pacs:11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.Dj
Strange and charm baryon masses with two flavors of dynamical twisted mass fermions
C. Alexandrou, J. Carbonell, D. Christaras, V. Drach, M. Gravina, M. Papinutto
Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Computation-based Science and Technology Research Center, Cyprus Institute,20 Kavafi Str., Nicosia 2121, Cyprus
CEA-Saclay, IRFU/Service de Physique Nucléaire, 91191 Gif-sur-Yvette, France
NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
Laboratoire de Physique Subatomique et Cosmologie, UJF/CNRS/IN2P3, 53 avenue des Martyrs, 38026 Grenoble, France
Dpto. de Física Teórica and Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain
The masses of the low-lying strange and charm baryons are evaluated using two degenerate flavors of twisted mass sea quarks for pion masses in the range of about 260 MeV to 450 MeV. The strange and charm valence quark masses are tuned to reproduce the mass of the kaon and D-meson at the physical point. The tree-level Symanzik improved gauge action is employed. We use three values of the lattice spacing, corresponding to , and with , and respectively. We examine the dependence of the strange and charm baryons on the lattice spacing and strange and charm quark masses. The pion mass dependence is studied and physical results are obtained using heavy baryon chiral perturbation theory to extrapolate to the physical point.
March 15, 2018
Lattice QCD simulations with two light degenerate sea quarks ) as well as with a strange sea quark () close to physical values of the pion mass are being carried out. Masses of low-lying hadrons are primary quantities that can be extracted using these simulations. Comparing the lattice and experimental values provides a check of lattice discretization effects. Such a comparison is necessary before one can use the lattice approach to study hadron structure. The European Twisted Mass Collaboration (ETMC) has generated a number of ensembles at four values of the lattice spacing, ranging from 0.1 fm to about 0.05 fm, at several values of the light sea quark mass and for several physical volumes with maximally twisted mass fermions. We will use ensembles generated at the three smallest lattice spacings to evaluate the masses of strange and charm baryons. The strange and charm quarks are added as valence quarks.
For heavy quarks the Compton wavelength of the associated heavy-light meson is comparable to presently attainable lattice spacings, which means that cut-off effects maybe large. The charm quark mass is at the upper limit of the range of masses that can be directly simulated at present. In order to obtain values for the masses that can be compared to experiment, it is important to assess the size of lattice artifacts. A first study of cut-off effects was carried out for light and strange baryons in Refs. Alexandrou et al. (2008, 2009). In this work we extend the study by including a finer lattice spacing and calculate besides the mass of strange baryons the masses of charm baryons. Having three lattice spacings the continuum extrapolation can be better assessed.
In this work we compare our results in the strange baryon sector with recent results obtained with Clover-improved Wilson fermions with different levels of smearing. The PACS-CS Aoki et al. (2009) and BMW Durr et al. (2008) collaborations evaluated the octet spectrum using two degenerate flavors of light quarks and a strange quark with mass tuned to its physical value. The PACS-CS has also computed the decuplet baryon masses. In addition, we compare with the LHPC that computed the octet and decuplet spectrum using a hybrid action with domain wall valence fermions on Kogut-Susskind sea quarks Walker-Loud et al. (2009)
Besides the strange baryons, we also study the ground state spectrum of charm baryons with spin and spin . Experimental searches of charm hadrons have received significant attention, mainly due to the experimental observation for candidates of the doubly charm baryons and by the SELEX collaboration Mattson et al. (2002); Russ (2002); Ocherashvili et al. (2005). The 60 MeV mass difference between the singly and doubly charged states is difficult to understand since it is an order of magnitude larger compared to what is expected. No evidence was found for these states by the BABAR experiment Aubert et al. (2006) and FOCUS Collaboration Ratti (2003). The BELLE Collaboration Chistov et al. (2006) finds -states lower in mass, that can be candidates of excited states of but no doubly charm . Additional experiments are planned at the new Beijing Spectrometer (BES-III) and at the antiProton ANnihilation at DArmstadt (PANDA) experiment at GSI, that can shed light on these charm baryon states. Several lattice QCD studies have been carried out to study charm baryons. We will compare the results of the current work with recent lattice QCD results all computed in a hybrid action approach where the charm valence quark was introduced on gauge configurations produced with staggered sea fermions by the MILC collaboration Bernard et al. (2001); Aubin et al. (2004); Bazavov et al. (2010).
