (right). The latter are OZI suppressed, and previous lattice calculations have determined their contributions to be consistent with zero disc (); disc2 (); disc2 (); disc3 ().

In this section, we calculate the charmonium splitting and the rest of the charmonium spectrum in the continuum limit, and we compare them with experimental and previous dynamical lattice results. We use the ratios of spin averages of and masses to baryon masses to tune the charm-quark mass for each ensemble; thus, the splitting between them is not fixed in our calculations. Any deviations from the well-measured experimental values give us an estimation of the final systematics.

In constructing the meson correlation functions, we restrict ourselves to the local interpolating operators shown in Table 3. In order to evaluate the full correlation functions of the charmonium spectrum, we need to perform two different types of propagator contractions, as depicted in Fig. 3, connected and disconnected diagrams. Disconnected diagrams would increase the number of propagators needed by approximately two orders of magnitude but are suppressed by the OZI rule OZI1 (); *OZI2; *OZI3; *OZI4. Previous lattice calculations at zero temperature have shown disconnected diagrams in the charmed sector are rather noisy, and their contributions to the hyperfine splitting are in the range of 1–4 MeV and consistent with zero disc (); disc2 (); disc2 (); disc3 (). Thus, we neglect contributions arising from disconnected diagrams here. Figure 4 displays examples of the effective-mass plots after performing the generalized Prony-Matrix method for the charmonium sector, and the charmonium masses for each ensemble are shown in Table 4 in lattice units.

Figure 4: Sample effective-mass plots of the charmonium spectrum from the various ensembles. The errorbar shown includes the statistical and systematic uncertainty (from varying the fitted range) added in quadrature.

For every hadron, we calculated the ratio of its mass to the mass, at the two different values of the charm-quark mass. After interpolating these to the physical charm-quark mass for each ensemble, we simultaneously extrapolated the five values of the hadron masses to the continuum and the physical . To perform the light-quark mass extrapolation we use the PT expressions, which up to is linear in :


where is the bare charmonium mass, and the -dependence is parametrized by .

Hadron Interpolator

Table 3: Interpolating operators for the charmonium spectrum. labels the the charm quark with color index .

1.86213(61)(21) [8–16] 1.85571(32)(5) [9–15] 1.37703(27)(18) [11–18] 1.36696(41)(26) [13–23] 0.93723(18)(11) [12–32]
1.83438(47)(24) [8–16] 1.80666(32)(5) [9–15] 1.34397(28)(20) [11–18] 1.32897(41)(15) [13–23] 0.92157(18)(9) [12–32]
1.86213(61)(19) [7–13] 1.85571(32)(4) [16–24] 1.37703(27)(15) [24–36] 1.36696(41)(39) [20–25] 0.93723(18)(14) [18–36]
1.83438(47)(27) [7–13] 1.80666(32)(4) [16–24] 1.34397(28)(20) [24–36] 1.32897(41)(13) [20–25] 0.92157(18)(16) [18–36]
1.91025(50)(29) [5–17] 1.90212(54)(21) [13–17] 1.41634(78)(19) [26–30] 1.40612(53)(18) [21–26] 0.96470(29)(38) [18–36]
1.88354(44)(30) [5–17] 1.85446(55)(15) [13–17] 1.38428(81)(29) [26–30] 1.36975(54)(9) [21–26] 0.94955(30)(22) [18–36]
2.1382(22)(19) [4–9] 2.1264(23)(34) [5–12] 1.5873(29)(27) [7–10] 1.5599(44)(22) [12–23] 1.0619(34)(20) [17–22]
2.1126(19)(17) [4–9] 2.0787(23)(28) [5–12] 1.5537(28)(25) [7–10] 1.5209(46)(21) [12–23] 1.0557(19)(20) [17–22]
2.164(11)(5) [11–16] 2.1574(56)(36) [9–12] 1.6121(26)(12) [3–9] 1.6001(53)(31) [12–23] 1.0966(37)(16) [12–18]
2.133(10)(4) [11–16] 2.1104(57)(44) [9–12] 1.5807(26)(16) [3–9] 1.5631(55)(33) [12–23] 1.0814(39)(10) [12–18]
2.1612(93)(60) [7–10] 2.1573(54)(62) [9–14] 1.6296(59)(45) [10–17] 1.6078(59)(45) [12–23] 1.0904(89)(36) [18–24]
2.1373(90)(65) [7–10] 2.1105(55)(35) [9–14] 1.5952(83)(74) [10–17] 1.5709(59)(47) [12–23] 1.0869(53)(20) [18–24]
1.20785(70)(38)[13–23] 1.20348(65)(29)[8–15] 0.89883(46)(40)[13–29] 0.88914(62)(46)[13–23] 0.61196(53)(37)[18–26]
1.19203(69)(40)[13–23] 1.17734(64)(26)[8–15] 0.88112(45)(36)[13–29] 0.86803(59)(45)[13–23] 0.60333(52)(28)[18–26]

