The static-light baryon spectrum from
twisted mass lattice QCD
Marc Wagner, Christian Wiese
Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, Germany
April 26, 2011
We compute the static-light baryon spectrum by means of Wilson twisted mass lattice QCD using flavors of sea quarks. As light valence quarks we consider quarks, which have the same mass as the sea quarks with corresponding pion masses in the range , as well as partially quenched quarks, which have a mass around the physical value. We consider all possible combinations of two light valence quarks, i.e. , , and baryons corresponding to isospin and strangeness as well as angular momentum of the light degrees of freedom and parity . We extrapolate in the light and in the heavy quark mass to the physical point and compare with available experimental results. Besides experimentally known positive parity states we are also able to predict a number of negative parity states, which have neither been measured in experiments nor previously been computed by lattice methods.
In this work we report on a lattice computation of the spectrum of baryons made from a heavy quark and two light quarks, which are , and/or .
Experimentally five baryon states have been observed. While has first been detected quite some time ago, , , and have only been discovered recently [1, 2, 3, 4, 5]. For the mass of there are two different results, which are not in agreement.
On the theoretical side there are a number of lattice studies of the spectrum of baryons. Some of these consider static heavy quarks [6, 7, 8, 9, 10, 11] using Heavy Quark Effective Theory (HQET) (cf. e.g. [12, 13]), while others apply heavy quarks of finite mass [14, 15, 16] mainly by means of Non-Relativistic QCD (cf. e.g. ). For a recent review of lattice results on baryon masses cf. .
In this work we treat the quark in leading order of HQET, which is the static limit. In this limit there are no interactions involving the spin of the heavy quark, i.e. states are doubly degenerate. Therefore, it is common to label static-light baryons by integer spin/angular momentum and parity of the light degrees of freedom. For the two light quarks we consider all possible combinations of , and , i.e. further quantum numbers are strangeness and isospin . We use flavors of dynamical quarks and study various ensembles with corresponding pion masses down to . Our lattice spacing is rather fine and we use the Wilson twisted mass formulation of lattice QCD at maximal twist, which guarantees automatically improved spectral results. We compute all five experimentally known baryon states. We also make predictions for , which has not yet been observed, as well as for a number of negative parity static-light baryons, which have neither been measured experimentally nor been computed by lattice methods.
The next-to-leading order of HQET, which removes the degeneracy with respect to the heavy quark spin, is , where is the mass of the heavy quark. This correction is expected to be relatively small for baryons, e.g. experimentally the mass difference between and is only around . Lattice methods to evaluate such contributions have been established and tested in quenched studies of mesons [19, 20, 21, 22]. We intend to explore these contributions using lattice techniques subsequently. An alternative way to predict the spectrum of baryons is to interpolate between charmed baryons, where the experimental spectrum is rather well known, and the static limit obtained by lattice QCD assuming a dependence as . Thus the splittings among baryons should approximately be of those among the corresponding baryons.
We try to determine the baryon spectrum as fully as possible, i.e. we consider all possible light flavor combinations corresponding to and as well as both parity and . This will help the construction of phenomenological models (cf. e.g. ), might contribute to resolve open experimental issues (e.g. the above mentioned mass discrepancy for ) and also provide valuable input for future experiments.
The paper is organized as follows. In section 2 we briefly recapitulate our lattice setup, which is discussed in more detail in . In section 3 we discuss static-light baryon trial states, corresponding correlation matrices and how we extract the static-light baryon spectrum from these matrices as well as our extrapolation procedure to the physical quark mass. In section 4 we interpolate between our static-light lattice results and experimental results for baryons, to account for the finite mass of the quark. We conclude with a brief summary and an outlook in section 5.
2 Lattice setup
We use flavor gauge field configurations generated by the European Twisted Mass Collaboration (ETMC). The gauge action is tree-level Symanzik improved ,
and are the gauge covariant forward and backward derivatives, and are the bare untwisted and twisted quark masses respectively, is the third Pauli matrix acting in flavor space and represents the quark fields in the so-called twisted basis. The twist angle is given by , where and denote the renormalized twisted and untwisted quark masses. has been tuned to by adjusting appropriately (cf.  for details). As argued in  this ensures automatic improvement for static-light spectral quantities, e.g. mass differences between static-light baryons and the lightest static-light meson (the “/ meson”), the quantities we are focusing on in this work.
The ensembles of gauge field configurations we are considering are listed in Table 1. They correspond to a single value of the lattice spacing , but various values of the pion mass in the range . The lattice extension is , which amounts to and . Details regarding the generation of these gauge field configurations and computation and analysis of standard quantities (e.g. lattice spacing or pion mass) can be found in [32, 33].
|in fm||in MeV||# of gauges|
We treat static-light baryons containing valence quarks in a partially quenched approach, where the mass of these quarks, , is approximately equal to the mass of the physical quark taken from a study of strange mesons using the same gauge field configurations [34, 35]. Note that partially quenched quarks can be realized in two ways, either with a twisted mass term or corresponding to the upper and the lower entry in the quark field doublet respectively. We consider both possibilities and denote them by .