As in the case of the other lattice QCD studies of heavy baryons, also in this work we use a mixed action approach. For the strange and charm sector we use an Osterwalder-Seiler valence quark, following the approach employed in the study of the pseudo scalar meson decay constants Blossier et al. (2008, 2009). The bare strange and charm valence quark mass is tuned by requiring that the physical values of the mass of the kaon and D-meson are reproduced after the lattice results are extrapolated at the physical value of the pion mass. The ETMC configurations Boucaud et al. (2007, 2008a) analyzed in this work correspond to pion masses in the range of 260 to 450 MeV and three values of the lattice spacing corresponding to and with and , respectively. The Sommer parameter is determined from the force between two static quarks, the continuum value of which is determined to be 0.462(5) fm. At we use two ensembles, one corresponding to the lowest value of the pion mass considered in this work and one to the upper pion mass range. We find that the baryon masses, in general, show a very weak dependence on the lattice spacing and are fully compatible with an behaviour with an almost vanishing coefficient of the term. This justifies neglecting the term in extrapolating results to the continuum limit.
An important issue raised by the twisted mass fermion formulation is isospin symmetry breaking. This symmetry, although exact in the continuum limit, is broken at a non-vanishing lattice spacing to . There are, however, theoretical arguments Frezzotti and Rossi (2007) and numerical evidences Dimopoulos et al. (2008); Jansen (2008) that these isospin breaking effects are only sizable for the neutral pseudo scalar mass whereas for other quantities studied so far by ETMC they are compatible with zero. In this paper we demonstrate that also in the baryon sector these isospin breaking effects are in general small or even compatible with zero. Small isospin breaking effects decrease as the lattice spacing decreases and they vanish at the continuum limit. This corroborates our previous findings Alexandrou et al. (2008, 2009). The isospin breaking effects are relevant not only for neutral pions but also for other particles, e.g. the kaons. However, since the mass of the kaon is higher, the relative splitting (between and ) is less drastic.
The paper is organized as follows: The details of our lattice formulation, namely those concerning the twisted mass action, the parameters of the simulations, the interpolating fields used and the tuning of the strange and charm quark masses are given in Section II. Section III contains the numerical results of the baryon masses computed for different lattice volumes, lattice spacings and bare quark masses. Lattice artifacts, including finite volume and discretization errors, and continuum extrapolation are also discussed in Section III, with special emphasis on the isospin breaking effects inherent to the twisted mass formulation of lattice QCD. The chiral extrapolations are analyzed in Section IV. Section V contains a comparison with other existing calculations. Our conclusions are finally drawn in Section VI.
Ii Lattice formulation
ii.1 The lattice action
For the gauge fields we use the tree-level Symanzik improved gauge action Weisz (1983), which includes besides the plaquette term also rectangular Wilson loops
with and the (proper) normalization condition . Note that at this action becomes the usual Wilson plaquette gauge action.
The fermionic action for two degenerate flavors of quarks in twisted mass QCD is given by
with the Pauli matrix acting in the isospin space, the bare twisted mass and the massless Wilson-Dirac operator given by
Maximally twisted Wilson quarks are obtained by setting the untwisted quark mass to its critical value , while the twisted quark mass parameter is kept non-vanishing in order to work away from the chiral limit. In Eq. (2) the quark fields are in the so-called ”twisted basis”. The ”physical basis” is obtained for maximal twist by the simple transformation
In terms of the physical fields the action is given by
In this paper, unless otherwise stated, the quark fields will be understood as “physical fields”, , in particular when we define the baryonic interpolating fields.
