Consistency between SU(3) and SU(2) chiral perturbation theory for the nucleon mass

Treating the strange quark mass as a heavy scale compared to the light quark mass, we perform a matching of the nucleon mass in the SU(3) sector to the two-flavor case in covariant baryon chiral perturbation theory. The validity of the $19$ low-energy constants appearing in the octet baryon masses up to next-to-next-to-next-to-leading order~\cite{Ren:2014vea} is supported by comparing the effective parameters (the combinations of the $19$ couplings) with the corresponding low-energy constants in the SU(2) sector~\cite{Alvarez-Ruso:2013fza}. In addition, it is shown that the dependence of the effective parameters and the pion-nucleon sigma term on the strange quark mass is relatively weak around its physical value, thus providing support to the assumption made in Ref.~\cite{Alvarez-Ruso:2013fza}.


I. INTRODUCTION
Chiral perturbation theory (ChPT) provides a model independent framework to explore the nonperturbative regime of strong interactions [3][4][5][6]. The formalism and main achievements of ChPT have been reviewed in Refs. [7][8][9][10][11][12][13]. As a low-energy effective field theory of quantum chromodynamics (QCD), it contains a finite number of low energy constants (LECs) up to a certain order, which encode high energy physics integrated out and can, in principle, only be determined by fitting to experimental data. The number of unknown LECs becomes large for high order studies, especially in three (u, d, s) flavors, and therefore, practical applications of ChPT are in most cases restricted to low orders. Fortunately, with the advancement of numerical algorithms and the continuous increase of computer power, lattice QCD (LQCD) simulations [14] have achieved great success in the study of nonperturbative QCD (see, e.g., Refs. [15,16]) and in addition provided an alternative way to help determine the values of the LECs present in high order chiral Lagrangians.
In Ref. [2], nucleon masses from the n f = 2 + 1 lattice simulations of BMW [52], PACS-CS [18], LHPC [19], HSC [20], NPLQCD [24], MILC [53], and RBC-UKQCD [54] were analyzed in SU (2)  assumption that the LECs depend only weakly on the strange quark mass around its physical value. If this assumption holds, because of the relatively faster convergence of SU(2) BChPT in comparison with its SU(3) counterpart, it would in principle provide a more reliable determination of the nucleon mass dependence on the u/d quark masses, and thus the pion-nucleon sigma term via the Feynman-Hellmann theorem.
In the present work, we wish to test the consistency between the SU(3) and SU (2) BChPT descriptions of the nucleon mass by matching the SU(3) BChPT to the SU(2) one. In particular, we compare certain combinations of the SU(3) LECs with their SU (2) counterparts. This can be achieved by treating the strange quark mass as a heavy scale compared to the light quark mass and expanding the SU(3) nucleon mass in terms of m q /m s , where m q is the average u and d quark masses and m s is the strange quark mass. Since the LECs in Ref. [1] and those in Ref. [2] are determined by fitting to different lattice QCD simulations with varying strategies, the consistency between them will provide a nontrivial check on the validity of the obtained LECs, particularly the SU(3) ones, and on the assumption made in Ref. [2] that the dependence of the SU(2) LECs on the strange quark mass is mild close to the physical point. Furthermore, the relevant pion-nucleon sigma term σ πN is also evaluated. But one should treat this value with care because none of recent simulations at the physical point Refs. [55][56][57][58] were available back when the studies of Refs. [1,2] were performed.
We note that in Refs. [59,60], the SU(3) baryon masses and meson-baryon scattering lengths were matched to their SU(2) counterparts with the aim of constraining the large number of unknown SU(3) LECs with the SU(2) inputs. In the present case, because of the abundant n f = 2+1 LQCD baryon masses, both the SU(3) and SU(2) LECs have been independently determined in Refs. [1,2]. This provides us a unique opportunity to study the flavor dependence of BChPT. 1 This paper is organized as follows. In Sec II, we describe the procedure and strategy used to match the SU(3) nucleon mass to the SU(2) one. Hereby we obtain an effective SU(2) expression for the nucleon mass deduced from the SU(3) one. In Sec. III, we compare the effective SU(2) nucleon mass and pion-nucleon sigma term with the original SU(2) and SU(3) ones, and study the dependence of the SU(2) effective parameters on the strange quark mass. This is followed by a short summary in Sec. IV.

