I Introduction

Hyperon and charmed baryons, Axial Charge, Lattice QCD

Axial charges of hyperons and charmed baryons using twisted mass fermions


[1ex]ETM Collaboration

C. Alexandrou, K. Hadjiyiannakou, C. Kallidonis,

Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus

Computation-based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus

The axial couplings of the low lying baryons are evaluated using a total of five ensembles of dynamical twisted mass fermion gauge configurations. The simulations are performed using the Iwasaki gauge action and two degenerate flavors of light quarks, and a strange and a charm quark fixed to approximately their physical values at two values of the coupling constant. The lattice spacings, determined using the nucleon mass, are  fm and  fm and the simulations cover a pion mass in the range of about 210 MeV to 430 MeV. We study the dependence of the axial couplings on the pion mass in the range of about 210 MeV to 430 MeV as well as the breaking effects as we decrease the light quark mass towards its physical value.

March 15, 2018

I Introduction

The axial charges of hyperons are important parameters of low energy effective field theories. The nucleon axial charge, the value of which is well known experimentally, is a crucial parameter entering in the description of many observables computed within chiral effective theories. It describes neutron -decay and sheds light on spontaneous chiral symmetry breaking. As a well-measured quantity, it has been traditionally used as a benchmark quantity for lattice QCD computations and it has been extensively studied by many lattice QCD collaborations, including using simulations with a physical value of the pion mass Abdel-Rehim et al. (2015a); Bali et al. (2015). For recent reviews see Refs. (Alexandrou, 2014; Alexandrou and Jansen, 2015; Alexandrou, 2015; Constantinou, 2015). In addition, the quark axial charge probes the intrinsic quark spin contribution to the total spin of a quark in the nucleon, and has been studied both theoretically and experimentally for a number of years.

While there has been an extensive work for the nucleon axial charge, the axial charges of hyperons or charmed baryons are less well studied. The knowledge of these axial charges is very important allowing us to examine the validity of SU(3) relations among them as a function of the pion mass. They are also important parameters for chiral expansions of baryonic quantities. Their experimental determination is difficult because most baryons are very short-lived as for instance the , which decays in . Therefore, lattice QCD can provide valuable information on these quantities and in general into the structure of these baryons.

In this work, we study the axial charges of hyperons and charmed baryons using twisted mass fermions with two light quark doublets as well as a strange and a charm quark with mass fixed to their physical values, denoted as ensembles. Results are obtained for the axial charges of the two 20-plets of spin-1/2 and spin-3/2 baryons that arise when considering the two light, the strange and charm quarks. Five ensembles of twisted mass fermions are analyzed spanning a pion mass range between 210 MeV and 450 MeV allowing us to examine the dependence of the axial charges on the pion mass, which is found similar to the one observed for the nucleon axial charge within this pion mass range.

Ii Lattice setup and simulation details

In this work, we analyze five ensembles of gauge configurations produced by the European Twisted Mass Collaboration (ETMC) Baron et al. (2008, 2009), with maximally twisted quark flavours. In summary, these gauge fields are produced using as a gauge action the Iwasaki improved gauge action Weisz (1983); Iwasaki (1985); Iwasaki et al. (1997), which includes besides the plaquette term also rectangular Wilson loops given by


is the bare inverse coupling, and the (proper) normalization condition . We note that for this action becomes the usual Wilson plaquette gauge action.

The twisted mass Wilson action used for the light degenerate doublet of quarks (,) is given by Frezzotti and Rossi (2004a); Frezzotti et al. (2001)


with the third Pauli matrix acting in flavour space, the bare untwisted light quark mass, the bare twisted light mass. The massless Wilson-Dirac operator is given by




The quark fields denoted by in Eq. (2) are in the so-called “twisted basis”. The fields in the “physical basis”, denoted by , are obtained at maximal twist by the transformation


In addition to the light sector, a twisted heavy mass-split doublet for the strange and charm quarks is introduced, described by the action Frezzotti and Rossi (2004b, c)


where is the bare untwisted quark mass for the heavy doublet, is the bare twisted mass along the direction and is the mass splitting in the direction. The quark fields for the heavy quarks in the physical basis are obtained from the twisted basis through the transformation


In this paper, unless otherwise stated, the quark fields will be understood as “physical fields”, , in particular when we define the interpolating fields of the baryons.

