Applications of baryon chiral perturbation theory. A topical example: The nucleon sigma terms
aa r X i v : . [ h e p - ph ] M a r Applications of baryon chiral perturbation theory.A topical example: The nucleon sigma terms
J. Martin Camalich ∗ Department of Physics and Astronomy, University of Sussex, BN1 9QH, Brighton, UKE-mail:
We present an overview of modern approaches to low-energy baryon structure based on baryonchiral perturbation theory. These are driven by the emergence of Lorentz covariant schemesand the systematic consideration of the effects of the lowest-lying decuplet resonances. In orderto illustrate the progress recently achieved in this field, we present the last developments on ourunderstanding of the nucleon sigma terms along these lines. In particular, we will show how thesemethods, in SU ( ) or SU ( ) settings, are reliable tools to process and maximize the informationon the physical structure of the nucleon one can obtain from either experimental data or latticeQCD results. The 7th International Workshop on Chiral Dynamics,August 6 -10, 2012Jefferson Lab, Newport News, Virginia, USA ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ pplications of B c PT: The nucleon s -terms J. Martin Camalich
One of the major scientific challenges in fundamental physics consists of understanding thestrong interactions and the nuclear structure phenomena, directly from QCD. The low-energy struc-ture of the nucleons and other baryons, and their interaction with the Goldstone bosons of chiralsymmetry (pions, kaons and eta’s), turns out to be an essential ingredient of this program. This is,in fact, a topic much revitalized in recent years, mainly due to the remarkable progress made bythe lattice QCD (LQCD) community on the calculation of key observables [1]. This is being facil-itated, at the same time, by the emergence of modern effective field theory approaches, based onbaryon chiral perturbation theory (B c PT), which allow to clear the LQCD simulations from somesystematics and check them against the experimental data, or to process efficiently the informationgenerated to provide reliable predictions. On the other hand, investigations involving accurate de-terminations of baryonic observables (such as the proton electric radius or the sigma term) havebecome part of the modern “new-physics rush”, and interest in this direction is rapidly spreadingacross the field.In this contribution I will present a snapshot of the improvements in B c PT by reporting theprogress recently made on the understanding of the nucleon sigma terms and p N scattering. Theseobservables can be studied using either SU ( ) or SU ( ) settings and can be extracted from ex-perimental data on p N scattering and the baryon-octet mass splittings or LQCD results on thelowest-lying baryon spectrum. Therefore, they represent a perfect example to study the potentialand limitations of B c PT for processing and correlating all this information in a model-independentand systematic manner. Moreover, the sigma terms epitomize the type of baryonic observableswith high physics interest, containing information on the origin of the mass of the ordinary matteras well as becoming the main uncertainty in the constraints derived from direct searches of darkmatter [2].One customarily introduces two independent observables, s p N and s s , which are known asthe pion-nucleon and nucleon strangeness sigma terms. These are defined in the isospin limit( m u = m d ≡ ˆ m ) as s p N = h N | ˆ m (cid:0) ¯ uu + ¯ dd (cid:1) | N i , s s = h N | m s ¯ ss | N i . (1)and contain information on the contribution of the quark-condensate to the masses of the baryonsand parametrize the flavour-structure of the nucleon scalar form factors at t =
0. The s s is ofparticular significance as it contains information on the virtual s ¯ s pairs and their contribution tothe nucleon mass. In the following we will briefly review the state-of-the-art in B c PT and, then,report recent determinations of these matrix elements using this approach in combination withexperimental data or LQCD results.