A crucial advantage of the twisted mass formulation is the fact that, by tuning the bare untwisted quark mass to its critical value , all physical observables are automatically improved. In practice, we implement maximal twist of Wilson quarks by tuning to zero the bare untwisted current quark mass, commonly called PCAC mass, Boucaud et al. (2008b), which is proportional to up to corrections. The value of is determined at each value at the lowest twisted mass used in our simulations, a procedure that preserves improvement and keeps small Boucaud et al. (2008a); Frezzotti et al. (2006). The twisted mass fermionic action breaks parity and isospin at non-vanishing lattice spacing, as it is apparent from the form of the Wilson term in Eq. (6). In particular, the isospin breaking in physical observables is a cut-off effect of Frezzotti and Rossi (2004a). To simulate the strange quark in the valence sector several choices are possible.
The strange and charm quarks are added as Osterwalder-Seiler valence quarks and their action reads
where and are the strange and charm valence quark masses. This is naturally realized in the twisted mass approach by introducing two additional doublets of strange and charm quarks and keeping only the positive diagonal component of . The value is taken to be equal to the critical mass determined in the light sector, thus guaranteeing the improvement in any observable. The reader interested in the advantage of this mixed action in the mesonic sector is referred to the Refs Frezzotti and Rossi (2004b); Abdel-Rehim et al. (2007, 2006); Blossier et al. (2008, 2009).
ii.2 Simulation details
The input parameters of the calculation, namely , and are summarized in Table 1. The corresponding lattice spacing and the pion mass values, spanning a mass range from 260 to 450 MeV, are taken from Ref. Alexandrou et al. (2011). At MeV we have simulations for lattices of spatial size fm and fm at allowing to investigate finite size effects. Finite lattice spacing effects are investigated using three sets of results at , and . These sets of gauge ensembles allow us to estimate all the systematic errors in order to have reliable predictions for the baryon spectrum.
|Statistics||144, 144||194, 193||201, 201|
|Statistics||4112, 310||545, 278||1817, 369||477, 475|
|Statistics||659, NA||242, NA|
ii.3 Tuning of the bare strange and charm quark masses
The dependence of the pseudoscalar meson mass on the valence and sea quarks can be written as a polynomial of the form Blossier et al. (2008)
where is the sea quark mass and are the valence quark masses, , and . For the ensembles we consider in total 164 pseudoscalar meson masses using all possible combinations of sea and valence quark masses. Namely, we have considered 150 combinations obtained from and independently taking the values
whereas takes the values
We have an additional 12 combinations coming from the combinations
plus two extra combinations from
For the tuning at we use the following 20 combinations
For the tuning at we consider 10 pseudoscalar meson masses:
In Figs. 1 and 2 we show representative fits to the pseudo-scalar masses in the range of the kaon and D-meson masses using the expression given in Eq. (8). The values of the strange and charm quark masses are varied until the resulting kaon and D-meson masses are matched to their physical values. The resulting fit parameters are listed in Table 2. We note that for two fitting ranges are used, one range spanning the the strange quark mass and one the charm quark mass, For and we fit all data together since we do not have enough mass combinations in order to apply Eq. (8). If one does the same for then the tuned value for the strange quark mass is compatible with the value of if we restrict the fit to the strange region. In addition, at each -value we can restrict the fit in the charm region using the Ansatz
The tuned charm quark value is found to be compatible with the one extracted using Eq. (8). This procedure can be carried out using either the lattice spacing determined from the nucleon mass or form . The difference in the tuned masses reflects the systematic error in setting the scale.
|(strange quark)||(charm quark)|
The tuned values of the strange and charm quark masses and obtained at the the physical pion are given in Table 3. In a previous paper, the ETMC computed pseudo-scalar meson masses for a number of sea and valence quark masses using the gauge configurations. Matching the experimental value of the mass ratio of the kaon to the pion, , the bare strange quark mass was determined Blossier et al. (2008). Depending on the polynomial fit used the values for at varied from to . Thus, our value of from matching the physical value of the kaon mass in combination with the lattice spacing determined from the nucleon mass is compatible with the value determined in Ref. Blossier et al. (2008). Such an agreement is satisfactory and shows that the two procedures lead to the same determination within the uncertainties associated with the extrapolation. The systematic error introduced from the way the lattice scale is fixed can be assessed by comparing the tuned values extracted using the lattice spacing determined from the nucleon mass and from the pion decay constant . The values of the lattice spacing determined using , taken from Ref. Baron et al. (2010), are fm, fm and fm. In Table 3 we give the tuned values for the charm and strange quark masses expressed in physical units. As can be seen, the values for the charm quark masses are in agreement, whereas for strange quark masses the differences are within about two standard deviations.
