Baryon masses and \sigma-terms in SU(3) BChPT x 1/Nc

Baryon masses and nucleon \sigma-terms are studied with the effective theory that combines the chiral and 1/Nc expansions for three flavors. In particular the connection between the deviation of the Gell-Mann-Okubo relation and the \sigma-term corresponding to the scalar density associated with the hypercharge is emphasized. The latter is at lowest order related to a mass combination whose low value has given rise to a \sigma- term puzzle. It is shown that while the nucleon \sigma-terms have a well behaved low energy expansion, that mass combination is affected by large higher order corrections non-analytic in quark masses. Adding to the analysis lattice QCD baryon masses, it is found that {\sigma}{\pi}N=69(10) MeV and {\sigma}s has natural magnitude within its relative large uncertainty.

2) There is a long lasting "puzzle" associated with a combination of baryon masses (in SU(3) ) in the iso-spin symmetric limit, to obtain the pion-Nucleon sigma term, assuming the contribution by strange quark mass to the nucleon mass is negligible (OZI).  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results. xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite the lack of the decuplet contributions leads to values of the terms which are not very di↵erent  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite the lack of the decuplet contributions leads to values of the terms which are not very di↵erent but somewhat larger than the ones obtained here (ˆ ⇠ 83 MeV, ⇡N ⇠ 76 MeV). So, where is  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite the lack of the decuplet contributions leads to values of the terms which are not very di↵erent  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite the lack of the decuplet contributions leads to values of the terms which are not very di↵erent but somewhat larger than the ones obtained here (ˆ ⇠ 83 MeV, ⇡N ⇠ 76 MeV). So, where is the di↵erence?. The answer is simple: in ordinary HBChPT the corrections to the axial currents  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite the lack of the decuplet contributions leads to values of the terms which are not very di↵erent Bali et al. (2016) Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
[7-11]. Note however that a more reliable value would require some more accurate and extensive LQCD results. Fig. (1) depicts the result for ⇡N from Fit 2 and its comparison with other results.
xi) The analysis also gives an estimate of the isospin-breaking terms 3 and u+d (p n). In addition one can extract the separate contributions q (N), q = u, d, N = p, n. The results are the following: u (p) = 26.23 MeV, d (p) = 42.42 MeV, u (n) = 23.82 MeV, d (n) = 46.48 MeV, which checks with ⇡N =m( u /m u + d /m d ). The relation u (p) = d (n) in the isospin symmetry limit is of course satisfied, but the naive quark model relation in the isospin limit u (p) = 2 d (p) is significantly violated due to contributions by the SU(2) singlet component of the quark masses. xii) Obviously, the discussion can be extended to the rest of the terms for the di↵erent baryons and their various relations [29]. xiii) One can compare with an analysis in ordinary HBChPT without the decuplet. In that case GMO requiresg A /F ⇡ to be significantly larger (corresponding to g N A = 1.48 at LO), which despite the lack of the decuplet contributions leads to values of the terms which are not very di↵erent but somewhat larger than the ones obtained here (ˆ ⇠ 83 MeV, ⇡N ⇠ 76 MeV). So, where is 3) The connection between the pion-Nucleon sigma term and size of the correction to the Gell-Mann-Okubo relation Can one explain these from the ChPT point of view ? Motivation 3 baryon matrix elements of scalar quark densities, for w determination. The definition of terms is through t for three flavors, through the physical baryon masses giv those associated with the SU(3) octet quark mass combin wherem is the average of the u and d quark masses. component m 0 = 1 3 (2m + m s ) require knowledge of bar which is made possible through lattice QCD (LQCD) c nucleon term ⇡N ⌘m 2m N hN |ūu +dd | Ni is accessib scattering via a low energy theorem [1; 2; 3]. Such a d through the availability of increasingly accurate data an of dispersion theory and chiral perturbation theory [4; 5 for ⇡N range from ⇠ 45 MeV [4; 5; 6] to & 58 MeV [ between the results of the di↵erent dispersive analyses the S-wave ⇡N scattering lengths a 1/2,3/2 used in the s Non relativistic version of the BChPT or HBChPT is based on the expansion in terms of the "baryon mass" Solution :

Power counting scheme in HBChPT
In the HBChPT, the derivative expansion for both mesons and bary expansion in powers of (k/⇤ ), where k is a momentum of the order of t and ⇤ is the chiral symmetry breaking scale. Therefore, the higher d in the e↵ective theory are suppressed by powers of (k/⇤ ). The bary can be revised into 1 a version where the mass dependence can be resided in the vertices w dered according to their power in 1/m B . The chiral dimension D for a diagram is given by [46], where, N L is the number of loops, I M is the number of internal meso In the heavy baryon chiral pertuebation theory (HBChPT), the sidered as heavy static fermions [40,59]. The velocity of the bar changed or e↵ectively conserved when it exchanges a small moment Therefore the baryon four-momentum can be decomposed into a m B v and a small residual momentum component k µ ,

Heavy Baryon Approach
In the heavy baryon chiral pertuebation theory (HBChPT), the baryons are considered as heavy static fermions [40,59]. The velocity of the baryon is nearly unchanged or e↵ectively conserved when it exchanges a small momentum with a meson.
Therefore the baryon four-momentum can be decomposed into a large component m B v and a small residual momentum component k µ , The issue of experiencing a slower rate of convergence compare to the Goldstone Boson Sector Consistency of pion-nucleon scattering implies an exact spin-flavor symmetry exists for baryons in the limit N c ! 1 Dominant diagrams for baryon-meson scattering amplitude Large N c consistency condition : [X ia 0 , X ib 0 ] = 0 Large N c QCD has a contracted spin-flavor symmetry SU c (2N f ) In baryon sector. Where N f is number of flavors.

/
inant diagrams for baryon-meson scattering amplitude e N c consistency condition : [X ia 0 , X ib 0 ] = 0 e N c QCD has a contracted spin-flavor symmetry SU c (2N f ) In bary r. Where N f is number of flavors. exists for baryons in the limit N c ! 1 Dominant diagrams for baryon-meson scattering amplitude Large N c consistency condition : [X ia 0 , X ib 0 ] = 0 Large N c QCD has a contracted spin-flavor symmetry SU c (2N f ) In bar sector. Where N f is number of flavors.