1. Power Counting and decuplet resonances in B c PT Unlike in the meson sector, in B c PT the power counting (PC) is violated by the presenceof M N as a heavy scale and the baryonic loop diagrams do not fulfill the chiral order prescribedby their topology [3]. A crucial observation follows from noticing that this naive PC arises fromconsidering the nucleons in a non-relativistic expansion, which eventually can be implemented2 pplications of B c PT: The nucleon s -terms J. Martin Camalich from the outset using heavy-baryon (HB) c PT [4]. The HB is an elegant formalism with a neatPC, but it alters the analytic structure of the baryon propagators. This has been argued to be thereason behind the problematic convergence showed by the HB expansion in some observables,motivating the study and emergence of Lorentz covariant approaches. An important developmentin this sense comes from realizing that the genuine non-analytical chiral corrections in a covariantformalism satisfy the PC and are identical to those obtained in the HB expansion. The PC-breakingpieces, on the other hand, are analytic and can be eventually absorbed into the local countertermsor low energy constants (LECs) in some (renormalization) prescription [5, 6]. There are two maincovariant approaches, the infrared (IR) [5] and the extended-on-mass-shell (EOMS) scheme [6].The former one, however, introduces unphysical cuts that can disrupt the chiral expansion. TheEOMS scheme, on the other hand, is nothing else than conventional dimensional regularization inwhich the finite parts of the counter-terms are adjusted to cancel the PC-breaking pieces. In thisway, it recovers the PC at the same time as it does not change the analytic structure dictated by S -matrix theory.Another difficulty in B c PT is the treatment of the lowest-lying decuplet resonances. In theconventional approach, these resonances and other heavier degrees of freedom are integrated outand accounted for by the LECs. This is a valid prescription as long as the energies probed in thetheory are well below the mass gap of these states with respect to the ground state octet baryons,or M N . However, in case of the decuplet resonances like the D ( ) , the mass gap is only of d ∼ M D − M N ∼
300 MeV and, moreover, this resonance couples strongly to the p N system. Ina SU ( ) -B c PT approach, the size of the perturbative parameter, ∼ M K / L c SB is even larger thanthis typical scale d / L c SB . Therefore, it becomes necessary to properly take the D ( ) and otherdecuplet resonances into account as explicit degrees of freedom. In order to do so, one needs todefine a suitable PC for the new scale d [7, 8], and also to tackle the consistency problem thatafflicts interacting spin-3/2 theories (see Ref. [9] and references therein). Once these two issuesare solved, one can apply any of the formalisms to cure the power counting problem and explicitlycalculate their contributions to low-energy baryon structure.
2. Experimental determinations of the sigma terms p N scattering amplitude and s p N The elastic scattering of pions and nucleons probes their scalar vertices, and this is formalizedin the nucleon case by the Cheng-Dashen theorem connecting s p N to the isospin-even scalar scatter-ing amplitude at the kinematical point ( s = m N , t = M p ) , which lies in the unphysical region of theprocess. The traditional extrapolation is done using an energy-dependent parameterization of thedata in partial waves (PW) supplemented by dispersion relations that impose strong analyticity andunitarity constraints onto the scattering amplitude at low energies. There are two “classic” determi-nations of s p N , the one based on the Karlsruhe-Helsinki (KH) group, s p N ≃ ( ) MeV [10, 11],and the other performed by the George-Washington (GW) group, s p N = ( ) MeV [12, 13].Although the difference between these two determinations is not too large, it leads to radicallydifferent interpretations of the strangeness content in the nucleon, as we will see below. Besides,a substantial reduction of the ∼
