I Introduction

Octet and Decuplet mass, Lattice QCD
pacs:
11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.Dj

DESY 09-160

SFB/CPP-09-91


Low-lying baryon spectrum with two dynamical twisted mass fermions

C. Alexandrou, R. Baron, J. Carbonell, V. Drach, P. Guichon, K. Jansen, T. Korzec O. Pène

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

Laboratoire de Physique Théorique (Bât. 210), Université de Paris XI,CNRS-UMR8627, Centre d’Orsay, 91405 Orsay-Cedex, France

The masses of the low lying baryons are evaluated using two degenerate flavors of twisted mass sea quarks corresponding to pseudo scalar masses in the range of about 270 MeV to 500 MeV. The strange valence quark mass is tuned to reproduce the mass of the kaon in the physical limit. The tree-level Symanzik improved gauge action is employed. We use lattices of spatial size 2.1 fm and 2.7 fm at two values of the lattice spacing with and . We check for both finite volume and cut-off effects on the baryon masses. We performed a detailed study of the chiral extrapolation of the octet and decuplet masses using SU(2) PT. The lattice spacings determined using the nucleon mass at the physical point are consistent with the values extracted using the pion decay constant. We examine the issue of isospin symmetry breaking for the octet and decuplet baryons and its dependence on the lattice spacing. We show that in the continuum limit isospin breaking is consistent with zero, as expected. The baryon masses that we find after taking the continuum limit and extrapolating to the physical limit are in good agreement with experiment.

March 15, 2018

I Introduction

In the last couple of years an intense and successful effort in extending unquenched lattice calculations towards realistic values of quark masses, small lattice spacings and large volumes has been undertaken using a variety of algorithmic techniques and lattice actions. A review of the salient features of the various discretization schemes currently employed can be found in Ref Jansen (2008). Of particular relevance to the current work are the calculations of the low-lying baryon spectrum using two degenerate flavors () of light dynamical quarks. Such studies have been carried out by the MILC collaboration Bernard et al. (2001); Aubin et al. (2004) using Kogut-Susskind fermions and by the European Twisted Mass Collaboration (ETMC) Alexandrou et al. (2008) for the nucleon () and baryons using twisted mass fermions. There are also baryon mass calculations using two degenerate flavors of light quarks and a strange quark with the mass tuned to its physical value () mainly using clover improved Wilson fermions with different levels of smearing, such as the calculation of the nucleon mass by the QCDSF-UKQCD collaboration Ali Khan et al. (2004), and the evaluation of the octet and decuplet spectrum by the PACS-CS Aoki et al. (2009) and BMW Durr et al. (2008) collaborations. The LHP Collaboration computed the octet and decuplet spectrum using a hybrid action with domain wall valence fermions on asqtad improved staggered sea quarks Walker-Loud et al. (2009). Preliminary results on the nucleon mass are also computed using domain wall fermions by the RBC-UKQCD collaboration Antonio et al. (2006a, b).

In this work we study the low-lying spectrum of the baryon octet and decuplet with twisted mass fermions at maximal twist. The light quarks are dynamical degrees of freedom while in the strange 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 valence quark mass is taken to be the same as the one determined in the meson studies tuned by requiring that the mass of the kaon at the physical point matches its physical value. Using the ETMC configurations Boucaud et al. (2007, 2008a) we calculate the baryon spectrum for pion masses in the range of 270 MeV to 500 MeV and at two values of the lattice spacing corresponding to and with and , respectively, where is determined from the force between two static quarks. Results are also obtained at a third -value, namely , which corresponds to . The latter results are not taken into account in the final analysis due to large autocorrelation effects observed in the Monte Carlo history for quantities like the PCAC mass and the plaquette at small sea quark masses. Data at are only used as a consistency check of the continuum extrapolation. For the nucleon mass we also performed the calculation at an even finer value of the lattice spacing corresponding to and to ensure that indeed the continuum extrapolation using a weighted average with results at and is valid. We find that the baryon masses considered here 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.

For a fixed value of the lattice spacing we have used up to five different light quark masses and two different volumes. The corresponding values are in the range 3.3 to 7.4, where is the spatial extent of the lattice. Using these various values of the lattice spacing, quark masses and volumes allows us to estimate the volume corrections and perform a continuum and chiral extrapolation. The good precision of our results on the baryon masses allows us to perform a study of chiral extrapolations to the physical point. This study shows that one of our main uncertainties in predicting the mass at the physical point is caused by the chiral extrapolations. Another source of systematic error is the partially quenched approximation that we have used.