Consistency of pion-nucleon scattering implies an exact spin-flavor symmetry exists for baryons in the limit N c ! 1 Dominant diagrams for baryon-meson scattering amplitude Large N c consistency condition : [X ia 0 , X ib 0 ] = 0 Large N c QCD has a contracted spin-flavor symmetry SU c (2N f ) In baryon sector. Where N f is number of flavors.  Consistency of pion-nucleon scattering implies an exact spin-flavor symmetry exists for baryons in the limit N c ! 1 Dominant diagrams for baryon-meson scattering amplitude Large N c consistency condition : [X ia 0 , X ib 0 ] = 0 Large N c QCD has a contracted spin-flavor symmetry SU c (2N f ) In baryon sector. Where N f is number of flavors.

Baryon Spin-Flavor Symmetry
Consistency of pion-nucleon scattering implies an exact spin-flavor symmetry exists for baryons in the limit N c ! 1 Dominant diagrams for baryon-meson scattering amplitude Large N c consistency condition : [X ia 0 , X ib 0 ] = 0 Large N c QCD has a contracted spin-flavor symmetry SU c (2N f ) In baryon sector. Where N f is number of flavors.

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Flavor Symmetry y of pion-nucleon scattering implies an exact spin-flavor symmetry baryons in the limit N c ! 1 diagrams for baryon-meson scattering amplitude consistency condition : [X ia 0 , X ib 0 ] = 0 QCD has a contracted spin-flavor symmetry SU c (2N f ) In baryon here N f is number of flavors.     Leading order Lagrangian order to render most of the LECs dimensionless. In the calculations ⇤ = m ⇢ will be osen.
The lowest order Lagrangian is [31]: he kinetic term is O(p N 0 c ), and the terms involving GBs (when the vector and axial vector urces are turned o↵) start with the Weinberg-Tomozawa term which is O(p/N c ). The cond term gives in particular the axial vector current and the GB-baryon interaction.g A the axial coupling in the chiral and large N c limits (it has to be rescaled by a factor 5/6 coincide with the usual axial coupling as defined for the nucleon, i.e., g N A = g A = 5 6g A ). ecause the matrix elements of where m = N c m 0 is the spin-flavor singlet baryon mass in chiral limit, and the magnetic moment terms consist of the one coming from the Dirac term 1 m (B 0 + +B a + T a ) ·S and the anomalous terms involving  0 (isoscalar) and  1,2 (isovector). Note that the magnetic transition between baryons of di↵erent spin are meadited only by the term  2 .Do we need w 1,2 terms?
this, the Oðξ 2 Þ Lagrangian is given by 3 : dditional terms not explicitly displayed are not needed in the present work. Note that there are also Oðξ 2 Þ ter ng from the 1=N c suppressed terms in the LECs of the lower-order Lagrangian. Similar comments apply to rder Lagrangians. Such terms require knowledge of the physics at N c > 3 to be determined, which can in princi ined using LQCD results at varying N c [58,59]. arly, the Oðξ 3 Þ Lagrangian needed here is given by Chiral Symmetry + Spin-flavor Symmetry Intermediate Octet and Decuplet baryon contributions are included he leading 1-loop correction to the baryon self energy, diagram in Fig. 1, can be calculated rough the matrix element ⌅B | ⇥ 1 loop | B⇧, with: here n indicates the possible intermediate baryon spin-isospin states in the loop, P n are e corresponding spin-flavor projection operators, ⇥m n = ⇥m(S n ), and the loop integral is lculated in dimensional regularization with the result, here Q = ⇥m n p 0 , ⌅ = 1 + log 4⌥, and µ is the renormalization scale which will be ken later to be of the order of m ⇤ . For the specific evaluation of ⇥ 1 loop for a given baryon The leading 1-loop correction to the baryon self en through the matrix element ⌅B | ⇥ 1 loop | B⇧, wi where Q = ⇥m n p 0 , ⌅ = 1 + log 4⌥, and µ taken later to be of the order of m ⇤ . For the specifi where ⇥m 1 loop+CT The leading 1-loop correction to the baryon self energy, diagram in Fig. 1, can be calculated through the matrix element ⌅B | ⇥ 1 loop | B⇧, with: where n indicates the possible intermediate baryon spin-isospin states in the loop, P n are re well defined for N c ! 3 as the numerators on The chiral Lagrangian to O(⇠ 3 ), including electromagnetic corrections to the baryon mass given by [29]: where terms not directly relevant to the baryon masses have been omitted. In addition to the known chiral building blocks, B represents the baryon spin-flavor multiplet field,Ŝ 2 is the sq of the baryon spin operator, G ia are the spin-flavor generators of SU(6), andQ is the ele charge operator. No baryon-spin dependent electromagnetic e↵ects are included. M 0 = O(N the spin-flavor singlet piece of the baryon mass and provides the large mass expansion param for HBChPT. The term proportional to C HF gives the leading order mass splitting between the 1/2 and 3/2 baryons.g A is identified with 6 5 g N A at the LO. The rest of the terms describe the q mass e↵ects. The combinationˆ + = N c 0 + +˜ + , where 0 + = 1 3 Tr + and˜ + is the traceless p of + , assures that the nucleon mass dependency on m s is at most O(N 0 c ) (OZI). ⇤ is an arbit 4 of he piece of χ + , assures that the nucleon mass dependency on m s is is an arbitrary scale, which is conveniently chosen to be m ρ . The baryon mass formula then reads (neglecting isospin breaking for now) [29]: where δm loop B can be obtained with some work using the results in [29], where the details on the mass renormalization and results for general N c can be found.