30 MeV uncertainty involved by these two determinations would3 pplications of B c PT: The nucleon s -terms J. Martin Camalich increase the significance of the constraints set on the parameter space of models from the experi-mental bounds on the dark-matter nucleon cross sections.In order to understand some of the systematic effects, one would wish to complement the dis-persive treatments with B c PT. Ideally, one would even dream of performing a completely model-independent analysis of the scattering data leading to a systematic study of the subthreshold region, s p N and all other related quantities without any further input. However these studies have facedimportant difficulties. The classical works of Fettes et al. in HB at O ( p ) [14] and O ( p ) [15] wereable to reproduce the S - and P -wave phase shifts in the threshold region, but they didn’t succeedto give a realistic description of the subthreshold region and, consequently, they overestimated thevalue of the sigma-term. They concluded that an order-by-order improvement in the extrapolationonto the subthreshold region was far from obvious [15]. The inclusion of the D as explicit degreeof freedom in the small-scale-expansion (SSE) ( d ∼ O ( p ) ) allowed to stretch significantly the re-production of the phase-shifts to larger energies [16]. However, large correlations among the LECswere reported, with values depending much on the PW analysis used as input. As a result a stablevalue of s p N and extrapolation to the subthreshold region could be only achieved using the Olssondispersive sum rules [17]. These problems were corroborated by the O ( p ) calculation withoutexplicit D ’s done in the IR scheme [18]. In this case, though, an inverse approach was followed.The subthreshold description was investigated and the extension into the physical region was thenattempted, without success.The situation has recently improved with a novel chiral analysis of the p N scattering ampli-tude introducing two main innovations over previous work. In the first place, a fully covariantapproach in the EOMS scheme is employed [19, 20, 21]. This proves to be not only importantin the extrapolation onto the subthreshold region (in comparison with HB) but also in extendingthe framework to higher energies (as the IR scheme becomes sensitive to its unphysical cuts). Thesecond essential ingredient is the inclusion of the D ( ) as an explicit degree of freedom inthe d -counting [8], which exploits the hierarchy M p < d < L c SB by counting d as O ( p / ) . Thisanalysis was performed up to O ( p ) in this counting, implying that the only explicit D contribu-tions are those stemming from the Born-Term diagrams. The strategy followed in this work wasto determine the different LECs with the S - and P -wave phase shifts provided by the KH, GW andMatsinos’ [22] groups and then discuss the resulting phenomenology.The quality of the corresponding fits to the GW PW analysis is shown in Fig. 1 where the phaseshifts are perfectly reproduced up to energies of W = √ s ≃ . D contribution, which achieves a gooddescription of the phase shifts only up to energies slightly above threshold. In fact, a comparisonbetween the contributions at different orders shows that only in the former case a good convergenceis obtained up to O ( p ) in all the low-energy region.Once the LECs are determined, one can predict and study different p N scattering observablesor to investigate the extrapolation onto the subthreshold region. In Table 1 we present the resultsfor some selected observables, that can be checked to be in good agreement with those reported bythe respective PW analyses. Then, one can see that a B c PT analysis of the phase shifts ratifies the The inclusion of the D explicitly up to O ( p ) in the d -counting introduces 3 new LECs through the Born-termdiagram. However, one of these parameters can be fixed with the D ( ) -resonance width and the other two can beshown to be redundant [23, 20]. pplications of B c PT: The nucleon s -terms J. Martin Camalich -14-12-10-8-6-4-2 0 1.08 1.1 1.12 1.14 1.16 1.18 1.2 2 3 4 5 6 7 8 9 10 11 1.08 1.1 1.12 1.14 1.16 1.18 1.2-6-5-4-3-2-1 0 1 1.08 1.1 1.12 1.14 1.16 1.18 1.2 -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0.2 1.08 1.1 1.12 1.14 1.16 1.18 1.2 0 10 20 30 40 50 60 1.08 1.1 1.12 1.14 1.16 1.18 1.2 -3.5-3-2.5-2-1.5-1-0.5 0 0.5 1.08 1.1 1.12 1.14 1.16 1.18 1.2 P S f r ag r e p l a ce m e n t s √ s (GeV) √ s (GeV) √ s (GeV) √ s (GeV) √ s (GeV) √ s (GeV) S S P P P P Figure 1:
Chiral analysis of the GW phase shifts (blue points) up to O ( p ) in the EOMS scheme without D (dashed green line) and with D (solid red line) in the d -counting. Table 1:
Physical observables obtained from the O ( p ) p N scattering amplitude in the EOMS renormal-ization scheme fitted to different PW analyses. We show the p N D coupling h A , the p N coupling and cor-responding Goldberger-Treiman discrepancy D GT , the scattering lengths, which are given in units of 10 − M − p and the pion-nucleon sigma term, which is given in units of MeV c . o . f . h A g p N D GT [%] a + + a − + s p N KH [10] 0.75 3.02(4) 13.51(10) 4.9(8) − . ( ) − . ( ) s p N . A comparisonof the values of some of these observables with the alternative determinations that can be obtainedfrom other sources like pionic atoms or NN -scattering favors the GW solution. The KH solutiongives rise to a value for h A that is not compatible with the D ( ) width and to a value for g p N thatleads to a significant violation of the GT relation. As for our study of the EM PW analysis, we founda value for the isovector scattering length that is too small as compared with the accurate valuesobtained from pion-atoms data [22]. However, the most important observable in the discussion of s p N concerns the scalar-isoscalar scattering length. While the KH result is compatible with theold negative values, it is not with the more recent determinations from modern pionic-atom data, a + + = − . ( ) − M − p [24]. These are, on the other hand, compatible with the scattering dataextractions from the GW and EM solutions. Finally, notice that the Matsinos and GW analyseslead to the same s p N . This is relevant because these are two completely different analyses thatincorporate the new high quality data collected over the 2 last decades, whereas the KH groupstopped updating theirs in the mid 80’s. With these considerations, we reported the value [19] s p N = ( ) MeV , (2.1) See Ref. [20] for details. pplications of B c PT: The nucleon s -terms J. Martin Camalich B c PT Dispersive d + ( M − p ) − . ( ) − . d + ( M − p ) 1 . ( ) . d + ( M − p ) 0.99(14) 1.14(2) d + ( M − p ) 0.004(6) 0 . b + ( M − p ) -5.1(1.7) − . ( ) d − ( M − p ) 1.63(9) 1.53(2) d − ( M − p ) -0.112(25) − . ( ) d − ( M − p ) -0.18(5) − . ( ) b − ( M − p ) 9.63(30) 10.36(10) Table 2:
Results for different subthreshold coefficients obtained from the fits to the KH analysis and inB c PT in the EOMS scheme and with explicit D contributions up to O ( p ) in the d -counting. The resultsobtained using dispersive techniques are included for the sake of comparison. where the error includes a theoretical uncertainty estimated with the explicit calculation of higher-order graphs added in quadrature with the one given by the dispersion of the values in the averageof the GW and EM results.The success of this modern calculation to provide a reliable determination of s p N from phaseshifts can be understood by analyzing the scattering amplitude in the subthreshold region, where itcomes usually characterized by the so-called subthreshold coefficients stemming from a kinematicexpansion about the point s = u = m N + M p and t =
0. In Table 2 we show the values of these quan-tities obtained after fitting the LECs to the KH phase shifts compared to those given by dispersivetechniques [20]. As it can be seen, in this approach to B c PT, the long-sought connection betweenthe physical and subthreshold regions is accomplished. In particular, d + and d + correspond to theleading orders of the expansion in t of the Born-subtracted scalar-isoscalar amplitude and so theyare essential to understand the extrapolation to the Cheng-Dashen point [11] and determination of s p N . For the details on the discrepancy on d + and its meaning we refer the reader to Ref. [20]. The contribution of the strange quark to the nucleon mass can be related with s p N and the SU ( ) F -breaking of the baryon masses in the octet. Namely, one can re-express the pion-nucleonsigma term as [25] s p N = s − y , (2.2)where y is the so-called “strangeness content” of the nucleon, y = h N | ¯ ss | N ih N | ¯ uu + ¯ dd | N i = m s s m s s p N = − s s p N , (2.3)and s = ˆ m h N | ¯ uu + ¯ dd − ss | N i . (2.4)Thus, s can be understood as the value of the pion-nucleon sigma term in case that the strange-quark contribution to the nucleon wave function is null (Zweig rule). The interest of s lies on the6 pplications of B c PT: The nucleon s -terms J. Martin Camalich fact that it can be calculated in SU ( ) F B c PT up to O ( p ) using the experimental baryon-octetmass splittings. Subsequently, using Eq. 2.2, one can obtain the strangeness content of the nucleonfrom a given experimental determination of s p N .At LO, s = ˆ m / ( m s − ˆ m ) ( M X + M S − M N ) ≃