An important issue is the restoration of the explicitly broken isospin symmetry in the continuum limit. At finite lattice spacing, baryon masses display isospin breaking effects. There are, however, theoretical arguments Frezzotti and Rossi (2007) and numerical evidences Dimopoulos et al. (2008); Jansen (2008) that these isospin breaking effects are particularly pronounced for the neutral pseudo scalar mass whereas for other quantities studied so far by ETMC they are compatible with zero. In this paper we will demonstrate that also in the baryon sector these isospin breaking effects are in general small or even compatible with zero. For a preliminary account of these results see Ref. Drach et al. (2008).

The paper is organized as follows: The details of our lattice setup, namely those concerning the twisted mass action, the parameters of the simulations and the interpolating fields used, 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 as well as the Gell-Mann Okubo relations that are supposed to be fulfilled in the exact SU(3) limit. Lattice artifacts, including finite volume and discretization errors are discussed in Section IV, with special emphasis on the isospin breaking effects inherent in the twisted mass formulation of lattice QCD. The chiral extrapolations are analyzed in Section V. Section VI contains a comparison with other existing calculations and conclusions are finally drawn in Section VII.

Ii Lattice setup

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

(1)

with and the (proper) normalization condition . Note that at this action reduces to the usual Wilson plaquette gauge action.

The fermionic action for two degenerate flavors of quarks in twisted mass QCD is given by

(2)

with the Pauli matrix acting in the isospin space, the bare twisted mass and the massless Wilson-Dirac operator given by

(3)

where

(4)

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 be 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

(5)

In terms of the physical fields the action is given by

(6)

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 , 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, , which is proportional to up to corrections. As detailed in Ref. Boucaud et al. (2008b), is conveniently evaluated through

(7)

where is the axial vector current and is the pseudo scalar density in the twisted basis. The large limit is required in order to isolate the contribution of the lowest-lying charged pseudo scalar meson state in the correlators of Eq. (7). This way of determining is equivalent to imposing on the lattice the validity of the axial Ward identity between the vacuum and the charged one-pion zero three-momentum state:

(8)

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. We consider a quenched Osterwalder-Seiler fermion Osterwalder and Seiler (1978) with the following action in the twisted basis:

(9)

This is naturally realized in the twisted mass approach by introducing an additional doublet of strange quark 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 270 MeV to 500 MeV, are taken from Ref. Urbach (2007). At MeV we have simulations for lattices of spatial size  fm and  fm at allowing to investigate finite volume effects. Finite lattice spacing effects are investigated using two sets of results at and . The set at is used only as a cross-check and to estimate cut-off errors. These sets of gauge ensembles allow us to estimate all the systematic errors in order to have reliable predictions for the baryon spectrum.

,  fm from  Urbach (2007),
,  fm 0.0030 0.0060 0.0080 0.012
No. of confs. 269 253 409 182
 (GeV) 0.3070(18) 0.4236(18) 0.4884(15) 0.6881(18)
3.31 4.57 5.27 7.43
,  fm, from  Urbach (2007),
,  fm 0.0030 0.0040 0.0064 0.0085 0.010
No. of confs - 782 545 348 477
 (GeV) - 0.3131(16) 0.3903(9) 0.4470(12) 0.4839(12)
3.25 4.05 4.63 5.03
,  fm 0.003 0.004
No. of confs 659 232
 (GeV) 0.2696(9) 0.3082(6)
3.74 4.28
,  fm
,  fm 0.0060 0.0080 0.0110 0.0165
No. of confs 215 302 248 244
 (GeV) 0.3667(17) 0.4128(16) 0.4799(9) 0.5855(10)
4.44 5.00 5.81 7.09
Table 1: Input parameters () of our lattice calculation and corresponding lattice spacing () and pion mass ().

ii.3 Tuning of the bare strange quark mass

In a previous paper from the ETM collaboration Blossier et al. (2008), pseudo scalar meson masses have been computed for different values of the sea and valence quark masses for the gauge configurations. Using the experimental value of the mass ratio of the kaon to the pion, , the bare strange quark mass can be set. We use the value of at taken from Table 2 of Ref. Blossier et al. (2008). In a more recent study of the pseudo scalar decay constant of kaons and D-mesons Blossier et al. (2009), the computation was extended to and . However, this is still a preliminary analysis and an ongoing analysis for the accurate extraction of quark masses is still in progress. One can obtain an estimate of the bare strange quark mass at a given value of by taking the results at as a reference and using the scaling relation Lubicz and Tarantino (2008):

(10)

The values we use for and given in Table 2 are obtained by applying Eq. (10). We use the value of the renormalization constant found in the preliminary analysis of Ref. Dimopoulos et al. (2007) within the RI’-MOM scheme. This value is in agreement with a complementary analysis given in Ref. Alexandrou et al. ().