In Table 1 we also list the number of gauge configurations, on which we have computed static-light baryon correlation functions.
3 The static-light baryon spectrum
With static-light baryons we refer to baryons made from a single static quark and two light quarks, which can either be , and/or .
3.1 Static-light baryon trial states
3.1.1 Static-light baryon creation operators in the continuum
We start by discussing symmetries and quantum numbers of static-light baryons and corresponding creation operators in the continuum.
The continuum analogs of our lattice static-light baryon creation operators are
where is a static quark operator and are light quark operators (in the usual physical basis). The upper indices , and are color indices, is the charge conjugation matrix and is a combination of matrices, i.e. a matrix acting in spin space.
Since there are no interactions involving the static quark spin, it is appropriate to label static-light baryons by the angular momentum of their light degrees of freedom . For creation operators (4) it is determined by and can either be or . states correspond to total angular momentum , while states correspond to degenerate pairs of states with total angular momentum and , respectively.
Parity is also a quantum number depending on . Either or .
The flavor quantum numbers are isospin and strangeness . To access all possible combinations, we consider light quark flavors (corresponding to , ), (corresponding to , ), (corresponding to , ) and (corresponding to , ).
Creation operators and the quantum numbers of their associated trial states are collected in Table 2. Note that certain combinations do not need to be considered, since the corresponding creation operators are identical zero due to the anticommutation property of quark operators. Such combinations are either omitted from the table or marked with “X”.
3.1.2 Static-light baryon creation operators in twisted mass lattice QCD
Twisted basis lattice static-light baryon creation operators are of similar form,
where physical basis quark operators have been replaced by their twisted basis lattice counterparts.
In the continuum the relation between the physical and the twisted basis is given by the twist rotation , where at maximal twist. At finite lattice spacing, however, issues are more complicated: the twist rotation only holds for renormalized operators and the QCD symmetries isospin and parity are explicitely broken by . Nevertheless, it is possible to unambiguously interpret states obtained from correlation functions of twisted basis operators in terms of QCD quantum numbers as we will explain and demonstrate below.
On the lattice rotational symmetry is reduced to symmetry with respect to cubic rotations. There are only five different representations of the cubic group corresponding to integer angular momentum . in the continuum corresponds to the representation on the lattice containing angular momenta , while corresponds to the representation containing
While in twisted mass lattice QCD the -component of isospin is still a quantum number, isospin and parity are explicitely broken by the Wilson term, which is proportional to the lattice spacing. Only a specific combination of both symmetries, light flavor exchange combined with parity, is still a symmetry in twisted mass lattice QCD. We denote this symmetry by acting on the light twisted basis quark doublet according to , where is the first Pauli matrix acting in flavor space. Consequently, the four QCD sectors labeled by and are pairwise combined. is a combination of and , while is a combination of and .
As explained in section 2 the partially quenched quark can be realized in two ways denoted by and , respectively. As a consequence baryons computed at finite lattice spacing on the one hand with quarks and on the other hand with quarks, but which are otherwise identical, may differ in mass. Due to automatic improvement of twisted mass lattice QCD this mass splitting, however, will only be , i.e. is expected to be rather small and will vanish quadratically, when approaching the continuum limit.
Since and do not commute, they cannot simultaneously be chosen as quantum numbers. An exception are states with , which can also be classified with respect to .
The lattice static-light baryon creation operators we have been using are collected in Table 3, Table 4 and Table 5. Creation operators are sorted according to the twisted mass lattice quantum numbers of their associated trial states, i.e. creation operators exciting states from different sectors are separated by horizontal lines. To interpret these twisted basis creation operators in terms of QCD quantum numbers, we have performed an approximate rotation to the physical basis (neglecting renormalization and using ). The resulting so-called pseudo physical basis creation operators together with their corresponding QCD quantum numbers are also listed in the tables.
|twisted basis lattice operator||pseudo physical basis operator|
|twisted basis lattice operator||pseudo physical basis operator|
|twisted basis lattice operator||pseudo physical basis operator|
3.1.3 Smearing of gauge links and quark fields
To enhance the overlap of the trial states to low lying static-light baryon states, we make extensive use of standard smearing techniques. This allows to read off static-light baryon masses from correlation functions at rather small temporal separation, where the signal-to-noise ratio is favorable.
Smearing is done in two steps. At first we replace all spatial gauge links by APE smeared versions. The parameters we have chosen are and . Then we use Gaussian smearing on the light quark fields , , and , which resorts to the APE smeared spatial links. We consider three different smearing levels, characterized by and . This amounts to light quark field operators with approximate widths of
(cf.  for details).
Smeared static light baryon creation operators are denoted by .
3.2 Correlation matrices
For each sector characterized by strangeness , angular momentum of the light degrees of freedom , -component of isospin , and in certain cases twisted mass parity we compute temporal correlation matrices
We consider all the creation operators listed in Table 3, Table 4 and Table 5 at three different smearing levels as explained in the previous subsection. This amounts dependent on the sector to , or correlation matrices.