Setting c 3 = 0, 2 the terms analytic in quark masses in Eqn. (2) lead to the exact GMO and Equal Spacing mass relations, which where terms not directly relevant to the baryon masses known chiral building blocks, B represents the baryon of the baryon spin operator, G ia are the spin-flavor g charge operator. No baryon-spin dependent electroma the spin-flavor singlet piece of the baryon mass and pr for HBChPT. The term proportional to C HF gives the le 1/2 and 3/2 baryons.g A is identified with 6 5 g N A at the L mass e↵ects. The combinationˆ + = N c 0 + +˜ + , wher of + , assures that the nucleon mass dependency on m 4 l of these relations are poorly satisfied. The ons are calculable and given by the nonanalytic utions to one-loop. In the physical case N c ¼ 3, eviations are numerically large for the first, third, urth relations above. This in particular affects the n strangeness σ term, and thus indicates that its ion from arguments based on tree level relations is to important corrections [63]. In terms of the octet nents of the quark masses, in addition to GMO relations one finds: it can be readily checked that they are well for N c → 3 as the numerators on the RHS are tional to ðN c − 3Þ. These relations are violated at To complete this section, fits to the octet and decuplet baryon masses including results from LQCD are presented. This in particular allows for exploring the range of validity of the calculation as the quark masses are increased. The mass formula for the fit is 4 : where, in the isospin symmetry limit, LECs of terms that depend on quark masses can be more completely determined by fits that include the LQCD results for different quark masses, e.g., c 2 and the various h 0 s. For this reason, such combined fits are presented here, in Table II and in Fig. 4. Also, some LECs are redundant at N c ¼ 3, and are thus set to vanish for the fit. The constant c 3 is also set to vanish as it turns out to be of marginal importance for the fit. A test of mass relations is shown in Table III.
The study of the fits show that at fixed M K ∼ 500 MeV, the physical plus LQCD results up to M π ∼ 300 MeV can 4 A useful formula for the term proportional to h 4 is [64]: Several of these relations are poorly satisfied. The deviations are calculable and given by the nonanalytic contributions to one-loop. In the physical case N c ¼ 3, those deviations are numerically large for the first, third, and fourth relations above. This in particular affects the nucleon strangeness σ term, and thus indicates that its estimation from arguments based on tree level relations is subject to important corrections [63]. In terms of the octet components of the quark masses, in addition to GMO and ES relations one finds: where it can be readily checked that they are well defined for N c → 3 as the numerators on the RHS are proportional to ðN c − 3Þ. These relations are violated at large N c as Oðp 3 N 0 c Þ. For both relations in the limit 2 To complete this section baryon masses including res This in particular allows for of the calculation as the qu mass formula for the fit is 4 where, in the isospin sy LECs. LECs of terms that d more completely determined results for different quark m h 0 s. For this reason, such co in Table II and in Fig. 4. Als N c ¼ 3, and are thus set to c 3 is also set to vanish as importance for the fit. A tes Table III.
The study of the fits show the physical plus LQCD res Several of these relations are poorly satisfied. The deviations are calculable and given by the nonanalytic contributions to one-loop. In the physical case N c ¼ 3, those deviations are numerically large for the first, third, and fourth relations above. This in particular affects the nucleon strangeness σ term, and thus indicates that its estimation from arguments based on tree level relations is subject to important corrections [63]. In terms of the octet components of the quark masses, in addition to GMO and ES relations one finds: where it can be readily checked that they are well defined for N c → 3 as the numerators on the RHS are proportional to ðN c − 3Þ. These relations are violated at large N c as Oðp 3 N 0 c Þ. For both relations in the limit To complete this section, fits to the octet and dec baryon masses including results from LQCD are prese This in particular allows for exploring the range of va of the calculation as the quark masses are increased mass formula for the fit is 4 : where, in the isospin symmetry limit, LECs. LECs of terms that depend on quark masses c more completely determined by fits that include the L results for different quark masses, e.g., c 2 and the va h 0 s. For this reason, such combined fits are presented in Table II and in Fig. 4. Also, some LECs are redund N c ¼ 3, and are thus set to vanish for the fit. The con c 3 is also set to vanish as it turns out to be of mar importance for the fit. A test of mass relations is show Table III.
The study of the fits show that at fixed M K ∼ 500 M the physical plus LQCD results up to M π ∼ 300 MeV 4 A useful formula for the term proportional to h 4 is the effective theory; as indicated earlier, this however depends on the value the LO g ∘ A , which to be independently determined requires the analysis of other observables, namely the axial currents. Along the same lines Δ ES can be analyzed, although in this case the experimental uncertainty is rather large.
hyperfine interaction that splits 8 and 10 baryons, and such deviation starts with the term proportional to h 2 which is Oðp 2 =N c Þ. In addition, the one-loop contributions to it are free of UV divergencies and the nonanalytic terms when expanded in the large N c limit give contributions Oð1=N 2 c Þ. To one-loop: where the last line corresponds to strictly expanding in the large N c limit. For the physical M π , M K , and C HF , the 1=N c expansion of Δ GR is, however, only reasonable for N c > 8: clearly the nonanalytic dependency in 1=N c is important, showing the need for the combined ξ expansion in the physical case, similarly to what occurs for Δ GMO . Still, the understanding of the smallness of the deviation is connected with the 1=N c expansion. Finally, it is important to emphasize, as indicated earlier, that all the relations are not explicitly dependent on N c , and their deviations are suppressed by powers of 1=N c at large N c .
The σ-terms are obtained following the Hellman-Feynman theorem, σ Bm q ≡ m q ∂m B =∂m q , where m q can be taken to bem; m s , or the SUð3Þ singlet and octet components of the quark masses, namely m 0 ¼ ð2m þ m s Þ=3 and m 8 ¼ 2= ffiffi ffi 3 p ðm − m s Þ. Naturally they will satisfy the same relations discussed above for the masses. In particular, σ terms associated with the same m q are related via those relations and their deviations are calculable as described before for the masses. In addition to the GMO and ES relations, the following tree level Oðξ 3 Þ relations hold,  whether the decuplet baryons ought to be included or not in the effective theory; as indicated earlier, this however depends on the value the LO g ∘ A , which to be independently determined requires the analysis of other observables, namely the axial currents. Along the same lines Δ ES can be analyzed, although in this case the experimental uncertainty is rather large. relation (26) is due to SUð3Þ breakin hyperfine interaction that splits 8 and such deviation starts with the term propor is Oðp 2 =N c Þ. In addition, the one-loop c are free of UV divergencies and the nonan expanded in the large N c limit give contri To one-loop: where the last line corresponds to strictly expanding in the large N c limit. For the physical M π , M K , and C HF , the 1=N c expansion of Δ GR is, however, only reasonable for N c > 8: clearly the nonanalytic dependency in 1=N c is important, showing the need for the combined ξ expansion in the physical case, similarly to what occurs for Δ GMO . Still, the understanding of the smallness of the deviation is connected with the 1=N c expansion. Finally, it is important to emphasize, as indicated earlier, that all the relations are not explicitly dependent on N c , and their deviations are suppressed by powers of 1=N c at large N c .
