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Chiral extrapolation of baryon mass ratios

We analyze lattice data for octet baryon masses from the QCDSF collaboration employing manifestly covariant Baryon Chiral Perturbation Theory. It is shown that certain combinations of low-energy constants can be fixed more accurately than before from this data. We also examine the impact of this analysis on the pion-nucleon sigma term, and on the convergence properties of baryon mass expansions in the SU(3) symmetry limit.


I. INTRODUCTION
Three-flavor Baryon Chiral Perturbation Theory (BChPT) [1,2] is an effective field theory for QCD in the low-energy regime where the lowest-lying meson-octet and the lowest-lying baryon octet (and possibly also the decuplet) are the relevant degrees of freedom.
It is now known for many years that chiral expansions derived from this framework are usually not under sufficient theoretical control because of an unsatisfying convergence behavior 1 : higher-order corrections from chiral loops can in general not be guaranteed to be much smaller than the leading terms. The quark mass expansion around the threeflavor chiral limit converges slowly because the strange quark mass, m s , is much larger than the light quark masses m = m u,d , and induces sizeable kaon loop corrections to the leading tree level results. Detailed discussions on the convergence behavior of BChPT from various perspectives can be found in [3][4][5][6][7][8][9][10][11][12]. The essential point, however, can be seen immediately from a comparison of mass scales: while √ 2B 0 m ∼ M π ≈ 140 MeV is small compared to a typical hadronic scale of ∼ 1 GeV, the three-flavor expansions can involve √ 2B 0 m s =: Ms s ≈ 700 MeV, which is not small compared to that reference scale.
In fact, loop graphs involving the eta meson are only suppressed by an additional factor of M η /(4πF 0 ) ≈ 0.5, which can easily be compensated by additional prefactors (F 0 is the meson decay constant in the three-flavor chiral limit, while B 0 is related to the quark condensate in the same limit).
The problem is particularly urgent for chiral extrapolation formulae for lattice simulations. There, the meson masses are in general even larger than for physical quark masses.
Consequently, the application of BChPT to such lattice data is not under satisfactory theoretical control. Though one can obtain fits that seem to describe the data fairly well, a stable determination of the corresponding low-energy constants (LECs) is not yet within reach, because the extracted values can be strongly affected by uncontrolled higher-order corrections. For some specific application, one might not even worry much about this, but one should recall that the main point of ChPT is to yield relations between many different observables, in terms of a limited set (at least at a fixed order in the low-energy expansion) of parameters. For example, the baryon mass in the chiral limit, m 0 , and its leading quark mass corrections will unavoidably appear in practically any complete loop calculation in BChPT, in baryon form factors, meson-baryon scattering lengths, etc. Therefore, one should try to somehow control the higher-order loop corrections.
Many alternative ways to overcome this problem have been proposed in the literature, e.g. long-distance regularization [6], reordering and resummation schemes [7,13], large-N cscaling arguments [14][15][16][17] and two-flavor expansions for hyperon properties [18,19], each having its own merits. In this work, we stick to a standard scheme (evaluating the chiral loops employing infrared regularization [20]) and thus analyze the quark mass expansions of the baryon octet masses in a fashion closely following [21]. We refer to that reference for most of the corresponding technical details (see also [22]). For studies in different schemes, see e.g. [3,[23][24][25][26][27][28][29][30][31][32][33][34]. We have reproduced the complete one-loop (O(p 4 )) calculation in this covariant scheme. Our aim here is to put to use calculations like [21], by analyzing the lattice data presented in [35] which covers a quark mass region where all the masses of the pseudoscalar bosons are smaller than the physical eta mass, and to give improved estimates for LECs featuring in these calculations. We consider this a first and necessary step toward a controlled application of BChPT to such data sets. While in most threeflavor lattice simulations, the (large) strange quark mass is fixed 2 , and the physical point is approached lowering m (see e.g. [36,37] for recent examples), the strategy in [35] is < 400 MeV at m s = m ). This is already a nice feature, but there is a second point that is probably even more important. In a surprisingly large region around the symmetric point, approximations neglecting all but linear symmetry-breaking effects were shown in [35] to yield a very good description of the data. In particular, quantities which do not receive linear symmetry-breaking contributions are observed to be very stable even down to the physical point (one example for such a quantity is the famous Gell-Mann-Okubo mass difference).
This behavior can not be understood from ChPT alone, but it puts tight constraints on certain combinations of LECs (e.g. by enforcing large cancellations in higher-order terms in the δm -expansion of such quantities). It is convenient for the purpose of studying the symmetry-breaking effects separately to analyze certain baryon mass ratios, where a large factor with the dimension of mass (essentially the baryon mass at the symmetric point) cancels out. The behavior of these ratios can be very well described by BChPT (see our Fig. 5, and also [38] for a corresponding illustration). Here we want to exploit the advantages just mentioned for the determination of the LEC-combinations occurring in octet baryon masses, in the hope that this will help e.g. when analyzing lattice data for other quantities with the same strategy.
This paper is organized as follows: After briefly introducing some basic notation in sec. II, we collect the input from the purely mesonic sector in sec. III in some detail. This is necessary, because the quark-mass dependence of the masses of the pseudoscalar mesons is and directly continue with sec. VI, in which we discuss in detail the extrapolation functions we use, and which combinations of LECs can be extracted more accurately than before from the data we consider (and which can not). In a series of tables, we present our fit results, obtained with different fit strategies and input parameter sets. We discuss our results in sec. VII and give a short conclusion of our findings.