The form of the fermion action in Eq. (2) breaks parity and isospin at non-vanishing lattice spacing, as it is also apparent from the form of the Wilson term in Eq. (3). In particular, the isospin breaking in physical observables is a cut-off effect of  Frezzotti and Rossi (2004a). For the masses of baryon isospin multiplets such isospin breaking effects have been found to be small for the ensembles considered in this work Alexandrou et al. (2014a).

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 to give a mass to the pions. 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 Frezzotti and Rossi (2004a, c). In practice, we implement maximal twist of Wilson quarks by tuning to zero the bare untwisted quark mass, commonly called PCAC mass, Boucaud et al. (2008); Frezzotti et al. (2006), which is proportional to up to corrections.

The gauge configurations analyzed in this work correspond to two lattice volumes and four values of the pion mass for and one volume and one pion mass for . The corresponding lattice spacings are respectively  fm and determined from the nucleon mass Alexandrou et al. (2014a).

For the heavy quark sector we use Osterwalder-Seiler valence strange and charm quarks. Osterwalder-Seiler fermions are doublets like the the u- and d- doublet, i.e. and , having an action that is the same as for the doublet of light quarks, but with in Eq. (2) replaced with the tuned value of the bare twisted mass of the strange or charm valence quark. Taking to be equal to the critical mass determined in the light sector, the improvement in any observable still applies. One can equally work with () or () of the strange (charm) doublets. In the continuum limit both choices are equivalent and in this work we opt for and . Since our interest in this work is the baryon spectrum we choose to tune the strange and charm quark masses to reproduce the physical masses of the and baryons, respectively. More details on the tuning procedure can be found in Ref. Alexandrou et al. (2014a). In Table 1 we summarize the parameters of the simulations used in this work, including the value, the spatial lattice extent in lattice units , the value of the bare twisted light quark mass as well as the pion masses.

,  fm,
,  fm 0.0025 0.0035 0.0055 0.0075
No. of Confs 199 200 200 200
 (GeV) 0.256 0.302 0.373 0.432
3.42 4.03 4.97 5.77
,  fm
,  fm 0.0015
No. of Confs 200
 (GeV) 0.213
Table 1: Input parameters () of our lattice simulations and corresponding lattice spacing () and pion mass (). The lattice spacings are determined using the nucleon mass.

In the following we will refer to the ensembles with as the B-ensembles, and to the ensemble with as the D-ensemble. We also use the notation B or D where denotes the value and denotes the spatial extent of the lattice, , e.g. B25.32 refers to our ensemble with , and .

Iii Lattice evaluation

iii.1 Matrix element Decomposition

We consider the 40 diagonal baryon matrix elements of the axial vector operator , where denotes a quark field of a given flavor. For baryons containing up and down quarks we consider the isovector combination where disconnected contributions vanish in the continuum limit, namely . The isoscalar matrix elements of these baryons receive disconnected contributions. While there has been a big progress in developing techniques to compute them Alexandrou et al. (2012a, b), the computational resources required are typically two orders of magnitude larger than those required for the connected. The disconnected contribution to the isoscalar axial charge of the nucleon has been computed for the B55.32 ensemble that corresponds to a pion mass  MeV Abdel-Rehim et al. (2014); Alexandrou et al. (2014b). It has also been computed for an ensemble of clover fermions with pion mass  MeV Bali et al. (2012). In both calculations they were found to be about 10% of the connected isoscalar axial charge. Preliminary results at the physical value of the pion mass increase the value of the disconnected contribution to to about 20% the value of the connected . For the same ensemble the strange axial charge is found to be while the charm axial charge is consistent with zero Abdel-Rehim et al. (2015b). Given the large computational effort needed to obtain a reliable signal, the computation of disconnected contributions to the matrix elements of the isoscalar current and to the strange and charm axial currents are neglected in the current work. Instead in this first study of the hyperon and charmed baryon axial charges, we compute the dominant connected contributions as well as combinations where the disconnected contributions cancel in the flavor symmetric limit. Preliminary results on these quantities were presented in Ref. (Alexandrou et al., 2014c).

