It has proven a significant challenge to experiment and phenomenology to
extract precise values of the nucleon sigma terms. This difficulty opens the
window for lattice QCD simulations to lead the field in resolving this aspect
of nucleon structure. Here we report on recent advances in the extraction of
nucleon sigma terms in lattice QCD. In particular, the strangeness component is
now being resolved to a precision that far surpasses best phenomenological
estimates.Comment: 6 pages, 1 figure; prepared for Proc. 4th Int Symposium on Symmetries
  in Subatomic Physics (SSP2009), Taipei, Taiwan, June 2-5 200

A.W. Thomas

Alexandrou

Aoki

Bernard

Borasoy

Bottino

Brown

Cheng

Dong

Donoghue

Dürr

Ellis

Fukugita

Gasser

Güsken

Ishikawa

Koch

Leinweber

Maiani

Michael

Nelson

Ohki

Pavan

Procura

R.D. Young

Takeda

Toussaint

Walker-Loud

Young

Nuclear Physics A

English

arXiv

It has proven a significant challenge to experiment and phenomenology to extract precise values of the nucleon sigma terms. This difficulty opens the window for lattice QCD simulations to lead the field in resolving this aspect of nucleon structure. Here we report on recent advances in the extraction of nucleon sigma terms in lattice QCD. In particular, the strangeness component is now being resolved to a precision that far surpasses best phenomenological estimates.Ross D. Young and Anthony W. Thoma

Young, R.

Thomas, A.

