Bled Workshops in Physics Vol. 19, No. 1 p. 10
A
Proceedings of the Mini-Workshop Double-charm baryons and dimesons
Bled, Slovenia, June 17-23, 2018

Heavy baryon spectroscopy from lattice QCD
M. Padmanath
Institüt für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
Abstract. In this report, the most recent and precise estimates of masses of ground state baryons using lattice QCD are discussed. Considering the prospects in the heavy baryon sector, lattice estimates for these are emphasized. The first and only existing lattice determination of the highly excited Oc excitations in relation to the recent LHCb discovery is also discussed.
1	Introduction
Since its inception, heavy hadron physics continues to be in the limelight of scientific interests in understanding the nature of strong interactions. While heavy mesons have been studied extensively both experimentally and theoretically [13], studies on heavy baryons remained dormant. In this respect, the year 2017 featured two important landmarks in the heavy baryon physics. First of this is the unambiguous observation by LHCb collaboration of five new narrow D.c resonances in E+K- invariant mass distribution in the energy range between 3000 — 3120 MeV [4]. Four out of these five resonances were later confirmed by Belle collaboration [5]. Second landmark is the discovery of a doubly charmed baryon, E+c(ccu) with a mass of 3621.40 ± 0.78 MeV by LHCb Collaboration [6]. Anticipating the discovery of many more hadrons (including baryons) from the huge data being collected at LHCb and Belle II, heavy hadron spectroscopy using ab-initio first principles methodology such as lattice QCD is of great importance.
Lattice QCD has been proven to be a novel non-perturbative technology in investigating the physics of low energy regime of QCD. Remarkable progress has been achieved over past ten years in making large volume simulations with physical quark masses, impressive statistical precision and good control over the systematic uncertainties [7-10]. In this report, a collection of lattice determinations of baryon masses that are well below allowed strong decay thresholds are summarized. A recent and only existing calculation of excited Hc baryons is discussed and a qualitative comparison with the experiment is made.
2	Lattice methodology
Hadron spectroscopy on the lattice proceeds through evaluation of Euclidean two point correlation functions,
(1)
Heavy baryon spectroscopy from lattice QCD
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between different hadronic currents (Oi(t)) that are carefully built to respect the quantum numbers of interest. A generic baryon current or interpolator has a structure
Oi (x,t) = eabcSf6(x)qa,a(x)qb,p(x)q3,6(x),	(2)
where qj are the quark fields, e is the color space anti-symmetrizing Levi-Civita tensor and S carries all the flavor and spatial structure of the interpolator that determines the quantum information. Cij(tf — ti) are evaluated on lattice QCD ensembles that are generated via Monte Carlo techniques. A general practice is to compute matrices of correlation functions between a basis of carefully constructed interpolating currents Oi(t) and solving the generalized eigenvalue problem (GEVP) [11-13]
Cij (t) vn (t — to) = An (t — to) Cij (to )vn (t — to).	(3)
Hadron energies (En) are extracted from non-linear fits to the large time behavior of the eigenvalues An(t — t0). The eigenvectors (vn(t — t0)) are related to the operator state overlaps (Zn = (Oi|n)) that carry the quantum information of the propagating state. Basic principles remain the same as above, while details of the methodology differ between different groups in the lattice community. e.g. lattice ensembles being used in the study, lattice formulation of action for the fermion and the gauge fields, the hadron interpolators, different degree of control over the lattice systematics, etc. The success of lattice investigations are reflected in mutual agreement of the results they provide and their agreement with experiments.
All results presented in this report are estimated within the single hadron approximation, where only three quark interpolators (as in eqn. 2) are considered in the analysis and neglects effects of any nearby strong decay thresholds. This is a justifying assumption for most of the baryons discussed in this report, considering the fact that all of them are deeply below the respective lowest strong decay thresholds. Results for those baryons, which might be influenced by any nearby threshold effects will be alerted in the respective discussions.
3 Results
Light, strange and singly charm baryons : We begin our discussion with some benchmark calculations of baryon ground states that are experimentally well determined. In Fig. 1, a summary of lattice QCD estimates for the positive parity light baryon ground states (figure adapted from Ref. [10]) are presented at the top and for positive parity singly charm baryon ground states are shown at the bottom. Most of the baryons being discussed are deeply bound and stable to strong decays. Their masses as determined from the discrete energy spectrum on the lattice agree quite well with experiments. Agreement between all the lattice estimates with varying degree of control over the systematics involved in respective calculations and with the experiments demonstrate the power of lattice QCD techniques in making reliable predictions. However, lattice estimates for masses of baryon resonances, such as A, I* and E* that can decay strongly, are less rigorous. They demand a computation of correlation matrices build out of baryon
12 M. Padmanath
1.8 1.7 1.6 1.5 1.4 1.3 1.2 I.I 1
0.9
ETMC Nf -2 ET MC N, = 2 + 1 + 1 QCDSF-UKQCD N, = 2+1 PACS-CS N, = 2+1 bmw N, = 2+1