The σ-terms are obtained followin Feynman theorem, σ Bm q ≡ m q ∂m B =∂m q be taken to bem; m s , or the SUð3Þ singl ponents of the quark masses, namely m and m 8 ¼ 2= ffiffi ffi 3 p ðm − m s Þ. Naturally the same relations discussed above for the ticular, σ terms associated with the sam via those relations and their deviations described before for the masses. In additio ES relations, the following tree level Oðξ 054010-9 e physical M K and M π , the shown expansion is within of the exact result, and the expansion gives a good ximation for N c > 5. Note the large cancellations that ar within the first line and within the second line of the tion, and also the tendency to cancel between the first econd lines. In the physical case and not expanding in , it is found that the numerical dependency of Δ GMO HF is not very significant. One also observes that only of Δ GMO is contributed by the octet baryons in the and thus the decuplet contribution is very important.
is therefore an important observable for assessing her the decuplet baryons ought to be included or not in ffective theory; as indicated earlier, this however nds on the value the LO g ∘ A , which to be independently Disregarding the term proportional to h 2 in L ð3Þ B Eq. (13), which gives SUð3Þ breaking in the hyperfine splittings, one additional relation follows, first found by Gürsey and Radicati [62], namely: which relates SUð3Þ breaking in the octet and decuplet, and which is valid for arbitrary N c . The deviation from that relation (26)  At tree level the GMO relation is exact for any Nc up to Several of these relations are poorly satisfied. The deviations are calculable and given by the nonanalytic contributions to one-loop. In the physical case N c ¼ 3, those deviations are numerically large for the first, third, and fourth relations above. This in particular affects the nucleon strangeness σ term, and thus indicates that its estimation from arguments based on tree level relations is subject to important corrections [63]. In terms of the octet components of the quark masses, in addition to GMO and ES relations one finds: To complete this section, fits to the octet and decuplet baryon masses including results from LQCD are presented. This in particular allows for exploring the range of validity of the calculation as the quark masses are increased. The mass formula for the fit is 4 : where, in the isospin symmetry limit, be fitted with natural size LECs. The LEC h 2 which enters in Δ GR is best determined by fixing it using Δ GR in the physical case, and then the rest of the LECs are determined by the overall fit. In this way, the deviations of the mass relations are one of the predictions of the effective theory, and can therefore be used as a test of LQCD calculations. At present the errors in the LQCD calculations are relatively large, and thus such a test is not yet very significant.

IV. VECTOR CURRENTS: CHARGES
In this section, the one-loop corrections to the vector current charges are calculated. The analysis is similar to that carried out in [65], except that in that reference higher-order terms in 1=N c in the GB-baryon vertices were included. In the ξ expansion and the order considered here, such higher-order terms are not required. At lowest order the charges are simply given by the generators T a , the one-loop corrections are UV finite,

Introduction
Baryon mass dependencies on quark masses, quantified the fundamental observables in baryon chiral dynamics. In pa baryon matrix elements of scalar quark densities, for which determination. The definition of terms is through the Fe for three flavors, through the physical baryon masses gives ac those associated with the SU(3) octet quark mass combination wherem is the average of the u and d quark masses. The component m 0 = 1 3 (2m + m s ) require knowledge of baryon m which is made possible through lattice QCD (LQCD) calcul nucleon term ⇡N ⌘m 2m N hN |ūu +dd | Ni is accessible thr scattering via a low energy theorem [1; 2; 3]. Such a determ through the availability of increasingly accurate data and the of dispersion theory and chiral perturbation theory [4; 5; 6; 7 for ⇡N range from ⇠ 45 MeV [4; 5; 6] to & 58 MeV [7; 8; between the results of the di↵erent dispersive analyses resid perturbation theory [4; 5; 6; 7; 8; 9; 10; 11]. The value [4; 5; 6] to & 58 MeV [7; 8; 9; 10; 11; 12], where the rent dispersive analyses resides mostly in the di↵eren ths a 1/2,3/2 used in the subtractions, cf. [12]. In addi ed: i (B) = m i @ @m i m B , where m i indicates a quark mass (i = u, d, s) baryon. When B is not explicitly indicated it is assumed to be a nu the fundamental observables in baryon chiral dynamics. In particular, they give in baryon matrix elements of scalar quark densities, for which there is no alternati determination. The definition of terms is through the Feynman-Hellmann th for three flavors, through the physical baryon masses gives access to only two suc those associated with the SU(3) octet quark mass combinations m 3 = m u m d and m wherem is the average of the u and d quark masses. The terms associated component m 0 = 1 3 (2m + m s ) require knowledge of baryon masses for unphysica which is made possible through lattice QCD (LQCD) calculations. On the other nucleon term ⇡N ⌘m 2m N hN |ūu +dd | Ni is accessible through its connection scattering via a low energy theorem [1; 2; 3]. Such a determination of ⇡N had through the availability of increasingly accurate data and the development of com of dispersion theory and chiral perturbation theory [4; 5; 6; 7; 8; 9; 10; 11]. The for ⇡N range from ⇠ 45 MeV [4; 5; 6] to & 58 MeV [7; 8; 9; 10; 11; 12], whe between the results of the di↵erent dispersive analyses resides mostly in the di the S-wave ⇡N scattering lengths a 1/2,3/2 used in the subtractions, cf. [12]. In Matrix elements of octet operators can provide definitive information for the resolution of this puzzle. Defining, one finds a simple relation between ⇡N ,ˆ and s , xpansion, that mass combination is a↵ected by large higher order corrections nonark masses. Adding to the analysis lattice QCD baryon masses, it is found that MeV and s has natural magnitude within its relative large uncertainty.
igma terms, nucleon mass, baryon masses, Gell-Mann-Okubo mass formula ion ass dependencies on quark masses, quantified by the di↵erent -terms, are among tal observables in baryon chiral dynamics. In particular, they give information on the elements of scalar quark densities, for which there is no alternative way for their . The definition of terms is through the Feynman-Hellmann theorem 1 , which, rs, through the physical baryon masses gives access to only two such terms, namely In this paper we study the higher order cor Gell-Mann-Okubo mass formula andˆ with th mulation of Chiral EFT. We generalize the re for an arbitrary number of colors. Then, we both, O((m s m) 3/2 ) corrections are of natura the case of GMO, necessary to recover the expe With this correctionsˆ comes out larger than th Ref. [15,16] 14; 15; 16; 17]. The relatively large range of value c of study, and in part motivates the present work. A scalar quark operator matrix elements, quantities th er detection [18; 19; 20], and of lepton flavor violatio h nuclei [21]. t has been emphasized for a long time [22] is the r y limit and the nucleon'sˆ ⌘ p 3m m 8 8 , namely ⇡ lue of s should give ⇡N ⇠ˆ . The origin of the p ⌅ ) (or other combinations related via the Gell-Man rder in quark masses, which givesˆ ⇠ 25 MeV. If t in scattering with nuclei [21]. A puzzle that has been emphasized for a long time [22] is the relation b isospin symmetry limit and the nucleon'sˆ ⌘ p 3m m 8 8 , namely ⇡N =ˆ + a natural size value of s should give ⇡N ⇠ˆ . The origin of the puzzle is 1 3 (2m N m ⌃ m ⌅ ) (or other combinations related via the Gell-Mann-Okubo valid at linear order in quark masses, which givesˆ ⇠ 25 MeV. If that relati approximation to the value ofˆ , the implication is that, contrary to expectati a very large contribution to the nucleon mass even for the smaller values of particularly striking for the larger values that have been obtained for ⇡N , w s ⇠ 0.5 GeV!. Indeed, this is clearly impossible if one considers that s = O( This work analyzes the terms through the octet and decuplet baryon mass chiral and 1/N c expansions BChPT ⇥ 1/N c . The emphasis is in that the e↵ecti at NNLO (one chiral loop) a natural description of baryon masses, including LQ with the axial couplings which have been obtained in LQCD at di↵erent quark lar, the resolution of the term puzzle is explained by the fact that 8 ⌘ 8 receives large non-analytic in quark mass corrections dominated by m s . It will 8 itself, and thusˆ , has a natural low energy expansion and therefore the o resides in the large non-analytic correction to the mass combination 1 3 (2m N m big part of that large correction stems from the contribution of decuplet baryon was found in Refs. [13; 23]. By analyzing LQCD baryon masses [24], it is foun ⇡N ⇠ˆ , with the results ⇡N = 69(8)(6) MeV, where the errors are respectively theoretical (expected NNNLO corrections) ones, and | s |. 50 MeV. The co the deviation from the GMO relation, GMO ⌘ 3m ⇤ + m ⌃ 2(m N + m ⌅ ), and at NNLO and given solely in terms of non-analytic loop contributions, is of par in the present work.

Introduction
Baryon mass dependencies on quark masses, quantified by the di↵erent -terms, are amo the fundamental observables in baryon chiral dynamics. In particular, they give information on baryon matrix elements of scalar quark densities, for which there is no alternative way for th determination. The definition of terms is through the Feynman-Hellmann theorem 1 , wh for three flavors, through the physical baryon masses gives access to only two such terms, nam those associated with the SU (3) 12], where the di↵ere between the results of the di↵erent dispersive analyses resides mostly in the di↵erent values the S-wave ⇡N scattering lengths a 1/2,3/2 used in the subtractions, cf. [12]. In addition to 1 The following notation will be used: i (B) = m i @ @mi m B , where m i indicates a quark mass (i = u, d, s) or comb tion thereof (0, 3, 8), and B is a given baryon. When B is not explicitly indicated it is assumed to be a nucleon. A long lasting puzzle ! 1 3 Tr + and˜ + is the traceless piece of + , assures that the nucleon mass dependency on m most O(N 0 c ) (OZI). ⇤ is an arbitrary scale, which is conveniently chosen to be m ⇢ . The baryo s formula then reads (neglecting isospin breaking for now) [29]: can be obtained with some work using the results in [29], where the details on th s renormalization and results for general N c can be found. Setting c 3 = 0 2 , the terms analytic in quark masses in Eqn. (2) lead to the exact GMO an al Spacing mass relations, which are unchanged at generic N c . On the other hand at gener he mass relation for 8 at tree level reads: (3 dominant contributions to GMO and 8 are calculable non-analytic contributions. GMO 4 ) and in large N c limit it is O (1/N c ). On the other hand, 8 is O(⇠) and it has a prefactor N phasized for a long time [22] is the relation between ⇡N in the nucleon'sˆ ⌘ p 3m m 8 8 , namely ⇡N =ˆ + 2m m s s , which for ld give ⇡N ⇠ˆ . The origin of the puzzle is the relation: 8 = ombinations related via the Gell-Mann-Okubo (GMO) relation) asses, which givesˆ ⇠ 25 MeV. If that relation is a reasonable ˆ , the implication is that, contrary to expectations, m s must give nucleon mass even for the smaller values of ⇡N . The puzzle is ger values that have been obtained for ⇡N , which would imply learly impossible if one considers that s = O( 1 N c ) ⇡N . rms through the octet and decuplet baryon masses in the combined hPT ⇥ 1/N c . The emphasis is in that the e↵ective