II. THEORETICAL FRAMEWORK
In this section, we explain in detail how one can match the SU(3) nucleon mass to the SU(2) one by assuming that the strange quark contribution can be integrated out, namely taking m q /m s as a small expansion parameter, where m q is the average u and d quark masses and m s is the strange quark mass. In the SU(3) EOMS BChPT, the chiral expansion of the nucleon mass up to O(p 4 ) can be written as where m 0 is the baryon mass in the chiral limit while m    [33], respectively. The pseudoscalar meson masses are denoted by m φ (φ = π, K, η); F φ is the pseudoscalar meson decay constant in the chiral limit, which is taken to be F φ = 0.0871 GeV [62]. Latin characters a, b, c, d, e represent the five Feynman diagrams shown in Fig. 1. The ξ coefficients denote combinations of the 19 LECs (m 0 , b 0,D,F , b 1,··· ,8 , and d 1,··· ,5,7,8 ) appearing in the octet baryon masses up to N 3 LO. They are given in Tables 1-5 of Ref. [33], where the corresponding loop functions H can also be found. Note that the loop functions H depend on the meson masses (obtained in leading order ChPT), the chiral limit baryon mass m 0 , and the NLO mass splittings induced by b 0 , b D , and b F .
It is convenient to isolate the ss contribution to the meson masses by introducing m 2 ss = 2B 0 m s . Using the leading order ChPT, the kaon and eta masses can then be expressed as, At the physical point, m ss = 2m 2 K − m 2 π = 683.2 MeV, where m K and m π are the isospin averages of the kaon and pion masses. Now one can approximate the kaon-and eta-loop contributions to the nucleon self-energy (Σ K, η ) by polynomials of the pion mass. Namely, one replaces m K, η with m π, ss and performs a perturbative expansion in terms of m π /m ss up to fourth order, where i denotes the different diagrams (i = a, · · · , e); the expansion coefficients (A K,η ) are given in the Appendix. For the pion-cloud contributions of diagram (e), Σ (e) π , because the leading-order correction to the nucleon mass, m ss , contains the strange quark contributions, it should be expanded as well where D and F are the axial-vector coupling constants, µ denotes the renormalization scale, and π , and C (e) π are given in the Appendix. Putting all pieces together, we obtain the SU(2) equivalent nucleon mass, where the tadpole contributions are separated in two terms proportional to m 4 π and m 4 π log(µ 2 /m 2 π ). The corresponding effective parameters, m eff 0 , c eff 1 , α eff , and β eff are combinations of the original SU(3) LECs (underlined) and the expansion parameters in Eqs. (3) and (4), These results, when expanded in 1/m 0 , are consistent with those of Refs. [59,60]. For comparison, the nucleon mass directly obtained in SU(2) BChPT is [2], where M 0 is the nucleon mass in the SU(2) chiral limit with m u = m d = 0 and m s fixed at its physical value; c 1,2,3 and α are the unknown LECs. In order to obtain the same form as Eq. (5), the above equation can be rewritten as with the following two combinations of the LECs,