IT^1"'
ik

N
h a
n
2800
d) s
' 2600
— : Expt '17		t	: HSC	15
I	: ILGTI'13-'18	1	: Brown et.al. '14	
1	: TWQCD '17	t	: PACS-CS '13	
+	: ETMC '17	t	: Briceno et.al. '12	
+	: RQCD'15		: Durretal. '12	
				
				
4		V*		
Ac	Zc Zc* Sc	77/		Oc o;
Fig. 1. Top: summary of lattice estimates for positive parity light and strange baryons from selected lattice investigations - ETMC Nf=2 [10], ETMC Nf=2+1+1 [14], QCDSF-UKQCD Nf=2+1 [15], PACS-CS Nf=2+1 [8] and BMW Nf=2+1 [7], Bottom: summary of lattice estimates for positive parity singly charm baryons : ILGTI '13-'18 [16,17], TWQCD '17 [18], ETMC '17 [10], RQCD '15 [19], HSC '15 [20,21], Brown et nl '14 [22], PACS-CS '13 [8], Briceno et nl '12 [23], Dürr et nl '12 [24],
interpolators (as in eqn. 2) plus baryon-meson interpolators (corresponding to the allowed strong decay modes). The masses of baryon resonances then have to be inferred from the infinite volume scattering matrices build from the discrete spectrum extracted from such correlation matrices. Such investigations are being practised extensively by many collaborations to understand various mesonic resonances (see Ref. [3]), while existing lattice investigations of baryon resonances in this direction are limited to a few [25,26].
Doubly heavy baryons : In Fig. 2, a summary of lattice QCD estimates for positive parity doubly charm baryon ground states at the top is presented. For the Ecc(1 /2+) baryon, good agreement between all lattice estimates (all of which predates the LHCb-discovery [6]) and with LHCb estimate is quite evident from the figure. At this point, the reader is reminded of the observation of another baryon resonance by SELEX collaboration in 2002 [27] at a mass of 3519(1) MeV,
Heavy baryon spectroscopy from lattice QCD
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S
O
Expt '17 ILGTI '13 - '18 TWQCD '17 ETMC '17 RQCD '15