For spin-1/2 baryons the matrix element of the axial-vector current in Euclidean space can be expressed as


where , () are the momentum and spin of the final (initial) spin-1/2 baryonic state (), is the momentum transfer and represents a Dirac (spin-1/2) spinor. For a Dirac spinor we have


The corresponding equation in Euclidean space for spin-3/2 baryons reads


where now represents a Rarita-Schwinger spin-3/2 spinor, with


The Rarita-Schwinger spinors satisfy the spin sum relation given by (Alexandrou et al., 2013a)


For both spin-1/2 and spin-3/2 baryons the axial charge is obtained from the forward matrix element i.e. setting in Eqs. 8 and 10, yielding and .

iii.2 Baryon interpolating fields

In the lattice formulation hadron states of interest are obtained by acting on the vacuum with interpolating fields constructed to have the quantum numbers of the hadron under study. For low-lying states, we usually consider interpolating fields that reduce to the quark model wave functions in the non-relativistic limit. Baryons made out of three combinations of the , , and quarks belong to SU(4) multiplets, and thus we use SU(3) subgroups of the SU(4) symmetry to identify their interpolating fields. In general, the interpolating fields of baryons can be written as a sum of terms of the form , apart from normalization constants. The structures and are such that they give rise to the quantum numbers of the baryon state in interest. For spin-1/2 baryons, we will use the combination and for spin-3/2 baryons we will use , taking spatial and is the charge conjugation matrix.

The multiplet numerology is . All the baryons in a given multiplet have the same spin and parity. The -plet consists of the spin-3/2 baryon states and can be further decomposed according to the charm content of the baryons into , where the is the standard decuplet and is the triply charm singlet. The singly charmed baryon states belonging to the multiplet are symmetric under the interchange of , and quarks, following the rule that the diquark is symmetric under interchanging . Finally, the doubly charmed -plet consists of the isospin partners and the singlet . The -plet is shown schematically in the left panel of Fig. 1. The corresponding interpolating fields of the spin-3/2 baryons are collected in Table LABEL:spin32_tab of Appendix A.

The -plet consists of the spin-1/2 baryons shown schematically in the center panel of Fig. 1. It can be decomposed as . The ground level comprises the well-known baryon octet, whereas the first level splits into two SU(3) multiplets, a and a . The states of the are symmetric under interchanging , and where the states of the are anti-symmetric. We show these states explicitly in the right panel of Fig. 1. We note that the diquark appearing the interpolating field of spin-1/2 baryons, is anti-symmetric under interchanging . The top level consists of the -plet with . The interpolating fields of the spin-1/2 baryons are collected in Table LABEL:spin12_tab of Appendix A. The fully antisymmetric -plet is not considered in this work.

Figure 1: The baryon multiplets constructed using the SU(4) group. The left diagram shows the spin-3/2 -plet, the center diagram shows the spin-1/2 -plet and in the right diagram we show the decomposition of the level of the spin-1/2 -plet of the center diagram. All diagrams are taken from the PDG Olive et al. (2014).

The interpolating fields of spin-3/2 baryons as defined in Table LABEL:spin32_tab can have an overlap with spin-1/2 excited states. To remove the unwanted contributions and isolate the desired spin-3/2 ground state we project the spin-3/2 components by acting with the 3/2-projector on the interpolating fields as


For non-zero momentum the projector is given by (Benmerrouche et al., 1989)


The spin-1/2 projector is obtained by yielding


In this work we are interested in correlation functions in the rest frame where thus the last term of Eqs. (14) and (15) involving momentum terms vanishes. The form of the two-point correlation functions when the projectors to spin-1/2 and spin-3/2 are applied to the corresponding interpolating fields is given by


where . For some of the spin-3/2 baryons, the inclusion of the spin-3/2 projector does not have a significant effect in the correlation function, since the spin-1/2 is an excitation with a large energy splitting from the spin-3/2 ground state. This is the case, for instance, for the . However, for other baryons, such as the s, the projector is required to isolate the ground state. Thus, in order to ensure that we measure the desired spin-3/2 ground state, we always apply the spin-3/2 projector to the interpolating fields of Table LABEL:spin32_tab. The reader interested in more details on the effects of these projectors on the baryon masses is referred to Ref. Alexandrou et al. (2014a).