ii.4 Interpolating fields
The low-lying baryons belonging to the octet and decuplet representations of are given in Figs. 4 and 4 respectively. They are classified by giving the isospin, , the third component of the isospin, , the strangeness (S), the spin and the parity. In order to extract their masses in lattice QCD, we evaluate two-point correlators. We use interpolating fields to create these states from the vacuum that have the correct quantum numbers and reduce to the quark model wave functions in the non-relativistic limit. The interpolating fields used in this work are collected in Tables 4 Ioffe (1981); Leinweber et al. (1991) and 5 Ioffe (1981); Leinweber et al. (1992) for the octet and decuplet respectively.
Charm baryons with no strange quarks are obtained from the interpolating fields of strange baryons by replacing the strange with the charm quark. There are additional charm baryons containing strange quarks, giving a 20-plet of spin-1/2 and a 20-plet of spin-3/2. In most of this work we do not consider the particles that contained both a strange and charm quarks. For the lattice with the smallest lattice spacing and at the smallest pion mass we also consider the spin-1/2 , , and and the spin-3/2 , and . The interpolating fields for these baryons are given in Table 6.
Local interpolating fields are not optimal for suppressing excited state contributions. We instead apply Gaussian smearing to each quark field, : using the gauge invariant smearing function
constructed from the hopping matrix,
Furthermore we apply APE smearing to the spatial links that enter the hopping matrix. The parameters of the Gaussian and APE smearing are the same as those used in our previous work devoted to the nucleon and masses Alexandrou et al. (2008).
ii.5 Two-point correlators
To extract masses in the rest frame we consider two-point correlators defined by
Space-time reflection symmetries of the action and the anti-periodic boundary conditions in the temporal direction for the quark fields imply, for zero three-momentum correlators, that . So, In order to decrease errors we average correlators in the forward and backward direction and define:
In order to decrease correlation between measurement, we choose the source location randomly on the whole lattice for each configuration. Masses are then extracted from the so called effective mass which is defined by
where is the mass difference of the excited state with respect to the ground mass .
In Fig. 5 we show representative examples of the effective masses of strange and charm baryons. As can be seen, a plateau region can be clearly identified. What is shown in these figures are effective masses extracted from correlators where smearing is applied both at the sink and source. Although local correlators are expected to have the same value in the large time limit, smearing suppresses excited state contributions yielding a plateau at earlier time separations and to a better accuracy in the mass extraction. We therefore extract the masses using smeared source and sink. Our fitting procedure to extract is as follows: The sum over excited states in the effective mass given in Eq. (14) is truncated keeping only the first excited state. Allowing a couple of time slice separation the effective mass is fitted to the form given in Eq. (14). This yields an estimate for the parameters and . The lower fit range is increased until the contribution due to the first excited state is less than 50% of the statistical error of . This criterion is in most of the cases in agreement with a . In the cases in which this criterion is not satisfied a careful examination of the effective mass is made to ensure that the fit range is in the plateau region.
Iii Lattice results
Before we extrapolate our lattice results on the strange and charm baryon masses to the physical point, we need to examine their dependence on the heavy quark mass as well as cut-off effects. We collect lattice results for the masses of the strange and charm baryons in the Appendix. The errors are evaluated using jackknife and the -method Wolff (2004) to check consistency.
In Figs. 6 and 7 we show the dependence of the strange and charm baryon masses on the the strange and charm quark mass, respectively. Overall, the data display a linear dependence on both the strange and charm quark mass. One can therefore interpolate between different values of quark masses, if needed.
iii.1 Strange baryon mass with strange quark mass tuned to its physical value
In this section we restrict our analysis only to the subset of data obtained at the tuned values of the strange quark mass. Namely, for and , we use the tuned value given in Table 3, whereas for we use which agrees with the tuned strange quark mass within error bars.