Table 4: Charmonium and masses in lattice units for the five ensembles and two charm masses. Errors listed are statistical and fitting window systematic. The fitting window is given in square brackets.
Figure 5: Our determination of the low-lying charmonium spectrum after extrapolating to the physical point, labeled as “Briceno et al.”, as well as a survey of previous unquenched lattice calculations meson1 (); disc3 (); meson3 (); meson4 (); meson5 (). Mohler et al. determined the spitting between and  meson1 (); in order to compare their results with ours, we have set to its physical value, while leaving their hyperfine splitting unchanged. The statistical uncertainty is shown as a thick inner error bar, while the statistical and systematic uncertainties (if estimated in the paper) added in quadrature are shown as a larger thin error bar. Our systematic uncertainties include errors originating from the fitting window, scale setting, pion mass determination, finite-volume effects, -corrections to the expressions used to extrapolate to the physical point, and the strange mass tuning (as discussed in Sec. IV.2). The orange line/band indicates the experimentally measured masses or splittings with their corresponding uncertainties pdg ().
Figure 6: PT and continuum extrapolations of the splitting. The red line indicates the fit of the data that has been extrapolated to . The blue band include the statistical and systematic errors added in quadrature.

Using this procedure, we have verified that our calculations reproduce the experimental low-lying charmed-meson spectrum. In Fig. 5 we show our results for the charmonium spectrum (as well as the hyperfine splitting ) after extrapolating to the physical point. As a result, our errorbars are larger than those of other calculations. For comparison, we show in Fig. 5 a sample of previous dynamical lattice calculations that have studied the charmonium spectrum. By comparing the level of precision of (see Table 4) and (see Table 4), one can see that it is the uncertainty of that dominates the overall uncertainty of the ratio.

The works by Bali et al. and Mohler et al. are far more extensive than the small sample that is being represented here. Both groups used the variational method over different sources and sinks to not only extract ground-state energies but also those of the excited states. Mohler et al. evaluated the spectrum for the systems for a range of six pion masses ranging from 702 MeV to 156 MeV at a single lattice spacing,  fm. On the other hand, Bali et al. evaluated the spectrum, including disconnected diagrams, at three lattice spacings but did not provide a continuum-extrapolated result for the spectrum or an estimate of the discretization error.

The conclusion of Fig. 5 is evident: these non-continuum results come with a large systematic error due to nonzero lattice spacing. This error decreases with lattice spacing, but from Fig. 5 it is clear that in order to reproduce the physical spectrum, it is necessary to extrapolate masses to the continuum. For example, in the upper figure in Fig. 5 we see that despite our masses having the largest uncertainties, ours are the only results that are consistently in agreement with experiment. We conclude that previous calculations that do not extrapolate their results to the continuum have underestimated their systematic errors.

When tuning the charm mass to the spin-averaged mass, , the most natural quantity to study is the hyperfine splitting . As a result, this splitting has received a great deal of attention in the community. One surprising feature is that for a finite lattice spacing, is underestimated meson1 (); meson4 (). In our calculations we find the value of agrees with experiment only after extrapolating to the continuum. This is consistent with the findings of the HPQCD/UKQCD Collaboration meson1 () and Fermilab Lattice and MILC Collaborations meson5 (), as shown in the lower part of Fig. 5. Therefore, it cannot be overstated that charmed quantities need to be evaluated at multiple lattice spacings to properly quantify the systematics.

In order to further test the strange- and charm-mass tuning, we evaluated the splitting. This is the binding-energy difference between the heavy-light and heavy-heavy systems; there is no reliable analytical procedure for calculating this quantity. Since the strange-charm meson, , has no light degrees of freedom, up to its mass is linear in , therefore the splitting can be extrapolated using


where denotes the bare splitting, and we extrapolate to the continuum using .