iii.3 Correlation functions

The matrix elements required for the calculation of the axial charges are extracted from dimensionless ratios involving two- and three-point correlation functions. The diagrams of the two-point function and the connected part of the three-point function involved in our calculations are depicted in Fig. 2. To extract the axial charges we consider kinematics for which the final and initial momentum are . Since we only compute diagonal matrix elements, we consider three-point functions with the same baryon state at both source and sink. The time-independent ratio is obtained by dividing the three-point function with the corresponding zero-momentum two-point function. For the case of spin-1/2 baryons the two- and three-point functions are given by (Alexandrou and Jansen, 2015; Alexandrou et al., 2011a)


For the case of spin-3/2 baryons the traces of the corresponding two- and three-point functions are given by (Alexandrou et al., 2013b, 2011b)


The projection matrices and are given by

Figure 2: The baryon two-point function (left) and the connected three-point function (right) are shown diagrammatically. The solid lines represent fully dressed quark propagators.

iii.4 Smearing techniques

In order to increase the overlap with the baryon ground state, we apply Gaussian smearing at the source and the sink Alexandrou et al. (1994); Gusken (1990). The smeared interpolating fields are given by




and is the hopping matrix given by


We also apply APE-smearing to the gauge fields entering the hopping matrix. The parameters for the Gaussian smearing and are optimized using the nucleon ground state Alexandrou et al. (2008). Various combinations of Gaussian smearing parameters, and have been tested and it was found that combinations giving a root mean square radius of about fm are optimal for suppressing excited states in the case of the nucleon. We adopt the same parameters here, which have the following values

: , , , ,
: , , , .

iii.5 Plateau method to extract axial charge

The computation of the axial charges proceeds through the evaluation of the diagrams shown in Fig. 2. As already mentioned, when taking the isovector combination of the axial current the disconnected diagrams are zero up to lattice artifacts and can be safely neglected when close to the continuum limit. In such cases the connected contribution depicted in Fig. 2 yields the whole contribution. For the rest of the cases the disconnected contributions are neglected in this first computation. The creation operator of the baryon of interest is taken at a fixed source position with zero-momentum. Since, as discussed above, the axial charges are extracted directly from the matrix elements at , the annihilation operator at a later time also carries momentum . The current couples to a quark at an intermediate time and carries zero momentum (). To compute the connected three-point function we use the so-called fixed-current method Martinelli and Sachrajda (1989) where the current-type and the time separation between the source and the current insertion, are fixed. The advantage of the sequential inversion through the current is that with one set of sequential inversions per choice of momentum and insertion time we obtain results for all possible sink times, any particle state and any choice of the projectors given in Eq. (21). An alternative approach that computes the spatial all-to-all propagator using stochastic methods was shown to be suitable for the evaluation of baryon three-point functions Alexandrou et al. (2013c). With this method one can include any current at the insertion point for any particle state and any projector at the sink without needing additional inversions. However, the disadvantage is that one introduces stochastic noise, so one has to check convergence as a function of the number of noise vectors. In this work, since we are only interested in the axial charges, we instead adopt the sequential method through the current also referred to as fixed current method.

A standard way of isolating the matrix elements, is to form appropriate ratios with the use of the two- and three-point functions of Eqs. 17 and 18 for the spin-1/2 baryons and Eqs. 19 and 20 for the spin-3/2 baryons.

In the limit and the unknown overlap terms and Euclidean time dependence cancel thus yielding a time-independent result as a function of the sink time, referred to as plateau region. A constant fit is then performed to extract the axial charge. The traces involved in the two- and three-point functions can be calculated using Dirac trace algebra. Since zero-momentum kinematics are employed the final relations acquire simple forms. Specifically, for the spin-1/2 baryons one obtains


while the corresponding expression for spin-3/2 baryons yields (Alexandrou et al., 2013b)


The plateau value thus yields the unrenormalised charges, which after renormalization with , gives directly the axial charge of the baryon. The renormalization constants used in this work are and , taken from Ref. Abdel-Rehim et al. (2015a).

iii.6 Flavor structure of the axial-vector current

The baryon axial charges govern processes like and . They can be extracted by considering the matrix elements where  Choi et al. (2010) and is the isovector combination for the axial-vector current. Given that we have four quark flavours, we can construct for the axial-vector current combinations corresponding to the generators of the gauge group. In this study besides the isovector that corresponds to the generator we consider combinations of the other two diagonal SU(4) generators, namely and .