27 MeV. The NLO or O ( p ) corrections werefirst calculated by Gasser in 1982 using an early version of B c PT regularized with phenomenolog-ical form factors, giving s = ( ) MeV. This result was later bolstered by an O ( p ) HB calcu-lation [26] which obtained s = ( ) . However, at O ( p ) several new unknown LECs contributeand they had to be determined in this calculation using model estimates, or even the value s p N = SU ( ) -flavor applications [27] (for a recent review see [28]). Nevertheless, and despite the possibleproblems in these numerical determinations, they have settled in the field, becoming an importantsource of distrust in “relatively large” values of s p N . Indeed, taking the Gasser result on s withthe value of s p N = ( ) MeV extracted from p N scattering data by the GW group, one obtains astrangeness contribution to M N of about 300 MeV, a scenario that is considered implausible. Table 3:
Values of s for different B c PT approaches.
Octet O ( p ) Octet+Decuplet O ( p ) Tree level O ( p ) HB Covariant HB-SSE Covariant s [MeV] 27 58(23) 46(8) 89(23) 58(8)These calculations were recently revisited in the context of B c PT framed in a covariant frame-work within the EOMS scheme and considering explicitly the contributions of the decuplet [29],which were neglected in previous works. The results of this analysis are summarized in Table 3,where we present those corresponding to the EOMS and HB approaches, with and without decupletcontributions. The errors are obtained by the explicit computation of O ( p ) diagrams stemmingfrom the SU ( ) -breaking of the baryon masses in the O ( p ) diagrams. As we can see, the correc-tions to the LO result on s studied are large. The HB expansion has severe problems of conver-gence in the description of the sigma terms at O ( p ) . Focusing in the following on the covariantresults, we see that considering only the octet contributions the result is reasonably close to theclassical result of Gasser [25], whereas the new contributions given by the decuplet baryons are notnegligible producing a ∼
10 MeV rise on s . In summary, this implies that a pion-nucleon sigmaterm of ∼
60 MeV is not at odds with a small strangeness content in the nucleon. In fact, pluggingthe result for s p N from p N -scattering reported in the previous section, we obtain y = . ( ) . (2.5)
3. Determinations from lattice QCD
A theoretical determination of the sigma terms is accessible through the LQCD simulations.There are two possible strategies. The most straightforward one consists of directly evaluatingthe matrix elements (1) in the lattice. However this is computationally very expensive due tothe evaluation of the contributions of the current coupled to disconnected quark lines, which areexpected to play an important role in the numerical determinations. The second and most widely7 pplications of B c PT: The nucleon s -terms J. Martin Camalich M D [ G e V ] m p 2 [GeV ] M B [ G e V ] W - X * S * D XSL N Figure 2:
Extrapolation of the PACS-CS results [39] on the lowest-lying baryon masses within the covariantformulation of SU ( ) F -B c PT up to O ( p ) [31]. used strategy is based on the Hellmann-Feynman theorem which relates the sigma-terms to thequark-mass derivatives of the nucleon mass. Therefore, one can obtain s p N and s s by interpolatingthe physical nucleon mass with determinations of M N at different values of quark masses. Oneneeds enough accurate determinations close to the physical point and the main difficulty lies inassessing the systematic effects originating from a specific choice of interpolators. In this sense, itis natural to use B c PT to perform these studies. Interpolators based on SU ( ) HB c PT up to O ( p ) and O ( p ) have become standard in the extrapolations of M N and determinations of s p N performedby the lattice collaborations.Two main difficulties have been encountered in the development