Static quarks are treated with the HYP2 static action [36, 37, 38], i.e. Wilson lines appearing in static quark propagators are formed by products of HYP2 smeared temporal links (cf.  for details).
Light quark propagators are estimated by means of stochastic timeslice sources (cf.  for details). On each gauge field configuration we invert 48 independently chosen sources, all located on the same timeslice, 12 for each of the four possible light quark propagators , , and . Multiple inversions of the same timeslice of the same gauge field configuration are beneficial with respect to statistical precision, because each correlation function contains two light quark propagators. This allows to form statistical samples, i.e. the number of samples is the square of the number of inversions (cf. ).
3.3 Determination of static-light baryon masses
From correlation matrices (6) we compute effective mass plateaus by solving generalized eigenvalue problems
with (cf. e.g. [39, 40]). Instead of using the full , or correlation matrices we have chosen “optimal submatrices” in a sense that on the one hand effective masses exhibit plateaus already at small temporal separations and that on the other hand statistical errors on are minimized. We found that with the following choice both criteria are adequately fulfilled:
use submatrices with smearing levels ;
use submatrices with smearing levels ;
use submatrices with smearing levels .
To demonstrate the quality of our lattice results, we show in Figure 1 examples of effective mass plateaus (at light quark mass ) corresponding to (, , ), (, , ) and its parity partner (, , ).
We extract static-light baryon masses by fitting constants to these plateaus in regions of sufficiently large temporal separation . We found that yields reasonable values, which are for all states investigated. on the other hand hardly affects the resulting static-light baryon masses (on the “-side” of the effective mass plateau statistical errors are rather large and, therefore, data points have a negligible effect on the fit). The resulting fits for the examples shown in Figure 1 are indicated by dashed lines. We checked the stability of all our results by varying by . We found consistency within statistical errors.
To assign appropriate QCD quantum numbers to the extracted static-light baryon states, we follow a method introduced and explained in detail in , section 3.1 (“Method 1: solving a generalized eigenvalue problem”). For the -th state the components of the corresponding eigenvector characterize the contribution of the -th static-light baryon creation operator entering the correlation matrix. After transforming these operators from the twisted basis to the pseudo physical basis by means of the twist rotation , (cf. the right columns of Table 3, Table 4 and Table 5), one expects and and also finds that for each extracted state operators corresponding to only one of the two QCD sectors corresponding to the investigated twisted mass lattice QCD sector clearly dominate, while the contribution from operators from the other sector are rather small. This allows to unambiguously assign a QCD label to each extracted static-light baryon state. An example, the identification of (, , ) and its parity partner (, , ), is shown in Figure 2 (cf. also Figure 1 for the corresponding effective masses both having twisted mass quantum numbers (, , )).
Since static-light baryon masses diverge in the continuum limit due to the self energy of the static quark, we always consider mass differences of these baryons to the lightest static-light meson (“ meson”). In such differences the divergent self energy exactly cancels. We take the mass values of the lightest static-light mesons from , where they have been computed using the same lattice setup. The mass differences (in lattice units) together with the pion masses (also in lattice units; cf. Table 1 and ) serve as input for the extrapolation procedure to the physical quark mass described in the next subsection.
3.4 Extrapolation to the physical quark mass
The mass differences obtained for the four ensembles listed in Table 1, which only differ in the value of the quark mass (both sea and valence), are plotted against in Figure 3 (, , i.e. baryons), Figure 4 (, , i.e. baryons), Figure 5 (, i.e. baryons) and Figure 6 (, i.e. baryons) and are collected in appendix A.
For the extrapolation to the physical quark mass one could use an effective field theory approach (Chiral HQET for example) as used e.g. to study static-light meson decay constants . However, this approach has not fully been developed to discuss mass differences between excited static-light baryon states and the lightest static-light meson so is not appropriate here. Instead we use the simplest assumption, which is supported by our results: a linear dependence in .
Data points are correlated via in case they correspond to the same ensemble, i.e. to the same value of the quark mass. We take that into account via a covariance matrix, which we estimate by resampling and all extracted static-light mass differences ( samples). Consequently, we do not fit straight lines to the data points individually for every static-light baryon state, but perform a single correlated fit of straight lines to the mass differences considered. During the fitting we take statistical errors both along the horizontal axis (errors in ) and along the vertical axis (errors in ) into account. The method for performing such two-dimensional fits is explained in detail in .
We find that a fit, which is linear in the light quark mass (represented by the mass squared of the light-light pseudoscalar meson ) is acceptable, i.e. yields . This fit enables us to extrapolate to the physical quark mass, in this work taken as and converted to lattice units by using the lattice spacing  resulting in (cf. Figure 3, Figure 4, Figure 5 and Figure 6).
Extrapolations of static-light mass differences to the physical quark mass are listed in Table 6 in MeV. Since there seems to be a controversy of around 10% regarding the value of the lattice spacing in physical units, when using on the one hand the pion mass and the pion decay constant