The generator gives the well-known isovector combination, which produces the axial coupling between the pion and the baryon effective fields. In the SU(4) limit disconnected contributions will cancel for all three combinations given by the currents


In what follows, for a given baryon, we denote the flavor combination of the current corresponding to as , to as and to as . In the case of at least one term in the current will yield a purely disconnected contribution, which will be neglected here. In addition, we consider the isoscalar combination


Having these combinations one can extract the axial charge corresponding to each quark flavor . In Eqs. III.6 and 28 and . Depending on the quark flavor content of the baryon some terms will give purely disconnected contributions and will be neglected.

We note that determines the intrinsic spin carried by the quark inside the given baryon.

iii.7 Fixing the insertion time

In the fixed current method that involves sequential inversion through the current, the time separation between the source and the current insertion, is fixed. Optimally, one would choose a source-insertion separation small enough to keep the statistical errors as small as possible and still large enough to ensure that excited state contributions are sufficiently suppressed.

Figure 3: Results for the axial charges of hyperons and charmed baryons for two choices of the current insertion time are shown, namely with red circles and with blue squares as a function of .

While recent studies have shown that the optimal source-sink time separation is operator dependent Alexandrou et al. (2011c); Dinter et al. (2011), for the axial charge the excited state contamination was generally found to be small at least for pion masses larger than physical Alexandrou et al. (2013a). Still we need to ensure that the insertion-source time separation is sufficiently large to be free of large excited state contaminations. We examine two values of the insertion time, namely and for our B-ensemble with or  MeV, with results shown in Fig. 3. As can be seen, the results are compatible for these two values of the current time insertion. Thus, we fix and seek a plateau as a function of .

A plateau region starting at is obtained confirming ground state dominance at a time separation of from the source and the sink. For the D-ensemble we take to keep the time separation in physical units about the same.

Iv Effective Lagrangian

Before we present our lattice QCD results, we discuss briefly the effective meson-baryon Lagrangians where these axial couplings are defined. Heavy baryon chiral perturbation theory (HBPT) is most commonly applied to the octet and decuplet baryons. The lowest order (tree-level) meson-baryon effective interaction for the octet can be written in terms of two SU(3) scalars. Arranging these two scalars into symmetric and antisymmetric combinations we have (Jenkins and Manohar, 1991a)


where is the traceless octet field


is written in terms of and it is the combination of meson fields that transform like an axial-vector current. Here we follow standard notation and take the matrix of the pseudoscalar mesons, the spin operator acting on the baryon fields, while we suppress the velocity index on and .

In the limit of flavour symmetry, the axial couplings are thus given in terms of the two low-energy constants and appearing in the Lagrangian. For the pion-baryon axial couplings we thus have


while for the octet -baryon couplings


There are five transition coupling constants in addition to the above, namely , , , and , which are also written in terms of and . These require the computation of non-diagonal matrix elements and are not considered in this work.

For decuplet baryons one can only construct one SU(3) scalar and thus the axial coupling constants are given in terms of one constant . The lowest order interaction Lagrangian involving diagonal terms is given by (Jenkins and Manohar, 1991b)


where we suppress the velocity index on the tensor . Suppressing the Lorenz index , is given by Butler et al. (1993)


In Eq. 33 we have not written the coupling of the decuplet to the octet baryons, which introduces another axial transition coupling constant since this will also involved non-diagonal matrix elements which are not computed here. In the SU(3) limit the decuplet axial couplings are given by Tiburzi and Walker-Loud (2008)


Only a few groups have considered charmed baryons within HBPT, see e.g. Refs Jiang et al. (2014); Li and Zhu (2012); Liu and Zhu (2012, 2013), and these studies focus only on the singly charmed baryons. For completeness we give here the Lagrangian for singly charmed baryons. The baryon fields for the symmetric -tet and the antisymmetric -plet of spin-1/2 charmed baryons are defined as follows


The definition of for the spin-3/2 -tet is similar to that of . The effective Lagrangian at tree-level can be written in terms of the couplings and reads Liu and Zhu (2012, 2013)


Similarly to the octet case, is written in terms of , where is the pseudoscalar meson field and is the spin matrix acting on the baryon fields.