Table 2: Bare strange quark mass used in the valence sector for different values.

ii.4 Interpolating fields

The low lying baryons belonging to the octet and decuplet representations of are given in Figs. 2 and 2 respectively. They are classified by giving the isospin, , the third component of the isospin, , the strangeness (s), spin and 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 3 Ioffe (1981); Leinweber et al. (1991) and  4 Ioffe (1981); Leinweber et al. (1992) for the octet and decuplet respectively.

Figure 1: The low lying baryons belonging to the octet representation labeled by the value of and hypercharge.

Figure 2: The low lying baryons belonging to the decuplet representation labeled by the value of and hypercharge.
Strangeness Baryon Interpolating field
Table 3: Interpolating fields and quantum numbers for the baryons in the octet representation.
Strangeness Baryon Interpolating field
Table 4: Interpolating fields and quantum numbers for baryons in the decuplet representation.

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

(11)

constructed from the hopping matrix,

(12)

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

(13)

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:

(14)

In order to decrease correlation between measurements, we choose the source location randomly on the whole lattice for each configuration. Masses are extracted from the so called effective mass which is defined by

(15)

where is the mass difference of the excited state with respect to the ground state mass .

In Figs. 4 and 4 we show the effective masses of the baryons in the octet and decuplet representation respectively. As can be seen a plateau region can be identified for all baryons. 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 a better accuracy in the mass extraction. Our fitting procedure to extract is as follows: The mass is obtained from the leading term in Eq. (15), i.e.from a constant fit to . A second fit, including the first excited state, allows us to estimate the systematical error of the previously determined due to excited states for a given plateau range. The plateau range is then chosen such that the systematical error on drops below 50% of its statistical error 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. The results for the masses of the octet and decuplet at are are collected in Tables 5 and 6 respectively. The corresponding results for the masses at are given in Table 7 and 8. The errors are evaluated using both jackknife and the -method Wolff (2004) to ensure consistency.

Figure 3: Effective masses of the octet states for , on a lattice using configurations.

Figure 4: Effective masses of the decuplet states for , on a lattice using configurations.

Iii Results

The bulk of the numerical results are presented in this section. Baryon masses are given in lattice units. Our procedure to convert the results to physical units will be discussed in the next section.

iii.1 Baryon masses

In Tables 5 to 8 we present the masses of the octet and decuplet states with the lattice input parameters given in Table 1. For the isospin multiplets we have computed separately the masses corresponding to each isospin components as well as their averaged value. These results (averaged values in case of isospin multiplets) are displayed in Figs. 6 and 6. The , data are linked by dotted lines to guide the eye. An inspection of the plots indicates that the lattice artifacts, studied in detail in the next section, are small. Notice that the natural order of the and states comes out to be correct for MeV while for larger masses this order is inverted.

stat.
782
545
348
477
652
232
Table 5: Baryon masses in the octet representation at in lattice units.
stat.
782
545
348
477
652 0.6234(139) 0.6497(133) 0.6859(96) 0.6838(93) 0.6859(106) 0.7027(101) 0.7058(50) 0.7097(58) 0.7032(53) 0.7926(49)
232
Table 6: Baryon masses in the decuplet representation at in lattice units.
stat.
269
253
409
Table 7: Baryon masses in the octet p representation at in lattice units.
stat.
269
253
409
Table 8: Baryon masses in the decuplet representation at in lattice units.

Figure 5: Octet states measured in our different gauge ensembles. Physical points are indicated by their name symbols (magenta). The data at , are connected by dotted lines to guide the eye.

Figure 6: The same as Fig. 6 but for the decuplet states.

iii.2 Strange quark mass dependence

The dependence of the masses of baryons with strangeness on the bare strange quark mass has been investigated at for . The results are given in Tables 9 and 10 and displayed in Figs. 8 and 8. The vertical dotted line indicates the value of the tuned bare strange quark mass as given in Table 2. The symmetric point is given by the nucleon and mass for the octet and decuplet respectively. As can be seen in the limit all the octet and decuplet masses converge to a single point up to cut-off effects and the fact that we only have simulations. For clarity we only show in Fig. 8 the mass of , and . They should be degenerate with the nucleon in the limit of . Indeed, if one computes the nucleon mass with the same statistics with that used for and , one finds them to be degenerate within the errors as can be seen in Fig. 8.

The corresponding results for the decuplet-baryons are displayed in Fig. 8. As can be seen, also in the case of the decuplet masses there is convergence to the mass as predicted in the exact limit .