II. PRELIMINARIES
Throughout we work in the isospin limit and set the ight quark masses to m u = m d =: m . In [35], the quark masses are varied along a trajectory T in the (m , m s )-space where the average quark massm := 1 3 (2m + m s ) is kept fixed, while δm = m −m is varied. The aforesaid trajectory T connects the symmetric point, where m = m s =m, and the physical point where the strange quark mass attains its physical value while m = 1 2 (m u + m d ). To get familiar with the notation, let us consider the leading terms in the quark-mass expansion of the squares of the pseudo-Goldstone-boson masses. We havė Here B 0 is related to the quark condensate in the chiral limit, see e.g. [39]. The leading term of the pseudo-Goldstone-boson mass at the symmetric point is then found by setting Our notation is set up such that quantities in the chiral limit will be denoted by a subscript 0, while quantities at the symmetric point on T are marked with a .
In [35], it was noted that some specific combinations of hadron masses are very stable when varying the quark masses along T , compare e.g. Tab. 2 and Fig. 13 in that paper. Below, we list three of these combinations which are relevant for our present study: The experimental values are given in round brackets.

Meson masses
The one-loop expressions for the masses of the pions, kaons and the eta can be found e.g.
in [39]. We give here the expansions of these formulae in terms ofm and δm . The meson mass squared at the symmetric point is given (to one-loop accuracy) by The expansion of the meson masses in the symmetry-breaking parameter δm reads Consequently, we have for the combinations X 2 π , X 2 η : The low-energy constants L i of course correspond to the renormalized constants L r i (µ) of [39]. In [35], instead of the quark mass difference B 0 δm , the variable is used to parameterize the SU(3) symmetry breaking. Inverting this relation and again expanding up to the second order in the symmetry-breaking variable δm , we get where we have also truncated the chiral expansion of the coefficient functions in this series.

Decay constants
Similarly, the expansion of the decay constants is given by where Note that if terms nonlinear in the symmetry-breaking variable are neglected, the combina- is stable at that order.

Quark masses
For the sake of completeness, we also give an expression for the quantity M 2 ss := 2B 0 m s in terms of the one-loop formulae for the meson masses: Keeping this quantity fixed, one can express the light quark mass dependence of the baryon masses through the pion mass dependence, using Numerical analysis For an overview on lattice determinations of the parameters F 0 , L i we refer to the recent discussion in [40]. We reproduce three sets of results in the table below. As central values in our numerical analysis, we adopt the set of parameters displayed in the first row (MILC2010).
The other three parameter sets are used only for the error estimates of our results. The numerical values of the L i are given at a renormalization scale of µ = 770 MeV. At the order we are working, we can replace F 0 in eq. (8) by F , see eq. (19). Given the observed stability of X π , and the fact that this quantity is even used to set the scale for meson observables in [35], we determine the parameter 2B 0m from the requirement that M ! = X phys π = 412 MeV.
We find We note that inserting the value for F 0 directly in eq. (8), one finds very similar values,Ṁ = 428 MeV and F = 112.5 MeV, while the experimental value for F X is F phys X = 105 MeV.
As the one-loop effects discussed above enter our calculation of baryon masses formally at NNLO (meson masses) or even at two-loop order (decay constants), the estimates given in eqs. (23), (24) are sufficient for our purposes.
We also note that the fan plots for the meson masses shown in [35] can be nicely reproduced with the values of the LECs employed above, while X π is indeed almost a constant along T .
For this to be the case, one should expect from eq. (12) that In fact, for our central values (MILC2010) we have so that terms in X π quadratic in the symmetry breaking are indeed tiny. For X η to be equally stable, we infer from eq. (13) that we should have L 7 ∼ −0.1 · 10 −3 . In Fig. 1, we show a "fan plot" for the meson decay constants, i.e. we plot the ratios F π /F X and F K /F X as functions of the variable ν which parametrizes the SU(3) symmetry breaking, see eq. (15). The derivative at the symmetric point (ν = 0) is essentially determined by the LEC L 5 , which we vary within the error according to MILC2010 to generate the bands around the central curves. One observes that the fan plot shows a linear behavior of the ratios with ν in the vicinity of the symmetric point, but some curvature is visible near the physical point, which seems to be necessary to reproduce the experimental values of F π and F K .
Inserting the experimental pion and kaon masses in eq. (21), one arrives at where again we use the MILC2010 parameters as central values. The uncertainty due to the variation of the input parameters is somewhat larger than in eq. (23). We have a clear hierarchy compare eqs. (23) and (27). While it is quite obvious that an expansion in Ms s (over a typical hadronic scale) will be ineffective, an expansion inṀ can be expected to work much better, since the expansion parameter is about 40% smaller.

IV. BARYON MASSES AT THIRD CHIRAL ORDER
From the relevant terms in the chiral Lagrangian, one derives the octet baryon masses to O(p 3 ) in BChPT (see e.g. [24]) from the second order quark mass insertions and the sunrise-type loop graph of Fig. 2. To this order, the mass of the eta can be eliminated by using the Gell-Mann-Okubo relation, InsertingṀ K → 495 MeV andṀ π → 140 MeV givesṀ η 566 MeV. In our numerical work, we shall use D = 3 4 , F = 1 2 which is consistent with the available literature [43][44][45][46], see also [15]. It is difficult to reliably obtain these two axial coupling constants from a fit to baryon masses. In [15], values close to the expectation from phenomenology were obtained imposing a large-N c counting and restricting the fit to certain mass combinations where the singlet piece drops out. See also [33] for a similar result. For the above choice of D, F , the p 3 -terms cancel completely in the combination This quantity vanishes in the SU(3) limit, however, it is not too small in the real world.
Inserting the physical values for the baryon masses (averaging over isospin multiplets), gives ∆ phys DF 292 MeV. Assuming that this choice for D, F is roughly correct, and that the O(p 4 ) terms are not too large, one would get 20 It is known, however, that the BChPT expressions for the baryon masses only converge slowly at the physical point. Moreover, the numerical values of D and F are also a source of uncertainty. We shall therefore require only that the estimate for this combination of LECs is valid within a ∼ 50% error, In contrast to ∆ DF , the well-known Gell-Mann-Okubo difference [47,48] 6.0 MeV with the baryon masses used above. It seems that the higher-order corrections in this combination are very small: the effect is almost entirely given by the leading one-loop correction. The fact that ∆ GM O is so small numerically could be traced back to the fact that it is of order (δm ) 2 in the symmetry-breaking variable.
Fitting the p 3 -expressions to the experimental baryon masses, with D = 3 4 , F = 1 2 , m eff 0 := m 0 − 6b 0 X 2 π leads to the results presented in the table below. We vary the input parameter F 0 appearing in the O(q 3 )-correction in a wide range to get a first impression of the uncertainty in these results. The convergence of the expansion of the baryon masses using these parameters is poor, however: For example, the third order correction to the nucleon mass is −423 MeV for F 0 = 80.3 MeV. This is a very large correction, even if one expects a generic suppression factor of ∼ M K 4πF 0 ∼ 0.5 for each additional chiral order. We infer from these results that a reliable determination of BChPT low-energy constants at (strange) quark masses higher than, or equal to, the value at the physical point is not feasible, due to much enhanced non-analytic loop corrections in this mass region. On the other hand, given that the above fit results can be regarded as a meaningful first approximation, we should at least expect the following pattern to be reproduced also in higher-order calculations:

Sigma term at third chiral order
The third-order formula for the pion-nucleon sigma term reads (see also [24]): where Numerically, the third order correction gives a contribution of at the physical point, to a positive quantity that is of the order 50 MeV ! This again casts strong doubts on the applicability of BChPT at the physical point to obtain reasonable predictions for sigma terms, without further input.

V. FOURTH ORDER CONTRIBUTIONS
At fourth chiral order, additional loop graphs have to be calculated. First, one has a graph of the same topology as the graph of Fig. 2, but with an additional quark mass insertion from the second order Lagrangian, proportional to b 0 , b D,F . This graph is shown on the l.h.s. of There are also seven new contact terms ∼ d i contributing to the masses at fourth order, they absorb divergences of the infrared-regularized loop integrals. The self-energy Σ (n) ( / p) of n-th chiral order is then calculated directly from those graphs, and gives rise to the mass shifts ∆m at the different orders: The calculation of the first graph in Fig. 3 is straightforward and does not introduce new free parameters, so we will not discuss it further here (see e.g. Ref. [21]). In the following, we are concerned with the contact terms and the tadpole graphs, which involve new unknown couplings not present in a simple leading one-loop calculation. These new parameters can be grouped in two sets: one which will simply represent new free parameters to be determined in the fits, and one which will be fixed using input from other sources. Let us discuss both sets in turn.

Contact terms
We use the fourth order Lagrangian of Frink and Meißner [21], The contact terms give the following contributions to the baryon masses: where We also recall that the combination X N has no linear symmetry-breaking terms. In particular, In the contact term contributions linear in the symmetry-breaking, only the combinations so these two parameters can also not be determined individually from the fan plots alone (again, the shift proportional to d 6 would be of higher chiral order in the tadpole graph involving b 0 ). For a later application, we also give the contact term contribution to the pion-nucleon sigma term. It is computed here employing the Hellmann-Feynman-theorem, Inserting the expressions from eq. (44), we find σ (ct)

Tadpole graphs
The second order Lagrangian we employ here differs slightly from the one given in [21] because since then one further term could be eliminated [49]. The Lagrangian is then given This form has also been used in [50,51], and we shall also take the numerical values of the LECs b 1−4 , b 8−11 from the latter references. Note, however, that in those references, the chiral potential derived from the above Lagrangian is iterated to infinite order in a coupled-channel Bethe-Salpeter equation, introducing a model-dependent uncertainty not controlled within strict ChPT. To estimate the impact of this uncertainty on our results, we perform all our fits with three different parameter sets from meson-baryon scattering (set 1-3). In addition, we also add one parameter set where we take all b 1−11 to vanish (set 4). The four parameter sets are collected in the table below (in GeV −1 ). b 5−7 do not Ref.

Renormalization
The infrared regularized loop integrals still contain UV divergences as d → 4, proportional The couplings b 0,D,F do not have to absorb divergences from the infrared regularized loop graphs. At third order, there are no counterterms contributing to the baryon masses, and therefore also no divergences. The divergences of the infrared singular parts of the loop integrals at fourth order can be absorbed by the following renormalization prescription for the counterterms d i : with We can then define scale-independent quantitiesd i by

VI. ANALYSIS OF FAN PLOT DATA
The main objective of this study is the analysis of the fan plot data collected in Tab. 22 and Fig. 20 of [35]. The following ratios of baryon masses are considered, There is therefore good reason to believe that the chiral extrapolation formulae provided by three-flavor BChPT will work better when applied to the present data set than in the usual applications, though meson masses of ∼ 400 MeV are probably still too high to assure a proper convergence behavior of the chiral expansions in general. Another indication of a better behavior of the quantities considered here is the observation (made in [35]) that finite-volume effects, which are mainly generated by the chiral loops, tend to cancel out in the baryon mass ratios f B . Here we only remark that finite volume effects might still be non-negligible for the data in question, and deserve further study. It is also interesting to note that, employing large-N c arguments and Heavy-Baryon ChPT, it was shown in [15] that the poor convergence behavior of the three-flavor chiral extrapolations could be traced back to the flavor-singlet sector, considering in particular the mass relation called R 1 in the latter reference. This again suggests that it might be a good idea to factor out a convenient flavor-singlet quantity (in our case, X N ) as is done in the baryon mass ratios relevant for the fan plots. In this section, we will see that certain combinations of low-energy constants can already be extracted quite reliably from the fan plot data for the ratios f B , namely, the LEC combinations which parameterize the leading symmetry-breaking contributions to the baryon masses. We consider this as a first and necessary step toward a theoretically controlled application of BChPT formulae to lattice data. We also make an attempt to determine the remaining LECs occuring in the baryon mass extrapolations, however, the constants parameterizing the singlet contributions can only be roughly estimated.
Though we shall use the full one-loop BChPT expressions for the baryon masses in the ratios f B in the end, it is instructive to consider the expansion of the latter ratios in the variable δm parameterizing the symmetry breaking. This will make clear which subsets and combinations of LECs can be more accurately determined than before, and what the qualifications and sources of uncertainties for the various determinations are.

Expansion of mass ratios f B
Employing the full one-loop ChPT calculation of the baryon masses, we are sensitive only up to terms (δm ) km2−k , while higher order terms will be modified by terms on the two-loop level. Therefore, the following equations are to be understood as resulting from a double expansion: the chiral expansion, on the one hand, and the expansion in δm , on the other hand. One finds where f (1) and f (2) The dots stand for corrections from loop graphs of fourth chiral order, which we do not display explicitly here. Let us concentrate first on the contribution from the contact terms.
We make the following observations: The last two items would lead to problems when only using the fan plots and ratios f B in the fitting procedure: First, since terms of order (δm ) 3 and higher are expected to be small in most parts of the fan plots, it is very likely that a stable determination of d 7 would be difficult. Second, we would like to use the values of D, F, b 1−4 , b 8−11 directly as input, and the baryon mass in the chiral limit, m 0 , is not accurately known. What is more or less accurately known, however, is the baryon mass at the symmetric point, for reasons discussed in the next subsection. We can eliminate m 0 in favor of m . Moreover, we can replace F 0 by F (see eq. (19)) at the order we are working. These replacements are rather natural when expanding in the symmetry breaking around the point m = m s =m.
Doing this, the previous expansions read after this slight reordering (in terms ofb D,F defined above): while the formulae for f we can see that b 2,9 enter there only for N, Ξ, while the combinations 9b 1 + b 3 , 9b 8 + b 10 enter for all four baryons.

Expansion of X N
The above ratios f B are all normalized to X N := 1 3 (m N + m Σ + m Ξ ). This combination was shown in [35] to be practically constant along a trajectory with constantm, and thus approximately equal to m num = 1150 MeV for the choice ofm corresponding to the experimental hadron masses. Subsequently, X N was even used to set the scale in the simulations leading to the fan plot for the baryon masses. Employing our ChPT formulae, the flat behavior of X N is not automatically guaranteed, but should be enforced in the fitting procedure.
To see how this can be done, we write down the expansion of X N in the symmetry-breaking variable δm : and X (2) Here, the LEC d 7 shows up at orderm 2 and (δm ) 2 , respectively. The baryon mass in the chiral limit, m 0 , and the combinations of LECs appearing here will be determined in the next section from the behavior of m (m), for which some lattice data points with 300 MeV < M < 500 MeV exist. We repeat that we dismiss any lattice data involving meson masses much above 500 MeV.

Linear fits to the fan plots
As a first step, we perform a linearized fit to the fan plot data only. Only the LECcombinations that enter terms linear in the symmetry breaking can be safely determined from such fits. As shown above, in a full one-loop calculation in ChPT, these are the combinationsb D,F . As an additional constraint, we impose that the mass combination X N does not deviate much from the physical value ∼ 1150 MeV. Approximating using eqs. (68) and (15), and truncating the coefficients after O((2B 0m ) 2 ), we can determine the following four combinations of LECs from such fits: The parameter combinationsb D,F will be fixed in the full fits for every set 1 − 4, while m 0 , b 0 , d 1−7 will be left free -the comparison with the valuesb 0 ,d 7 from the tables above will only serve as a consistency check afterwards. Please note that the numerical values given in the previous four equations relate to the fixed values for µ and 2B 0m specified above.    In the next step, we use the full one-loop expressions for the baryon masses in the fit functions. We fit to the fan plot data for the baryon octet masses (Table 22 of [35]) and the lowest three points for X N (m, ν = 0) = m (M ) (from Table 19 of where the average quark mass is fixed to its physical value, we shall use the numerical values from eqs. (23,24). The meson-LECs will be taken from the MILC2010 set, see Tab. I.
The two combinationsb D,F will then be fixed to the corresponding values determined in the previous subsection, for each of the four sets of b 1−11 from Tab. II.
With given values forb D,F for √ 2B 0m = 0.42 GeV, we can determine seven more parameters from the present data set. First, the (δm ) 2 -terms in the baryon mass formulae can be fixed by four free parameters, which may be taken as (see e.g. eq. (63)) andd 7 defined in eq. (74). Furthermore, inserting in eq. (71), we can determine the combinations and m 0 from the running of X N (δm = 0) with B 0m . The combinationb 0 used in the previous subsection can then be computed, for √ 2B 0m = 0.42 GeV, as The results are given below, in Tab Tab. XI. We observe a good overall agreement of all these fits: varying the input parameters b i , F , L i always leads to similar results. We will discuss our results in detail in the next section.     The dimensionless suppression factor could naively be expected to be of the order Of course, we already know, e.g. from the considerations in sec. IV, that the leading nonanalytic loop correction ∼ M 3 is much enhanced w.r.t. this expectation. Using eq. (71), we  find The result for m is very close to m num in all cases due to the constraint included in the fit.
The coefficient in square brackets giving the fourth order contribution can be read off from eq. (71). We see that the observed convergence behavior is still inconclusive, though the fourth order contributions employing the LECs b 1−11 from the meson-baryon scattering amplitudes are roughly in line with a suppression factor of ∼ 0.5. A similar pattern (at physical quark masses) has e.g. been found in [28], where some fourth order effects were estimated.
In contrast, the fits for b 1−11 = 0 result in expansions of m which seem meaningless in the sense of perturbation theory.
To further study the convergence issue, we set the average quark mass to 2B 0m = (300 MeV) 2 (this entails F → 104 MeV using eq. (19)). Then we would have The third order term is still comparatively large, cancelling most of the second order corrections, but the fourth order contributions already seem to be well under control for this low average quark mass. Unfortunately, there is no data available for such low average quark masses yet.
The parameters which are mostly determined from the behavior of the singlet sector can therefore not be fixed very accurately from the data set considered here. In particular, As a consequence of the results in Tab. XI, we revise the lower limit for the allowed range of the combinationb 0 (see eq. (77)), In Fig. 4, we show m as a function of M . One immediately sees that the uncertainty grows rapidly above the region where the curves are fixed by the fit to data (note that the fourth data point shown here is not included in the fits). Recall that the only difference between the fits leading to the full and the dashed curves in this figure is the O(p 5 )-effect F → F 0 .
Beyond M 500 MeV, this higher-order effect has a very large impact on m . Of course one could obtain good fits to lattice data in this region with both choices for the decay constant, but in our opinion, these fits would not be reliable in the sense of a stable determination of the fitting parameters (LECs). The finding that chiral extrapolations should not be trusted for meson masses much above 500 MeV is consistent with related studies [8,52]. Beyond this regime, the extrapolations depend very strongly on the details of the input and fine-tuning of input and fitting parameters.

Symmetry breaking
Let us now discuss the parameters which determine the SU(3) symmetry breaking visualized by the fan plots, as seen in Fig. 5. First, it is reassuring to observe that it does not matter whetherb D,F are used as fixed input in the full fits or not: Tables VII and IX   and Tables VIII and X are  large variation of the meson decay constant (from F ∼ 112 MeV to F 0 ∼ 80 MeV) does not invalidate this picture, as can be seen from the fourth and the last column of Tab. XI. It is also interesting to note that the combination 20 3b D −b F falls into the bound given in eq. (37) for b D,F for all fits from Tabs. VII-X, and is slightly above that range for the fits with set 1-3 from Tab. XI. From all these observations we can conclude that the determination of the LEC combinations parameterizing the linear part of the flavor symmetry breaking is very stable, thanks to the impressive new data leading to the fan plots. This is also illustrated by the error band and the three different fit curves shown in Fig. 5.
It remains to discuss the parameters determined from the nonlinear symmetry breaking effects. They appear at fourth chiral order in our formulae, which is the highest order we can compute exactly within the present one-loop framework. Therefore, we cannot expect a very high precision here. From our results, we extract the following allowed ranges for the remaining parameter combinations (in units of GeV −3 ): −0.10 ≤d 4 ≤ 0.20 , −0.75 ≤d 7 ≤ −0.05 .
We stress again that these estimates concern the numerical values of the renormalized LECs at µ = 1150 MeV, and should be evolved by eqs. (52) when using infrared regularization at a different scale µ.
Obviously, the purely linear fits lead to an underestimation of the uncertainty ind 7 : Here we had to shift the lower limit of our previous constraint, mostly due to the fit results shown in Tab. XI, compare eqs. (78) and (91). In Fig. 6, we show the constrained behavior of the function X N (ν), which is mostly secured by the combinationd 7 . Here, the expectation of convergence at low orders must fail because the higher order terms in eq. (72) have to compensate for the leading (O(p 3 )) effect. This might partly explain why the range ford 7 in eq. (91) turns out to be comparatively broad.
Including the experimental masses in the fit mainly (and strongly) affects the result for  how accurate the estimates from resonance saturation can probably be here, in particular because these LECs are not of "dynamical" nature (see also the discussion in sec. 5.2 of [21]). What is more, the masses and σ πN (0) = 45 MeV at the physical point have been input in the estimates of [3], and we have already seen that in general the convergence properties of the chiral expansions are far from satisfying there due to the large m s -corrections.
In [31], several parameterizations for the baryon masses were used to test the model dependence of the resulting fits to lattice data, and of the determination of the sigma terms.
While this model dependence is found to be an important source of systematic uncertainty, we note that the LECs b 0,D,F obtained from the O(p 3 )-BChPT approach in [31] agree very well with the values we find for the corresponding parametersb 0,D,F in our analysis.