It is interesting to examine the degree of isospin splitting as a function of the lattice spacing. The splitting is expected to be zero in the continuum limit. In Fig. 8 we show the mass of , and at and 4.2. As expected the mass splitting among the three charge states of the decreases with the lattice spacing. The same behavior is observed for the other strange particles studied in this work. This is shown in Fig. 9 where we plot the mass difference as a function of at our smallest and heaviest pion mass. As can be seen, the mass difference is consistent with zero for all particles at the smallest lattice spacing. The small non-zero values seen for the and particles are just outside one standard deviation. Therefore, the general conclusion is that indeed isospin splitting is small at these values of the lattice spacing and it vanishes at the continuum limit. Since for finite there are small differnces, for the chiral extrapolation where we use all lattice data we do not average the masses for the different charge states of the , and .
Volume effects can be studied at where we have simulations at two volumes at pion mass of about 300 MeV. As can be seen in Figs. 8, 10 and 11 results at different volumes are consistent. Therefore, we conclude any volume effects are smaller than our statistical accuracy.
In order to examine the continuum limit we interpolate our lattice result at a given pion mass in units of . For we have simulations at only two values of the pion mass at the upper and lower range of pion masses considered in this work, namely for GeV and GeV. Therefore we interpolate the results for the other two values of to these two pion masses. In Figs. 10 and 11, we show results for the octet and decuplet strange baryon masses, respectively, at our three values of the lattice spacing. We perform a continuum extrapolation by perfroming a linear fit in as well as to constant. As can be seen from Fig. 10, the values obtained in the continuum limit agree for all octet baryons. In the case of the decuplet the statistical errors are larger and the value obtained at with the linear fit carries a large error. The value obtained using a constant fit has a smaller error and it is compatible with one the one obtained using a linear fit. Therefore, for a given charge state and within the current statistical accuracy, the term can be taken as negligible. Therefore, we can use results at all -values to extrapolate to the physical point since cut-off are small for a given charge state. There are two exceptions in the case of the decuplet. At the mass of the and at the lowest pion mass are systematically higher than that at the other two -values. Since the results at , with larger lattice space are consistent with those at we conclude that this is not a cut-off effect.
iii.2 Charm baryon mass with charm quark mass tuned to its physical value
As in the previous subsection we consider results obtained at the tuned charm mass given in Table 3. The only exception is at at the heavy pion mass, where we have results close to the tuned value, namely at . As we have seen, the dependence on the heavy quark is linear and therefore the charm baryon masses at the tuned value can be easily determined by a linear interpolation.
We follow the same analysis as in the case of the strange baryon sector. In Fig. 12 we show the mass difference between different charged states as a function of the lattice spacing at the smallest and largest pion masses used in this work. As can be seen, the mass splittings are zero at the smallest value of the lattice spacing for all particles confirming restoration of isospin symmetry in the continuum limit. Furthermore, except for the case of the mass, the mass splitting is consistent with zero also at the other two -values. Therefore, for all particles except the one may average over the mass of different charge states.
In order to examine the continuum limit we interpolate our results at the three -values at a given pion mass in units of . In Figs. 13 and 14 we show the mass in the octet and decuplet charm sector as a function of lattice spacing for a given charge state, at the smallest and largest value of the pion mass. A linear fit in and a constant fit yield consistent results at the continuum limit, albeit with large errors in the case of th elinear fit. We also note that at the largest pion mass, although results at are in agreement with those at indicating neglegible -dependence, at the results are systematically below. We note that we show only statistical errors. Systematic errors due, for example, to the matching are not shown. As discussed in the next section these are (5-10)%. Therefore, a reasonable way to extrapolate our results in the charm sector is to compare the chiral extrapolation using all lattice data to those using results at and . We will take the different between the two values at the physical point as an estimate of a systematic error.
Iv Chiral extrapolation
Having determined that effects are small for the lattice spacings considered here we can combine our lattice results at the various -values to extrapolate to the physical pion mass (physical point).
For the strange baryon sector, we consider SU(2) heavy baryon chiral perturbation, which was found to describe lattice data satisfactorily Alexandrou et al. (2009). To leading one-loop one can described the pion mass dependence using
respectively, and for the decuplet baryons , and :
In addition we consider next to leading order SU(2) PT results Tiburzi and Walker-Loud (2008). The expressions are included here for completeness:
and for the decuplet baryons: