The nucleon mass and pion-nucleon sigma term from a chiral analysis of Nf = 2+1 lattice QCD world data

Fits of the p^4 covariant SU(2) baryon chiral perturbation theory to lattice QCD nucleon mass data from several collaborations for 2 and 2+1 flavors are presented. We consider contributions from explicit Delta(1232) degrees of freedom, finite volume and finite spacing corrections. We emphasize here our Nf=2+1 study. We obtain low-energy constants of natural size that are compatible with the rather linear pion-mass dependence of the nucleon mass observed in lattice QCD. We report a value for the pion-nucleon sigma term of 41(5)(4) MeV for the 2 flavor case and 52(3)(8) MeV for 2+1 flavors.


Introduction
Through lattice QCD simulations (lQCD) it is possible to study QCD at current quark masses which are not restricted to their physical values. Data points obtained for such a scenario represent a powerful pool of information to fix free low-energy constants (LECs) of the (baryon) chiral perturbation theory (BχPT); LECs that also enter calculations for experimental observables. Thus, adjusting the BχPT to lQCD data in the unphysical region allows for predictions at the physical point.
In our study we performed such a matching for the quark-mass (m u = m d = m) dependence of the nucleon mass M N (m). We fit lQCD data for the N f = 2 flavors, two light degenerated quarks, as well as for the N f = 2 + 1 ensembles, two light degenerated and one heavy quarks. Here, we concentrate on our N f = 2 + 1 results and refer to Ref. 1 for more details on the N f = 2 ones.
A measure of the contribution from the explicit chiral symmetry breaking to the nucleon mass is given by the so-called σ πN -term. It is the nucleon scalar form factor at zero momentum transfer squared which can be isolated from πN -scattering experiments. Alternatively, the σ πN can also be obtained by the Hellmann-Feynman theorem from M N (m): As a result, the function M N (m), which has mainly unphysical values, connects lQCD, BχPT and experiment.

Nucleon mass and covariant baryon chiral perturbation theory
The BχPT up to p 4 describes the nucleon mass by: with M 2 π ∼ m and f π as the pion mass and pion decay constant. We fit the three LECs c 1 , α and M 0 to lQCD data and obtain through Eq. (1) a σ πN value. Explicit ∆(1232) contributions can appear in the loop-contributions Σ (3+4) loops and we refer to Ref. 1 for all technical details. Generally, lQCD data is given in the dimensionless form (aM π , aM N ) where each collaboration sets its scale through a specific value for the lattice spacing a. In the N f = 2 + 1 case, we fit the data after converting to physical units. Since all three quantities are subject to statistical uncertainties, the a uncertainty translates into a correlated uncertainty for the normalization of data points belonging to the same set. We take this correlation into account by defining a correlation matrix V containing all these uncertainties, and minimize the function: with d i being the data and M N and Σ (4) the BχPT infinite volume nucleon mass and its finite volume corrections. The constants c i parametrize finite spacings effects for each lQCD-action separately In our SU (2) BχPT analysis we only include N f = 2 + 1 lQCD data for which the strange quark mass is kept constant approximately at its phenomenological value. For this case the strange quark mass corrections to the nucleon mass can be seen as being integrated out into the LECs we are fitting. Furthermore, to ensure controllable finite volume corrections and a reasonable chiral convergence of our results, we only account for data points with M π L > 3.8 and M π < 415 MeV, with L being the lattice box size. Following this, we include data from the BMW 2 , HSC 3 , LHPC 4 , MILC 5 , NPLQCD 6 , PACS 7 and RBCUK-QCD 8 collaborations.

Results and conclusion
The Fig. 1 shows several BχPT fit strategies to the above listed N f = 2 + 1 lQCD data. We obtain consistent descriptions of the pion-mass dependence of the nucleon mass where the slope-variations are below the resolution of the current data. However, by looking to the pion-mass dependence of the σ πN term, left graph of σ πN value at the physical point: where the second (systematical) uncertainty spans all the above central values.
The right graph of Fig. 2 compares our BχPT results from separate fits to lQCD data from N f = 2 and N f = 2 + 1 ensembles. Also here, differences are within the spread and uncertainties of the input data but result in a slightly smaller σ πN value for the N f = 2 case: Concerning the uncertainties, these two values are compatible, however, show a slight tension which we mostly trace back to the different data point distributions.
On the one hand, the current N f = 2 data does not constrain the low-M π region as much as the N f = 2 + 1 data. This region is dominated by the LEC c 1 which is the σ πN at leading order. Consequently, we obtain by ∼ 13 % different c 1 -values with correspondingly different contributions to the total σ πN . On the other hand, in the N f = 2 case one direct data point 9 of the σ πN at M π ≈ 290 MeV could be included in the fit. A positive outcome was that the above spread of our results with and without inclusion of ∆(1232) contributions is lifted. Since no such data point at low-M π is available for the N f = 2 + 1 case, a spread of 9 MeV remains. Note that in the HBχPT formalism 10 this spread is more than 40 MeV. To compare our above numbers also with phenomenology, we cite the latest value extracted from pure π-N scattering data: σ πN = 59(7) MeV 11 . In conclusion, we performed fits of the p 4 SU (2) covariant BχPT with and without explicit ∆(1232) contributions to lQCD nucleon mass data. Even though the present data set is extensive, we spotted systematic effects coming mainly from the distribution of the data points.
These systematic effects would further be reduced by new data for: a) more N f = 2 data points in the low-M π region, b) already one direct calculation of the σ πN at M π < 300 MeV for the N f = 2 + 1 case. Fig. 2. Left: Pion-mass dependence of the σ πN term from N f = 2 + 1 fits. The color code is the same as in Fig. 1. The red-diamond is our predicted σ πN value at the physical point. Right: Comparison of BχPT p 4 fits with ∆(1232) to N f = 2 (red-triangles) and N f = 2+1 (blue-squares) lQCD data. The blue-circle is the physical nucleon mass.