HSC '15 Brown et.al. '14 PACS-CS '13 Briceno et.al. '12 Durr et al. '12
I
"'I

TT*
—cc
Qc
G
AS —♦—
A5 •—
— experiment • QCD+QED ■j prediction
AD —♦—
AN
Fig. 2. Top: summary of lattice estimates for positive parity doubly charm baryons. References as given in Fig. 1 caption. Bottom: Hadron isospin splittings as determined by BMW collaboration [9].
which is addressed as a Ecc(l /2+) baryon. All lattice estimates, being well above this energy, disfavors this observation. The bottom figure shows a summary of baryon isospin splittings as calculated by BMW collaboration [9]. This calculation involved lattice QCD and QED computations with four non-degenerate fermion flavors to estimate the isospin mass splitting in the nucléon, I, E, D and Ecc isospin multiplets. Precise estimation of the neutron-proton isospin splitting and the other known splittings demonstrate the reliability of these estimates. In this calculation, the isospin splitting of Ecc(l/2+) baryon was estimated to be 2.16(11)(17) MeV. This excludes the possibility that LHCb and SELEX candidates for Ecc(l/2+) baryon are isospin partners.
Estimates for other doubly charm baryons, that are yet to be discovered, can also be observed to be very well determined and consistent between different lattice calculations from the top of Fig. 2. Anticipating a near future discovery of the charmed-bottom hadrons at LHCb, at the top of Fig. 3 lattice predictions for such hadrons from a recent investigation [28] are shown. The lattice prediction for only know charmed-bottom hadron, Bc meson, is found to be in agreement with the
14 M. Padmanath
experiment, while the lattice predictions for other channels considered are consistent with another preceding calculation [22] with less control over systematics.
0.4
0.3
0.1
ac h
3/2* 1/2*
Expt.
Lattice
3/2~
S, K (S)
Fig. 3. Top: summary of lattice estimates for low lying charmed-bottom hadrons as determined in Ref. [28]. Bottom: Comparison plot from Ref. [32] between the lattice estimates and the experimental values for the energies of Oc excitations.
SK
Excited baryons : As discussed in the introduction, one of the major landmark in the year 2017 is the LHCb discovery of five narrow Hc resonances in E+K-invariant mass distribution in the energy range between 3000 — 3120 MeV [4]. Following this discovery, Belle collaboration has confirmed four out of these five excited states [5]. Many more highly excited baryons are coming into light with more discoveries. e.g. the observation of a H*-(3/2-) candidate with a mass of 2012.4(9) MeV by Belle collaboration [29], which is in very good agreement with lattice prediction for such a baryon [30,31]. Below we discuss the first and only existing lattice investigation of highly excited Hc resonances (Ref. [32]) that predicts the five excited D.c baryons as observed by LHCb.
Heavy baryon spectroscopy from lattice QCD	15
Following a detailed baryon interpolator construction procedure as invented in Ref. [33,34] a large basis of baryon interpolators, that is expected to extensively scan the radial as well as orbital excitations, are built. By solving the GEVP for correlation matrices constructed out of these interpolators on a lattice ensemble with mn ~ 391 MeV (for details see [21]), one extract the Hc baryon spectrum on the lattice. The bottom of Fig. 3 shows a comparison of the lattice energy estimates for the lowest nine Hc excitations with the seven experimentally observed nc resonances. The relevant strong decay thresholds in the infinite volume are shown as black lines at the top, whereas the black lines at the bottom indicate the relevant non-interacting levels on the lattice. The lowest two levels represent the well known 1/2+ and 3/2+ excitations. Lattice estimates for these excitations agree well with the experiment. In the energy region, where the five narrow resonances were observed, lattice predicts exactly five levels. Of these five excitations four are in good agreement with the experiment, while the fifth is possibly a 5/2-baryon related to the remaining higher lying experimental candidate. Identifying the quantum information of these lattice levels from the Z^s, these five states are argued to be the p-wave excitations [32].
Considering the exploratory nature of this first study, investigating Hc baryon spectrum on multiple lattice ensembles with close to physical mn and larger volumes would be an immediate extension. It would also be an interesting direction to extract the infinite volume scattering matrices considering the allowed baryonmeson scattering channels in the analysis of desired quantum channels in appropriate lattice ensembles. However, the presence of a valence heavy quark, the absence of any valence light quarks and the resonance widths being quite narrow (< 10 MeV) [6] indicates our estimates to be robust with such extensive investigations.
4	Summary
Over the past decade, lattice QCD has availed multiple precision determinations of the ground state baryon masses using full QCD lattice ensembles with good control over the systematic uncertainties. A summary of lattice determinations of various baryons along with their masses from experiment, where available, are given in Figs. 1, 2 and 3. The only existing exploratory lattice determination of the highly excited D.c states in relation to the recent LHCb discovery and its possible extensions are also discussed.
5	Acknowledgements
I would like to thank my collaborators N. Mathur, R. G. Edwards, M. J. Peardon and S. Mondal. I also express my thanks to the organizers and the participants of the Bled workshop for various interesting and insightful discussions. I acknowledge the support from EU under grant no. MSCA-IF-EF-ST-744659 (XQCD-Baryons) and Deutsche Forschungsgemeinschaft Grant No. SFB/TRR 55.
16 M. Padmanath
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