Note added
Shortly after a first version of the present work appeared on the web, two other analyses of the lattice data discussed here became available [17,34]. Of those, [34] uses a framework similar to the one used here (which is essentially given in [21]). It therefore seems reasonable to attempt a comparison of the resulting fitting parameters, though there is still a minor difference in the renormalization schemes used. Let us first have a look at the most important parameter in the singlet sector, the baryon mass in the chiral limit. The range of possible values for this parameter we found is quite broad, see eq. (84). In Tab. XII we show their result in the first column, together with those of three other references in which this parameter is determined. We see that only the result of [30] is not in line with our broad estimate. Going on to the other LECs, and forming the combinations determined here with the results given in Table 6 (Fit I) of [34], we note that they are all consistent with the ranges we gave above, at least within the errors given there, with one exception, which is given byb D (we have checked that the shift from µ = 1 GeV used in the latter reference to the value µ = 1.15 GeV used here does not induce large effects on our ranges). However, one should note that the parameter b D in [34] jumps from a small value ∼ 0.05/GeV at O(p 2 ) and O(p 3 ) (consistent with our bounds onb D ) to a comparatively large value ∼ 0.22/GeV in the O(p 4 ) fits. We suppose that this parameter is particularly afflicted by convergence problems, given that a large part of the data set analysed in [34] are at the border, or even outside, of the region where the three-flavor expansions work in a reliable manner.

Sigma term
Rewriting our formula for the contact term contribution to the sigma term, eq.
This shows that we can write the sigma term resulting from our formulae as where the first term is written in terms of our fitting parameters. The unknown constants  [31,53]. We also mention that the above consistency bounds for d 3,5 are obeyed by the results for these two parameters given in [34].
However, it is to be noted that we do not see a clear sign of convergence for the sigma term at physical quark masses, so one should not overinterprete these last results. Basically we have to conclude that the sigma term could only be reliably determined from BChPT extrapolations when more data also for smaller average quark masses (so that e.g. M ∼ 300 MeV) become available.

VIII. CONCLUSIONS
In summary, we have analyzed lattice data for octet baryon masses from [35] employing manifestly covariant BChPT with three dynamical flavors. We were able to give bounds on the numerical values of certain combinations of low-energy constants (LECs), which are, in our opinion, more reliable and accurate than it was possible before data for fixed average quark mass became available in [35]: As we have seen, the simulation strategy used in that work offers some advantages for this purpose. Eqs. (75,76), (84,85,86) and (88,89,90,91) should be considered as our main results. We hope that these bounds will be useful in the near future, e.g. when studying the chiral behavior of other hadron observables with the same simulation strategy. Though we have fixed the running of the baryon masses on the two trajectories {m = const., δm } and {m, δm = 0}, there are still two undetermined parameters, chosen here as d 3 and d 5 , which would have to be fixed in order to make an O(p 4 ) prediction for the sigma term. However, even if those parameters were fixed accurately, the uncertainty in the determination of the sigma term would still be very large due to the very slow convergence near the physical point. For an investigation of sigma terms, a two-flavor version of ChPT is certainly superior; we refer to [54] for a recent determination. In closing, we stress again that additional data points for lower average quark masses would be very helpful in order to reach a truly controlled chiral extrapolation.