theory can give tural description of baryon masses, including LQCD results, along ave been obtained in LQCD at di↵erent quark masses. In particupuzzle is explained by the fact that 8 ⌘ 8 1 3 (2m N m ⌃ m ⌅ ) uark mass corrections dominated by m s . It will also be shown that ural low energy expansion and therefore the origin of the puzzle correction to the mass combination 1 3 (2m N m ⌃ m ⌅ ). In fact, a stems from the contribution of decuplet baryons in the loop, as it analyzing LQCD baryon masses [24], it is found that as expected lue ofˆ , the implication is that, contrary to expectations, m s must give to the nucleon mass even for the smaller values of ⇡N . The puzzle is the larger values that have been obtained for ⇡N , which would imply this is clearly impossible if one considers that s = O( 1 N c ) ⇡N . he terms through the octet and decuplet baryon masses in the combined ns BChPT ⇥ 1/N c . The emphasis is in that the e↵ective theory can give p) a natural description of baryon masses, including LQCD results, along which have been obtained in LQCD at di↵erent quark masses. In particuterm puzzle is explained by the fact that 8 ⌘ 8 1 3 (2m N m ⌃ m ⌅ ) tic in quark mass corrections dominated by m s . It will also be shown that s a natural low energy expansion and therefore the origin of the puzzle nalytic correction to the mass combination 1 3 (2m N m ⌃ m ⌅ ). In fact, a rection stems from the contribution of decuplet baryons in the loop, as it 23]. By analyzing LQCD baryon masses [24], it is found that as expected ⇡N = 69(8)(6) MeV, where the errors are respectively the statistical and NLO corrections) ones, and | s |. 50 MeV. The connection between MO relation, GMO ⌘ 3m ⇤ + m ⌃ 2(m N + m ⌅ ), and 8 , both calculable ly in terms of non-analytic loop contributions, is of particular importance esults from the analyses of ⇡N scattering, LQCD calculations extrapolated to or at oint obtain di↵erent results, with values consistent with the recent ⇡N results [13] ⇡N ⇡ 40 MeV [14; 15; 16; 17]. The relatively large range of values obtained for s an active topic of study, and in part motivates the present work. An additional m he relevance of scalar quark operator matrix elements, quantities that are relevant irect dark matter detection [18; 19; 20], and of lepton flavor violation through µ n scattering with nuclei [21].
A puzzle that has been emphasized for a long time [22] is the relation betwee sospin symmetry limit and the nucleon'sˆ ⌘ p 3m m 8 8 , namely ⇡N =ˆ + 2m m s natural size value of s should give ⇡N ⇠ˆ . The origin of the puzzle is the re (2m N m ⌃ m ⌅ ) (or other combinations related via the Gell-Mann-Okubo (GM alid at linear order in quark masses, which givesˆ ⇠ 25 MeV. If that relation is pproximation to the value ofˆ , the implication is that, contrary to expectations, m very large contribution to the nucleon mass even for the smaller values of ⇡N . T at NNLO (one chiral loop) a na with the axial couplings which h lar, the resolution of the term receives large non-analytic in q 8 itself, and thusˆ , has a nat resides in the large non-analytic big part of that large correction was found in Refs. [13; 23]. By ⇡N ⇠ˆ , with the results ⇡N = theoretical (expected NNNLO the deviation from the GMO rel at NNLO and given solely in te in the present work. A puzzle that has been emphasized for a long time [22] is the rela isospin symmetry limit and the nucleon'sˆ ⌘ p 3m m 8 8 , namely ⇡N a natural size value of s should give ⇡N ⇠ˆ . The origin of the puz 1 3 (2m N m ⌃ m ⌅ ) (or other combinations related via the Gell-Mannvalid at linear order in quark masses, which givesˆ ⇠ 25 MeV. If tha approximation to the value ofˆ , the implication is that, contrary to ex a very large contribution to the nucleon mass even for the smaller valu particularly striking for the larger values that have been obtained for s ⇠ 0.5 GeV!. Indeed, this is clearly impossible if one considers that This work analyzes the terms through the octet and decuplet baryon chiral and 1/N c expansions BChPT ⇥ 1/N c . The emphasis is in that the at NNLO (one chiral loop) a natural description of baryon masses, includ with the axial couplings which have been obtained in LQCD at di↵erent lar, the resolution of the term puzzle is explained by the fact that 8 ⌘ receives large non-analytic in quark mass corrections dominated by m s . 8 itself, and thusˆ , has a natural low energy expansion and therefore resides in the large non-analytic correction to the mass combination 1 3 (2m big part of that large correction stems from the contribution of decuplet [7; 8; 9; 10; 11]. Note however that extensive LQCD results. Fig. (1) d  other results. xi) The analysis also gives an estim addition one can extract the separat following: u (p) = 26.23 MeV, d which checks with ⇡N =m( u /m u limit is of course satisfied, but the n is significantly violated due to contr xii) Obviously, the discussion can b and their various relations [29]. xiii) One can compare with an ana GMO requiresg A /F ⇡ to be significa the lack of the decuplet contributio but somewhat larger than the ones the di↵erence?. The answer is simp couplings have large N c power viol ofg A /F ⇡ required by GMO lead to a di↵erent quark masses [32], in parti There is a (hidden) large correction ~ 44 MeV from non-analytic contributions from baryon self-energies as an active topic of study, and in part motivates the prese the relevance of scalar quark operator matrix elements, qu direct dark matter detection [18; 19; 20], and of lepton flav in scattering with nuclei [21].