In Table 4, the and meson masses are shown for each ensemble. Figure  6 shows the values for the splitting after continuum extrapolation, along with their corresponding statistical and systematic uncertainties (see Sec. IV.2). Figure 6 shows that the -dependence of is sizable; in fact, continuum extrapolation is necessary in order to reproduce the physical value. In performing the continuum extrapolation of , we find the -dependent LEC to be . Since our determination of the and spectrum is in agreement with experiment, we believe that the estimates of the systematics in Sec. IV.2 accurately reflect the sources of systematic error of the calculation presented in this paper.

Figure 7: Sample effective-mass plots from the various ensembles of the charmed-baryon sector. The errorbar shown includes the statistical and systematic uncertainty (from varying the fitted range) added in quadrature.

Table 5: The interpolating operators for the positive-parity baryons UKQCD (). respectively denote the up-, down- and strange-quark annihilation operators, denotes the charm-quark operator, are color indices, while denote polarization indices. are the antisymmetric and symmetric spin matrices , where is the charge-conjugation matrix. In order to have the best possible overlap with the state of interest, we have used the spin projection operators and , and the positive-parity projection operator .

Iv Charmed-Baryon Spectrum

With confidence that our tuning reproduces the low-lying spectrum within our systematics, we proceed to evaluate the positive-parity charmed-baryon spectrum. Heavy-quark symmetry dictates that the quantum numbers of the light degrees of freedom of any heavy-light system are conserved. One can identify approximately degenerate multiplets by these quantum numbers. For singly charmed baryons, the light degrees of freedom can have total spin equal to zero or one. Under chiral symmetry, the spin-singlet multiplet transforms as a irrep. The spin triplet is a irrep when the total angular momentum is and a irrep when the total angular momentum is . In the heavy-quark limit, these are degenerate. The doubly charmed baryons form a irrep when the total angular momentum is and a irrep when the total angular momentum is . The triply charmed baryons are singlets under . This algebra was manifested by the interpolating operators used in this calculation, as shown in Table III UKQCD (). Figure 7 displays examples of the effective-mass plots for various correlation functions. Table 6 lists the baryon masses in lattice units for each charm-quark mass and ensemble along with the statistical and fitting-window systematic uncertainties and the chosen fitting window.