of this program based onB c PT. First, the extension of the formalism into a SU ( ) setup, describing the quark-mass de-pendence of the masses of all the octet (and decuplet) baryons and giving access to s s , has beentroubled by the problematic convergence of the HB approach in this sector of the theory. Onlyafter the application of approaches with cut-off regularization prescriptions [30] or in the covari-ant formalisms [31, 32, 33, 34, 35, 36], it has been possible to perform reliable SU ( ) F -B c PTextrapolations. Second, the systematic effects given by the decuplet degrees of freedom in the ex-trapolation of the baryon masses and on the value of the sigma-terms remains unclear. While theeffect of D pieces on s p N at O ( p ) in a SU ( ) calculation has been claimed to be negligible [37], amore thorough calculation of these effects up to O ( p ) contradicted this statement [38].In order to address these two issues, we report on the results of the extrapolation of the octet(and decuplet) baryon masses obtained in SU ( ) -B c PT in the EOMS scheme. In contrast with theresults obtained using the HB expansion, it has been found that a good description of the LQCDresults can be achieved within the Lorentz covariant approach to SU ( ) -B c PT up to O ( p ) andconsidering the explicit inclusion of decuplet degrees of freedom [31, 32, 35, 36]. Moreover, anorder-by-order improvement in the description of the lattice results on the octet baryon masses wasfound [31, 36]. Similar efforts in self-consistent formalisms up this accuracy have been reported8 pplications of B c PT: The nucleon s -terms J. Martin Camalich
Table 4:
Predictions on the s p N and s sN terms (in MeV) of the baryon-octet in covariant SU ( ) F -B c PT byfitting the LECs to the PACS-CS [39] or LHPC [38] results. The errors are only statistical.
PACS-CS LHPCNo Dec. Dec. No Dec. Dec. s p N s sN − − c PT up to O ( p ) for the case of the analysis of the PACS-CS re-sults [39]. The strategy followed was to fit the 4 (3) LECs appearing at this order for the octet(decuplet) baryons using the results of different LQCD collaborations. As it can be seen from thefigure, the lattice results are well reproduced and the extrapolation to the physical point agreeswith the experimental values within errors . The improvement obtained at this order in covariant SU ( ) -B c PT in comparison with the description provided by the Gell-Mann-Okubo approach at O ( p ) , stresses the relevance of the leading chiral non-analytical terms in the understanding of thenucleon mass from quark masses as light as those reached by PACS-CS [39] ( M p ≃
156 MeV),whereas the comparison with HB [31] highlights the fact that the relativistic corrections greatlyimprove the description of the LQCD results on the baryon masses at heavier quark-masses.As a result of the determinations of the LECs from the fits, one can predict the nucleon sigmaterms. In Table 4 we present the results on s p N and s sN after fitting the LECs to the PACS-CSand LHPC results. We also present the results that are obtained in fits with (Dec.) and without(No Dec.) the inclusion of decuplet resonances to discuss the systematic effects stemming fromthe treatment of these contributions. As we can see from this table, the results on s p N in eithercase are very consistent with the analysis of the experimental data described in the previous sectionin the case of approximate fulfillment of the Zweig rule. This confirms, in a highly non-trivialfashion, the conclusions at O ( p ) in SU ( ) -B c PT in the EOMS scheme derived exclusively fromexperimental data. In particular, it confirms that a scenario with a s p N ≃
60 MeV can not be ruledout on the grounds of a small strangeness content of the nucleon at this order and that an irreducibleuncertainty of about 15 MeV originates from the omission of the decuplet.In order to settle the question of the strangeness s s it is clear that one needs calculations at O ( p ) . However, at this order a staggering amount of 15 new unknown LECs enter the calculationand determining their values in a statistically sound fashion becomes a challenge. In fact, it seemsimpossible to constraint their values resorting to experimental data only and results from LQCDcalculations have to be massively used. Nevertheless, first steps