A puzzle that has been emphasized for a long time [2 isospin symmetry limit and the nucleon'sˆ ⌘ p 3m m 8 8 , a natural size value of s should give ⇡N ⇠ˆ . The orig 1 3 (2m N m ⌃ m ⌅ ) (or other combinations related via th valid at linear order in quark masses, which givesˆ ⇠ 25 approximation to the value ofˆ , the implication is that, c a very large contribution to the nucleon mass even for the particularly striking for the larger values that have been o s ⇠ 0.5 GeV!. Indeed, this is clearly impossible if one con This work analyzes the terms through the octet and de chiral and 1/N c expansions BChPT ⇥ 1/N c . The emphasis at NNLO (one chiral loop) a natural description of baryon m with the axial couplings which have been obtained in LQCD keeps it ivation is tudies of nversion ⇡N in the hich for n: 8 = relation) asonable ust give puzzle is ld imply ombined can give lts, along part motivates the present work. An additional motivation is tor matrix elements, quantities that are relevant in studies of ; 20], and of lepton flavor violation through µ e conversion ized for a long time [22] is the relation between ⇡N in the leon'sˆ ⌘ p 3m m 8 8 , namely ⇡N =ˆ + 2m m s s , which for ive ⇡N ⇠ˆ . The origin of the puzzle is the relation: 8 = inations related via the Gell-Mann-Okubo (GMO) relation) s, which givesˆ ⇠ 25 MeV. If that relation is a reasonable e implication is that, contrary to expectations, m s must give leon mass even for the smaller values of ⇡N . The puzzle is values that have been obtained for ⇡N , which would imply ly impossible if one considers that s = O( 1 N c ) ⇡N . through the octet and decuplet baryon masses in the combined ⇥ 1/N c . The emphasis is in that the e↵ective theory can give description of baryon masses, including LQCD results, along ass formula -terms, are among information on the ative way for their n theorem 1 , which, such terms, namely d m 8 = 1 p 3 (m m s ), ted with the singlet sical quark masses, her hand, the pionion to pion-nucleon ad a long evolution combined methods ied with the e↵ective theory that combines the chiral rticular the connection between the deviation of the associated with the scalar densityūu +dd 2ss is elated to a mass combination whose low value has hat while the nucleon terms have a well behaved n is a↵ected by large higher order corrections nonalysis lattice QCD baryon masses, it is found that tude within its relative large uncertainty. on masses, Gell-Mann-Okubo mass formula sses, quantified by the di↵erent -terms, are among dynamics. In particular, they give information on the ities, for which there is no alternative way for their through the Feynman-Hellmann theorem 1 , which, masses gives access to only two such terms, namely ass combinations m 3 = m u m d and m 8 = 1 p 3 (m m s ), k masses. The terms associated with the singlet dge of baryon masses for unphysical quark masses, (LQCD) calculations. On the other hand, the pionis accessible through its connection to pion-nucleon . Such a determination of ⇡N had a long evolution ate data and the development of combined methods heory [4; 5; 6; 7; 8; 9; 10; 11]. The values obtained 58 MeV [7; 8; 9; 10; 11; 12], where the di↵erence e analyses resides mostly in the di↵erent values of ed in the subtractions, cf. [12]. In addition to the @ m i m B , where m i indicates a quark mass (i = u, d, s) or combinais not explicitly indicated it is assumed to be a nucleon.
April 4, 2018 tive theory that combines the chiral ection between the deviation of the he scalar densityūu +dd 2ss is combination whose low value has cleon terms have a well behaved large higher order corrections non-D baryon masses, it is found that lative large uncertainty.
Mann-Okubo mass formula y the di↵erent -terms, are among ticular, they give information on the there is no alternative way for their ynman-Hellmann theorem 1 , which, cess to only two such terms, namely m 3 = m u m d and m 8 = 1 p 3 (m m s ), terms associated with the singlet asses for unphysical quark masses, tions. On the other hand, the pionugh its connection to pion-nucleon ination of ⇡N had a long evolution development of combined methods ; 8; 9; 10; 11]. The values obtained 9; 10; 11; 12], where the di↵erence es mostly in the di↵erent values of tions, cf. [12]. In addition to the cates a quark mass (i = u, d, s) or combinaicated it is assumed to be a nucleon.

April 4, 2018
where terms not directly relevant to the baryon masses have been omitted. M 0 = O(N c ) is the spin-flavor singlet piece of the baryon mass that provides the large mass expansion parameter for HBChPT. In addition to the well known chiral building blocks, B represents the baryon spin-flavor multiplet field,Ŝ 2 is the square of the baryon spin operator, G ia are the spin-flavor generators of SU(6), andQ is the electric charge operator. No baryon-spin dependent electromagnetic e↵ects are included. The term proportional to C HF gives the leading order mass splitting between the spin 1/2 and 3/2 baryons.g A is identified with 6 5 g N A at the LO, whose physical value is 1.2723 ± 0.0023. The term h 1 is only relevant if baryons with higher spin than 3/2 appear, which requires N c 5. The rest of the terms describe the quark mass e↵ects. The combinationˆ + = N c 0 + +˜ + , where 0 + = 1 3 Tr + and˜ + is the traceless piece of + , assures that the nucleon mass dependency on m s is at most O(N 0 c ) (OZI). ⇤ is an arbitrary scale, which is conveniently chosen to be m ⇢ . The baryon mass formula then reads (neglecting isospin breaking for now) [29]: where m loop B can be obtained with some work using the results in [29], where the details on the mass renormalization and results for general N c can be found.
Setting c 3 = 0 2 , the terms analytic in quark masses in Eqn.
(2) lead to the exact GMO and Equal Spacing mass relations, which are unchanged at generic N c . On the other hand at generic N c the mass relation for 8 at tree level reads: The dominant contributions to GMO and 8 are calculable non-analytic contributions. GMO is O(⇠ 4 ) and in large N c limit it is O (1/N c ). On the other hand, 8 is O(⇠) and it has a prefactor N c , and 8 is O(⇠ 2 ) also with a prefactor N and to 8 at O(⇠ 4 ), both being beyond the determined by the meson masses and by th on C HF , and drives to a large extent the det the ratio 8 / GMO (⇠ 13.5 for N c = 3), independent of the value of C HF in a very : time [22] is the relation between ⇡N in the m m 8 8 , namely ⇡N =ˆ + 2m m s s , which for he origin of the puzzle is the relation: 8 = via the Gell-Mann-Okubo (GMO) relation) ˆ ⇠ 25 MeV. If that relation is a reasonable that, contrary to expectations, m s must give for the smaller values of ⇡N . The puzzle is been obtained for ⇡N , which would imply one considers that s = O( 1 N c ) ⇡N . t and decuplet baryon masses in the combined phasis is in that the e↵ective theory can give aryon masses, including LQCD results, along Sigma Terms 13 which is made possible through lattice QCD (LQCD) c nucleon term ⇡N ⌘m 2m N hN |ūu +dd | Ni is accessib scattering via a low energy theorem [1; 2; 3]. Such a d through the availability of increasingly accurate data an of dispersion theory and chiral perturbation theory [4; 5 for ⇡N range from ⇠ 45 MeV [4; 5; 6] to & 58 MeV between the results of the di↵erent dispersive analyses the S-wave ⇡N scattering lengths a 1/2,3/2 used in the s 1 The following notation will be used: i (B) = m i  Dürr et al. [14] Yang et al. [15] Abdel-Rehim et al. [16] Bali et al. [17] This work  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% confidence interval.