1.4561(42)(70)[8–16] 1.4228(77)(73)[9–15] 1.0808(42)(33)[11–15] 1.0328(102)(79)[16–25] 0.7339(56)(15)[19–22]
1.4401(42)(70)[8–16] 1.3976(76)(69)[9–15] 1.0643(41)(35)[11–15] 1.0136(98)(62)[16–25] 0.7258(56)(15)[19–22]
1.5333(24)(28)[8–15] 1.5120(31)(20)[8–15] 1.1438(37)(21)[14–18] 1.1115(37)(47)[14–27] 0.7747(48)(10)[26–29]
1.5174(24)(27)[8–15] 1.4871(31)(21)[8–15] 1.1274(37)(19)[14–18] 1.0922(33)(20)[14–27] 0.7665(48)(10)[26–29]
1.5521(40)(30)[8–11] 1.5286(50)(54)[8–16] 1.1703(43)(25)[11–16] 1.1351(80)(78)[13–23] 0.7968(32)(54)[13–22]
1.5359(40)(30)[8–11] 1.5028(50)(51)[8–16] 1.1530(43)(29)[11–16] 1.1134(74)(52)[13–23] 0.7883(32)(54)[13–22]
1.6178(43)(48)[7–11] 1.5760(91)(44)[9–15] 1.1979(83)(53)[13–19] 1.1731(105)(167)[13–20] 0.8055(83)(29)[19–24]
1.6020(43)(50)[7–11] 1.5516(91)(42)[9–15] 1.1812(82)(55)[13–19] 1.1569(97)(76)[13–20] 0.7975(83)(29)[19–24]
1.5878(60)(78)[12–23] 1.5820(55)(54)[11–18] 1.1925(51)(14)[16–22] 1.1682(49)(34)[15–21] 0.8089(23)(22)[12–23]
1.5717(60)(86)[12–23] 1.5564(55)(51)[11–18] 1.1753(50)(15)[16–22] 1.1471(44)(16)[15–21] 0.8005(22)(23)[12–23]
1.662(3)(14)[8–18] 1.6388(58)(41)[10–14] 1.2314(65)(41)[15–21] 1.2060(54)(48)[14–21] 0.8328(54)(17)[20–24]
1.646(3)(14)[8–18] 1.6142(57)(39)[10–14] 1.2157(64)(39)[15–21] 1.1896(51)(9)[14–21] 0.8248(53)(17)[20–24]
1.6487(69)(16)[16–24] 1.6393(22)(24)[8–14] 1.2280(45)(17)[19–23] 1.2129(28)(3)[15–19] 0.8341(25)(25)[18–24]
1.6322(69)(16)[16–24] 1.6138(22)(24)[8–14] 1.2112(45)(16)[19–23] 1.1919(25)(3)[15–19] 0.8262(24)(24)[18–24]
1.6960(38)(52)[11–20] 1.6882(27)(29)[8–14] 1.2567(64)(34)[19–26] 1.2493(32)(17)[14–19] 0.8567(24)(24)[15–30]
1.6805(38)(52)[11–20] 1.6638(27)(28)[8–14] 1.2408(64)(29)[19–26] 1.2313(29)(7)[14–19] 0.8489(23)(25)[15–30]
2.2349(33)(42)[11–25] 2.2194(67)(61)[15–22] 1.6628(21)(13)[6–16] 1.6413(46)(17)[17–25] 1.1298(25)(12)[19–29]
2.2037(33)(39)[11–25] 2.1701(66)(56)[15–22] 1.6394(48)(50)[6–16] 1.6070(39)(21)[17–25] 1.1139(25)(12)[19–29]
2.3053(26)(27)[8–16] 2.2455(115)(72)[15–19] 1.6381(55)(47)[18–26] 1.6801(66)(37)[17–22] 1.1570(91)(32)[32–41]
2.2744(25)(27)[8–16] 2.1970(114)(73)[15–19] 1.6808(29)(44)[18–26] 1.6459(56)(27)[17–22] 1.1416(91)(34)[32–41]
2.2893(28)(9)[17–26] 2.2739(22)(12)[15–27] 1.7008(18)(2)[18–26] 1.6786(33)(14)[24–28] 1.1562(14)(4)[19–29]
2.2580(28)(10)[17–26] 2.2247(21)(12)[15–27] 1.6677(18)(3)[18–26] 1.6417(28)(6)[24–28] 1.1403(14)(4)[19–29]
2.3385(66)(29)[11–19] 2.3178(31)(19)[15–23] 1.7331(43)(10)[23–29] 1.7180(38)(23)[20–26] 1.1796(21)(6)[26–31]
2.3078(66)(29)[11–19] 2.2694(31)(19)[15–23] 1.7001(43)(9)[23–29] 1.6799(35)(16)[20–26] 1.1641(21)(6)[26–31]
2.9621(16)(9)[16–24] 2.9466(15)(17)[16–24] 2.1953(15)(7)[32–39] 2.1788(18)(2)[21–28] 1.4921(22)(8)[38–43]
2.9161(16)(8)[16–24] 2.8753(15)(17)[16–24] 2.1472(16)(8)[32–39] 2.1239(17)(2)[21–28] 1.4690(23)(4)[38–43]
Table 6: Charmed-baryon masses for the five ensembles in lattice units, statistical and fitting window systematic uncertainties, and fitting windows.

iv.1 Chiral and Continuum Extrapolation

As discussed in Sec. II.3, the ratios of each charmed-hadron mass to the mass are interpolated to the physical charm mass, defined by . After this is done for each ensemble, it is necessary to extrapolate the ratios to the physical light-quark mass and continuum. Due to the rather large expansion parameter of PT and poorer convergence rate, we use HHPT to extrapolate the baryon masses to the physical pion mass. Previous HHPT calculations of the singly charmed-baryon masses used the static limit,  hhchipt6 (); hhchipt7 (). At new operators are introduced that explicitly break the - degeneracy hhchipt5 (), resulting in three independent bare splittings . We extend previous work to include the corrections for the and multiplets by evaluating the contribution arising from the two self-energy diagrams depicted in Fig. 8.

Figure 8: Two of the self-energy diagrams contributing to the masses of a singly charmed baryon in the 6 irrep. The first depicts contributions arising from loops containing a pion and a member of the 6 irrep, while the second correspond to loops containing a pion and a member of the irrep. There are similar self-energy diagrams for baryons in the irrep.

First, consider the multiplet. Up to , the dependence of the ratio of the particle masses to can be written as


where , and label the bare masses and splittings, and ’s and ’s are the LECs of the theory. The chiral function is defined as