in this direction have been givenand stable fits to global LQCD results on the baryon masses have been obtained. In particular, inthe works by Semke and Lutz [33, 34], reported also in this conference, the usual chiral expansionis supplemented with another one in 1 / N c which allows to uncover hierarchies among the LECsand to reduce their total number. More general fits taking into account all the 19 LECs and also Notice that in these fits the experimental baryon masses are not included in the fit, so the results obtained at thephysical point are a prediction of the extrapolation. pplications of B c PT: The nucleon s -terms J. Martin Camalich M B [ G e V ] M p [GeV ] PACS-CS M B [ G e V ] M p [GeV ] LHPC M B / X B M p /X p QCDSF-UKQCD M B [ M e V ] M p [GeV ] HSC
Figure 3:
Extrapolation of the PACS-CS results [39] on the lowest-lying baryon masses within the covariantformulation of SU ( ) F -B c PT up to O ( p ) and without decuplet degrees of freedom [35]. finite volume corrections have been presented in the EOMS scheme without [35] and with the de-cuplet degrees of freedom [36] and using a total of 11 configurations at different quark masses andvolumes (each of which contains four points for the N , L , S and the X ). The resulting good descrip-tion of the quark mass dependence of the lowest lying octet baryons in this approach is illustratedin Fig. 3 where the results of these fits is plotted against PACS-CS [39], LHPC [38], HSC [40]and UKQCD [41] results. The NPLQCD [42] results are also used but not plotted and the BMWresults [43] are not included in the analysis. As for the sigma terms, the situation is at the momentunclear. In this calculation the values s p N = ( )( ) MeV and s sN = ( )( ) MeV are re-ported [35], whilst in the calculation at the same order by Semke et al. the values s p N = ( ) MeV and s sN = ( ) MeV are given [34] (see M. Lutz’s talk in this conference). Therefore,further efforts are required to understand these inconsistencies and to assess the convergence of thechiral expansion of these observables. Agreement with the results obtained with cut-off regulariza-tion [30] in the context of dimensional regularized approaches shall also be pursued.
4. Conclusions
We have reviewed the status and potential of the modern approaches to B c PT by showingdifferent recent determinations of the sigma terms. Besides being very important properties of thenucleon, they can be determined from different perspectives, based on SU ( ) or SU ( ) approaches10 pplications of B c PT: The nucleon s -terms J. Martin Camalich along one direction but also using either experimental data or LQCD results along an orthogonalone. We have seen how the situation in our understanding of the p N scattering data in a chiralframework has been greatly improved thanks to the application of modern Lorentz covariant tech-niques and dealing rigorously with the contributions of the D ( ) . Although at the moment theresulting phenomenology still depends on the PW used as input, this progress offers the possibilityof extracting the p N scattering observables, and in particular s p N , in B c PT using directly the scat-tering cross-section data. As for the LQCD determinations, there has been much progress both inthe quality of the LQCD results as well as on the accuracy of the B c PT calculations. Nevertheless,further work is needed to settle this issue from c PT perspective. On one hand, it would be desir-able to revisit the lattice world data on M N using a SU ( ) framework to determine s p N . On theother, more data and, ideally, calculations and extrapolations on other observables in SU ( ) seemnecessary to understand better the strangeness content of the nucleon at O ( p ) .
5. Acknowledgments
I would like to thank the organizers for inviting me to this extremely interesting meeting. AlsoI would like to thank my collaborators J. M. Alarcón, L. S. Geng, J. Meng, J. A. Oller, X. L. Ren,H. Toki and M. J. Vicente Vacas who have contributed to the work presented in this talk. This workis funded by the Science Technology and Facilities Council (STFC) under grant ST/J000477/1,the Spanish Government and FEDER funds under contract FIS2011-28853-C02-01 and the grantsFPA2010-17806 and Fundación Séneca 11871/PI/09.
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