(2m N m ⌃ m ⌅ ) (or other combinations related via the Gell-Ma alid at linear order in quark masses, which givesˆ ⇠ 25 MeV. If t pproximation to the value ofˆ , the implication is that, contrary to very large contribution to the nucleon mass even for the smaller va articularly striking for the larger values that have been obtained fo s ⇠ 0.5 GeV!. Indeed, this is clearly impossible if one considers that This work analyzes the terms through the octet and decuplet bar hiral and 1/N c expansions BChPT ⇥ 1/N c . The emphasis is in that t t NNLO (one chiral loop) a natural description of baryon masses, inc ith the axial couplings which have been obtained in LQCD at di↵ere ar, the resolution of the term puzzle is explained by the fact that eceives large non-analytic in quark mass corrections dominated by m 8 itself, and thusˆ , has a natural low energy expansion and theref esides in the large non-analytic correction to the mass combination 1 3 ig part of that large correction stems from the contribution of decup as found in Refs. [13; 23]. By analyzing LQCD baryon masses [24] ⇡N ⇠ˆ , with the results ⇡N = 69(8)(6) MeV, where the errors are re Table 1 Results from fits to baryon masses. Fit 1 uses only the physical octet and decuplet masses, Fit 2 uses the physical and the LQCD masses from Ref. [24] with M π 300 MeV, and Fit 3 uses only those LQCD masses and imposes the value of phys G M O determined by the physical masses. The renormalization scale µ and the scale are taken to be equal to m ρ . * indicates an input. An estimated theoretical error of 6 MeV is indicated for σ and σ π N .
• The value ofg A /F ⇡ can be fixed by GMO , and it is consistent with the other calculations.
• Octet baryons in the intermediate states contribute 43% to GMO and 33% to 8 .
• One can realize that this is a well behaved expansion by considering the contribution to the baryon mass from each LEC.
• GMO and 8 can be determined only byg A /F ⇡ , C HF and the meson masses, whereas the ratio 8 / GMO doesn't depend ong A /F ⇡ .
• Fit 2 is compatible with Fit 1: implies that the chiral extrapolation of the LQCD to the physical case is consistent.
• LQCD baryon masses have an issue of describing the hyperfine mass shifts between the octet and decuplet.
• Bothˆ and ⇡N has mild dependence on M K .
• Determination of s was not precise because the LQCD results are at approximately fixed m s .
• Our result for ⇡N is consistent with the larger values obtained from ⇡ N scattering analyses.
• Iso spin breaking sigma terms 3 and (u+d) were estimated.
• With the information we have we can determine the contribution of Nucleon mass due to the mass di↵erence of m u d and therefore m Proton and m Neutron di↵erence.
The intermediate spin 3/2 baryons play an important role in enhancingˆ and thus ⇡N .
f these relations are poorly satisfied. The are calculable and given by the nonanalytic ns to one-loop. In the physical case N c ¼ 3, ations are numerically large for the first, third, relations above. This in particular affects the trangeness σ term, and thus indicates that its from arguments based on tree level relations is important corrections [63]. In terms of the octet ts of the quark masses, in addition to GMO To complete this section, fits to the octet an baryon masses including results from LQCD are This in particular allows for exploring the range of the calculation as the quark masses are incre mass formula for the fit is 4 : what larger than the ones obtained here (σ ∼ 83 MeV, σ π N ∼ 76 MeV). The difference lies in the fact that in ordinary HBChPT the corrections to the axial currents couplings have large N c power violating contributions, which compounded with the larger value of g A /F π required by G M O lead to a failure in describing the axial couplings obtained in LQCD at different quark masses [32], in particular their observed small quark mass dependencies. xiii) Although the approach followed in recent work [33] should be expected to give a result for σ π N similar to the one obtained here, it is actually much smaller. It is not clear to the authors whether this may be entirely due to the different set of LQCD data. However, since σ is accurately obtained with only the physical masses, the result of [33] would require a large negative σ s , which seems to be unlikely within the present framework.

Summary
The σ terms of nucleons were calculated using SU(3) BChPT × 1/N c . From the physical octet and decuplet baryon masses a value of σ is obtained which is much larger than the one predicted by a tree level baryon mass combination, in agreement with similar observations in calculations that included the decuplet baryons as explicit degrees of freedom. The "σ term puzzle" is understood as the result of large non-analytic contributions to that mass combination, while the higher order corrections to the σ terms have natural magnitude. The intermediate spin 3/2 baryons play an important role in enhancing σ and thus σ π N . The analysis carried out here shows that there is compatibility in the description of G M O and the nucleon σ terms. The value of σ π N = 69 ± 10 MeV obtained here from including LQCD baryon masses agrees with the more recent results from π N analyses, where the increase in value  Table (1): physical and LQCD masses from [32]. The squares are the results from the fit and the error bands correspond to 68% • The LQCD masses do not describe correctly the hyperfine mass shifts between the octet and decuplet .
• Bothˆ and ⇡N has mild dependence on M K .
• Determination of s was not precise because the LQCD results are at approximately fixed m s .
• Our result for ⇡N is consistent with the larger values obtained from ⇡ N analyses.
• Iso spin breaking sigma terms 3 and (u+d) were estimated.
• With the information we have we can determine the contribution of Nucleon mass due to the mass di↵erence o↵ m u d and therefore m Proton and m Neutron di↵erence.