III. RESULTS AND DISCUSSION
In this section, we evaluate the effective parameters, m eff 0 , c eff 1 , α eff , and β eff , and compare them with the SU(2) LECs appearing in Eq. (11). In Ref. In order to better constrain the large number of unknown LECs, the strong isospin breaking effects on the octet baryon masses are also taken into account. As the LQCD data are still limited, it is worthwhile to investigate the consistency of the extracted LECs [1]. For this purpose we compare the SU(2) equivalent nucleon mass with the SU(2) one. As a first check, the four (2) ) of Eq. (11). In Ref. [2], these SU(2) LECs have been obtained from the n f = 2 + 1 LQCD data for the nucleon mass, with the strange quark mass close to its physical value. Therefore, they should implicitly incorporate the strange quark contribution that is apparent in Eqs. (6)(7)(8)(9).  In Table I, we tabulate the values of the effective parameters appearing in Eq. (5), with the strange quark mass fixed at its physical value (m ss = 683.2 MeV). For comparison, the corresponding SU(2) LECs, Eq. (11) and Ref. [2], are listed in the second column. We find that m eff 0 and c eff 1 agree well with M 0 and c 1 . 2 At O(p 4 ), we obtain larger discrepancies: α eff is consistent with α SU(2) because of the large error bar of the latter; instead, β eff and β SU(2) disagree. Although the SU(3) and SU(2) LECs have been obtained with different renormalization scales, µ = 1 GeV in Ref. [1] and µ = M 0 in Ref. [2], this only affects the comparison for α SU (2) , which receives from loop (e) and the β term contributions that are small (smaller than the error bar in α SU(2) quoted in Table I) To illustrate the impact of these similarities and differences on the nucleon mass, m 2 π , m 4 π and m 4 π log(µ 2 /m 2 π ) terms are separately plotted as a function of the leading order m 2 π in Fig. 2. The contributions of the two loop diagrams in Fig. 1 are also given. It should be mentioned that the upper limit in the pion mass is set at 500 MeV to guarantee a reasonable expansion in powers of m π /m ss for m ss close to its physical value. The agreement is very good for loop (b) and the m 2 π term but less so in the rest of terms. This is due to the differences in the central values of α and β parameters but also to the above mentioned difference in renormalization scales that reshuffles strength within O(p 4 ) terms.
The pion mass dependence of the nucleon mass for the effective SU(3) → SU(2), SU(2) [2] and SU(3) [1] approaches is presented in Fig. 3. One can see that the m π /m ss expansion truncated at O(m π /m ss ) 4 is a good approximation to the SU(3) case up to rather high m 2 π . The large error bars in the SU(2) fit make it consistent with both the SU(3) result and the SU(3) → SU(2) projection but there are clear differences in the central values which increase with m 2 π . In both Ref. [2] and [1], the LQCD pion masses are identified with the next-to-leading order pion masses M π but the way to express M N in terms of M π is different. In Ref. [2], higher order terms are neglected by taking m N (m π ) while in Ref. [1] these terms are included by numerically expressing O(p 4 ) meson masses in terms of O(p 2 ) ones. Although formally equivalent, these two procedures lead to numerically different nucleon masses at high pion masses (for a given set of parameters). However, we have checked that these differences are largely compensated by the different µ adopted in the two studies: if a given set of LQCD data for the nucleon mass are fitted with Eq. (11) using m (4) N (m π ) and µ = M 0 and, on the other hand, applying the numerical inversion of Ref. [1] with µ = 1 GeV, the resulting parameters are remarkably close. From this we conclude that the tension between the SU(2) nucleon mass from Ref. [2] and the SU(3) one from Ref. [1], or in the LEC comparison of Table I, predominantly follows from the use of different data sets, once the SU(3) study incorporates LQCD output for the other octet baryon masses.
As mentioned in the introduction, Ref. [2] reported a global analysis of the n f = 2 + 1 lattice nucleon mass from the BMW [52], PACS-CS [18], LHPC [19], HSC [20], NPLQCD [24], MILC [53], and RBC-UKQCD [54] collaborations by using the SU(2) nucleon mass with the assumption that LECs depend weakly on the strange quark mass around its physical value. In Fig. 5, the strange quark masses employed in the above LQCD simulations are given in the lower panel. It can be seen that the strange quark mass adopted in the LQCD simulations (0.55 GeV < m ss < 0.80 GeV) indeed is close to its physical value , therefore, it is interesting to explore the dependence of the SU(2) equivalent LECs on the strange quark mass. For this, we define the relative deviation R as with X = m eff 0 , c eff 1 , α eff , β eff . In the upper panel of Fig. 5, the relative deviation R for the four effective parameters is shown as a function of the strange quark mass. It is observed that the values of the parameters change very little, with |R| < 5%, in the range of the strange quark mass employed by LQCD simulations. This study gives an estimate about the range of the strange quark masses employed in the n f = 2 + 1 LQCD simulations suitable for an SU(2) BChPT study. It is also interesting to consider the m ss dependence of the πN sigma term. Figure 6 shows deviations of at most 10% from the value at the physical point (see the band), in the m s range of LQCD simulations. Both Figs. 5, 6 show asymmetries in the slope of some effective LECs and σ πN above and below the physical m ss value. The relatively faster growth of these values could reflect a slower convergence of BChPT for heavier strange quark masses and might introduce biases in SU(2) analyses of n f = 2 + 1 LQCD data.

IV. CONCLUSION
We have checked the consistency between the SU(2) and SU(3) baryon chiral perturbation theory for the nucleon mass. It is shown that although the number of LECs in the SU(2) and the SU(3) cases is quite different, and the strategy to fix them using LQCD simulations varies, the so-obtained LECs are largely consistent with each other. In addition, we have shown that the SU(2) equivalent LECs indeed depend rather weakly on the strange quark mass close to its physical value. This result further supports the idea that LQCD simulations provide an new alternative way to determine unknown LECs in baryon chiral perturbation theory, which might be hard to fix otherwise.
With the SU(2) equivalent chiral expansion reported here, we find a σ πN = 57(6) MeV, which is consistent with the results of Refs. [1,2]. On the other hand, one should take this value with caution, because neither the present study nor Refs. [1,2] include the latest LQCD simulations at the physical point which appeared after Refs. [1,2] were published. The current tension between the large sigma term obtained in π-N scattering analyses and the present study and those of the latest LQCD simulations calls for a new global analysis that includes these physical point lattice data in the fits. In addition, in Ref. [78], the chiral convergence of σ 0 was discussed in detail, emphasizing the breakdown of the chiral expansion in case of a large nucleon sigma term. We expect to gain further insight into this issue from a systematic study of all the state of the art LQCD simulations.

APPENDIX
In this section, we provide explicitly the expansion coefficients appearing in Eqs. (3) and (4).