For hyperons breaking arises as a result of the larger strange quark mass and lattice QCD provides a framework to study the breaking as a function of the quark mass. Although a similar approach can be used for charmed baryons, the much larger mass of the charm-quark can make symmetry patterns more difficult or even impossible to disentangle.

V Lattice results

In this section we present our results on the axial charges for the four B- and the one D-ensembles. Comparisons with other lattice calculations are shown for the axial charge of spin-1/2 hyperons results wherever available. A study of the SU(3) flavour breaking for the octet and decuplet baryons is also presented. A similar analysis is carried out for charmed baryons where corresponding relations hold when replacing the strange with the charm quark although the breaking is expected to be larger. All the lattice data on the axial couplings considered in this work are collected in Tables 2-9 of Appendix B.

v.1 Axial charges of octet and decuplet baryons

Octet baryons

As already pointed out, the axial charge of the nucleon is well measured and it is thus considered as a benchmark quantity within lattice QCD. Before discussing results on the axial charges of other baryons, we first compared the nucleon axial charge, , using the fixed current approach adopted here with the results obtained with the fixed sink method. The latter approach is the one routinely used to extract the nucleon axial form factors. In a previous work we calculated for the B55.32 and the D15.48 ensembles using the fixed sink method. In Fig. 4 we show results as a function of the pion mass for both the isovector and the isoscalar axial charges using the fixed current approach, as well as using the fixed sink method Alexandrou et al. (2011a, 2013a). One can see that the two methods give compatible results. A general observation is the underestimation of the nucleon isovector axial charge for larger than physical pion masses. A recent computation using twisted mass clover-improved fermions at a physical value of the pion mass yields a value consistent with the experimental value albeit with large statistical uncertainty Abdel-Rehim et al. (2015a). Similarly, the connected part of the isoscalar charge is overestimated for larger pion masses. Disconnected contributions are found to be negative and will thus decrease this value. As expected, as we approach the physical pion mass larger statistics are required in order to obtain a more robust result.

Figure 4: Axial charge for the nucleon as a function of the pion mass for the isovector (left) and isoscalar (right) combinations. With red circles we show the results of this work, in blue triangles we show results from twisted mass fermions (TMF) from Ref. Alexandrou et al. (2011a) and with open green circles we show results using TMF ensembles (B55.32 and D15.48) but obtained with using the fixed sink method from Ref. Alexandrou et al. (2013a) (shifted to the left for clarity). The experimental values shown with the black asterisk are taken from PDG Olive et al. (2014).

The other particles within the octet are the and isospin multiplets and the singlet. Contrary to the nucleon, the short lifetime of these baryons makes the experimental determination of their axial couplings difficult, and therefore very limited experimental data are available. Additionally, theoretical estimates are rather imprecise. On the lattice, only a handful of other calculations have considered the octet axial charges Lin and Orginos (2009); Gockeler et al. (2010); Erkol et al. (2010). We compare previous lattice QCD results with our values in Fig. 5, where we show the renormalization independent ratios and . There is an agreement among all the data within the whole pion mass range despite the different discretizations, lattice spacings and volumes, indicating that lattice artefacts are small for the parameters used in these simulations. An estimate of and from PT is found in Ref. Jiang and Tiburzi (2009), giving values of and , not far from our lattice values. These couplings have been also obtained from relativistic constituent quark model (RCQM) calculations in Ref. Choi et al. (2010), yielding values and . The value of from the latter calculation notably overestimates our lattice result 1.

In Figs. 6 and 7 we show the pion mass dependence for the , the and multiplets for the flavor combination of the two diagonal generators and . As can be seen, results are fully compatible between isospin partners, indicating that the isospin symmetry breaking effects, due to the finite lattice spacing, are small. All data exhibit weak dependence on the pion mass over the range of pion masses studied in this work. Comparing the results using the B25.32 and D15.48, which have similar pion mass, we observe consistent values for the axial charges, an indication that cut-off effects are small.

In order to obtain an estimate of the axial charges of the hyperons at the physical pion mass, we perform a chiral extrapolation according to the Ansatz , where and