Figure 7: Masses for octet baryons at and on a lattice of size as a function of . The vertical dashed line indicates the value of the tuned bare strange quark mass. The dotted lines are to guide the eye.

Figure 8: The same as for Fig. 8 but for the decuplet baryons.
stat.
597
316
316
315
308
311
316
314

Table 9: Octet masses for , on a lattice as a function of .
stat.
597
316
316
315
308
311
316
314
Table 10: Decuplet masses for , on a lattice as a function of

The dependence of the strange baryon masses provides an estimate of systematic errors due to the uncertainty in the tuning of the strange quark mass. As already explained, the kaon mass at the physical point is used to fix . This gives . The % uncertainty leads to a corresponding error in the strange baryon masses that can be estimated by the variation of their masses in the vicinity of . At we estimate an error that is comparable to the statistical error. In what follows we will analyze our results taking into account only statistical errors. This analysis shows that when the statistical error is given on the final results of strangeness non-zero baryon masses one must bear in mind that there is a systematic error of about the same magnitude due to the strange quark mass determination.

iii.3 Gell-Mann-Okubo relation

Assuming a small SU(3) breaking, Okubo derived interesting relations among baryons masses. We examine in this section how well the Gell-Mann-Okubo (GMO) relations Donoghue et al. (1992) are fulfilled for the baryons masses obtained on our lattices at different pion mass values. As we will discuss in detail in the next Section, volume and discretization effects are small, and therefore it suffices to analyze the and results. For this study we use the lattice spacing determined from to convert to physical units.

For the octet the GMO relation can be written in the form:

(16)

The results are displayed in Fig. 9 where the left and right hand side terms of Eq. (16) are separately plotted as a function of . The difference between the two terms are compatible with zero at any pion mass. The experimental values, shown by the squares, are respectively 254 MeV and 248 MeV. These results are similar to those presented in Ref. Beane et al. (2007) using a mixed action setup with valence domain wall fermions on rooted staggered sea fermions.

For the decuplet, the GMO relations predict equal mass difference among two consecutive () isospin multiplets:

(17)

The results for the decuplet baryons are displayed in Fig. 9. As can be seen, the equalities of Eq. (17) are strongly violated; the three mass differences of Eq. (17) are spread over about 200 MeV for the range of pion masses that have been computed. The experimental values for these mass differences are  MeV, shown in the plot by the squares. In the lattice results the larger deviation comes from , while for and the mass differences are smaller. The mass difference is increasing as the pion mass decreases. Unfortunately, with our present statistics it is unclear whether this increase is sufficient to bring this mass difference in agreement with experiment but the trend is definitely in the right direction.

A third relation exists, that connects the octet masses with the decuplet masses, which reads as:

(18)

Experimentally, this relation is fulfilled at the 10% level yielding 276 MeV for the left hand side and 305 MeV for the right hand side of Eq. (18). These values are again shown by the filled squares in Fig. 9. The corresponding lattice results are shown in the same figure. One can see that, as in the octet case, the relation of Eq. (18) is satisfied within our statistical uncertainties at each pion mass. It also approaches the experimental results with decreasing pion mass.

(a)

(b)

(c)

Figure 9: Gell Mann Okubo relations for the baryon octet (a) decuplet (b) and mixed octet-decuplet (c) as a function of . Vertical lines correspond to the physical results. Data were obtained from simulations at and . The lattice spacing determined from is used to convert to physical units.

Fulfillment of the GMO relations is considered a success of symmetry. Violations of these relations indicate that breaking is not small. Therefore one would expect that these relations are better satisfied as we approach the limit , up to discretization effects. This corresponds to about  GeV. For the decuplet mass relation given in Eq. (17) it is unclear if this would be indeed satisfied by the lattice data whereas the other two relations are fulfilled at all masses.

Iv Systematics

In order to compare our lattice results collected in Tables 567 and 8 to the physical masses we need to check for finite volume effects, cut-off effects and the extrapolation to the physical light quark masses. The strange quark was fixed to the physical value using the kaon mass with the light quarks extrapolated to the physical point as explained in Section II.B. A check of the effect of this tuning on baryon masses has been discussed in Section III.B. In this section we discuss finite volume and cutoff-effects, in particular the isospin breaking.

iv.1 Finite volume effects

Finite volume corrections to the nucleon mass in lattice QCD have been studied in Ref. Ali Khan et al. (2004) within the expansion which assumes that finite size effects originate from pions that propagate around the spatial box. Using relativistic baryon chiral perturbation theory Procura et al. (2004) the finite volume corrections to the nucleon mass to