Adelaide Research & Scholarship

arXiv:0911.1757v1  [hep-lat]  9 Nov 2009 Recent results on nucleon sigma terms in lattice QCDR. D. Younga and A. W. ThomasbaPhysics Division, Argonne National Laboratory, Argonne, Illinois 60439, USAbCSSM, School of Chemistry and Physics, University of Adelaide, Adelaide, SouthAustralia 5005, AustraliaIt has proven a significant challenge to experiment and phenomenology to extract precisevalues of the nucleon sigma terms. This difficulty opens the window for lattice QCDsimulations to lead the field in resolving this aspect of nucleon structure. Here we reporton recent advances in the extraction of nucleon sigma terms in lattice QCD. In particular,the strangeness component is now being resolved to a precision that far surpasses bestphenomenological estimates.1. NUCLEON SIGMA TERMSThe nucleon sigma terms are important quantities in resolving the dynamics of QCD,where they help to understand the role of explicit chiral symmetry breaking in the massof the nucleon. Indeed, the strangeness component plays a unique role in that is purelya vacuum polarization effect, analogous to the Lamb shift in QED. Further, the strangequark is light enough that it probes the nonperturbative distance scales of QCD. Beyondthe fundamental importance to nucleon structure, the strange quark condensate in thenucleon is of significant interest in studies of the QCD phase structure at large baryondensity [1] and in constraining predicted cross sections for dark matter detection [2,3].The sigma terms of the nucleon are defined by the scalar form factors in the limit ofvanishing momentum transfer, σq = mq〈N |q¯q|N〉, where q denotes the quark flavour ofinterest. The light-quark sigma term (or pion-nucleon sigma term), σ` = m`〈u¯u + d¯d〉(withm` ≡ (mu+md)/2), is related to pi–N scattering through a chiral low-energy relation[4,5,6],ΣpiN ≡ σ` = ΣCDpiN −∆R −∆σ , (1)where the experimental input required is the Born-subtracted, isoscalar piN scatteringamplitude evaluated at the (unphysical) Cheng-Dashen point, ΣCDpiN . The smallness of theup and down quark masses ensures that both the remainder term, ∆R [5,7], and the formfactor correction, ∆σ [8], can be reliably estimated. An early analysis of experimentalresults [9] led to the value σ` = 45± 8MeV [8], while a more recent analysis suggested ahigher value, 64± 7MeV [10].Unlike the light quarks, the strange quark is too heavy to invoke a low-energy relation.Therefore σs has generally been estimated by studying the breaking of SU(3) within the2 R. D. Young and A. W. Thomasbaryon octet [1,11,12]. Here, the observed spectrum has been used to derive a constrainton the non-singlet combinationσ0 = m`〈N |u¯u+ d¯d− 2s¯s|N〉 , (2)where chiral effective field theory studies lead to a value σ0 = 36± 7MeV [12], buildingon earlier estimates by Gasser [11]. The difference between σ0 and the extracted σ`then defines the strange-quark contribution, σ` − σ0 = 2m`msσs. Equivalently, the relativestrangeness component is often discussed in terms of the parameter y,σ0σ`= 1−2〈N |s¯s|N〉〈N |u¯u+ d¯d|N〉≡ 1− y . (3)Taking the above pion-nucleon sigma term values and the estimated σ0 gives values y '0.2 ± 0.2 and 0.44 ± 0.13, and using a strange-to-light quark mass ratio of ∼ 25 givesstrangeness sigma terms, σs = 110± 130 and 350± 120MeV, respectively. This approachclearly leads to a result for σs which is very sensitive to the precise value of σ`. We alsonote that even with perfect pi–N data to better constrain σ`, the 7MeV uncertainty in σ0alone leads to a ∼ 90MeV uncertainty in σs. This limitation clearly opens the way forlattice QCD to offer significant improvement.2. LATTICE CALCULATIONS2.1. Light quarksIn early lattice simulations, limited computing power forced the use of heavy quarksand the quenching of the light quarks in the theory [13,14]. The first direct calculationsof the disconnected contributions (though still on a quenched gauge ensemble), includingan estimate of the strange quark contributions, were performed in the work of Fukugita etal. [15] and Dong et al. [16]. In these works, the pi–N sigma was found to be consistentwith observation — see Figure 1 for a summary of results.The first determination using (2 flavours of) dynamical quarks found a value for σ`that is smaller than phenomenological expectations [17]. A notable feature of this cal-culation is that consistent results were demonstrated between the direct calculation ofthe disconnected loops and that extracted from the fit to the nucleon mass, using theFeynman-Hellman relation1σq = mq∂MN∂mq. (4)While this calculation appeared to yield a value that was rather low, it was soon identifiedthat the sigma term is particularly sensitive to chiral extrapolation [19]. After this time,most extractions of σ` have come by application of the Feynman-Hellman relation to themass extrapolation [20,21,22,23,24,25,26].It was common in the early calculations that the quoted error only reflected the sta-tistical uncertainty in the underlying extraction. The work of Procura et al. [22] madea detailed investigation of the systematic errors associated with the input parameters of1Commonly the quark masses in this relation are replaced by the meson mass squared using the Gell-Mann–Oakes–Renner relation.Recent results on nucleon sigma terms in lattice QCD 3òòòòòòòòòòòàààààààææ0 20 40 60 80 1000 20 40 60 80 100Fukugita et al. 1995Dong et al. 1996SESAM 1998Leinweber et al. 2000Leinweber et al. 2003Procura et al. 2003Procura et al. 2006ETM 2008JLQCD 2008Young & Thomas 2009PACS-CS 2009Σl HMeVLòòòòòòààææ0 100 200 300 400 5000 100 200 300 400 500Fukugita et al. 1995Dong et al. 1996SESAM 1998JLQCD 2008Young & Thomas 2009Toussaint & Freeman 2009Σs HMeVLFigure 1. Summary of lattice QCD calculations of the light-quark sigma term (left panel)and strangeness sigma term (right panel). Triangles, squares and circles denote quenched,2-flavour and 2+1-flavour lattices, respectively. We caution that different calculationshave invested differing degrees of effort into quantifying their error estimate.their functional form. Contrasting this with their earlier work [21], where only the statis-tical error was quoted, is a useful guide as to the potential size of systematic effects. Ofcourse, various systematics will continue to be better controlled as the simulation resultsimprove.While it is not obvious from Figure 1(a) that the lattice results are showing convergence— without taking bias — there does appear to be consensus that the lattice results areconsistent with the two phenomenological extractions discussed above.2.2. Strange quarkEarly lattice estimates of the strangeness sigma term supported the rather large valuesthat had been inferred from phenomenology [15,16]. Going beyond the quenched ap-proximation, the 2-flavour dynamical calculations of the SESAM Collaboration [17] alsosupported the larger strangeness values, though with substantial uncertainty. Anotherearly calculation highlighted some of the challenges of resolving these matrix elementsand concluded that the strangeness component is most likely rather small [18].While the strange quark in the 2-flavour simulations is essentially quenched, it is inter-esting to consider the light-quark disconnected diagrams, which are truly dynamical. Onecan consider the ratio of the disconnected to connected (or sea to valence) contributions4 R. D. Young and A. W. Thomasto the sigma term,Rd/c =〈N |u¯u+ d¯d|N〉disc〈N |u¯u+ d¯d|N〉con(5)In the quenched simulations a relatively large value for this ratio was observed, withRd/c = 2.23(52) [15] and 1.79(7) [16]. We note that the effect of unqenching appears toreduce this ratio, with the corrsponding ratio found to beRd/c = 1.26(57) in the dynamicalsimulation [17]. This indicates a decrease in the relative strength of the disconnectedcomponent.Extending this discussion, a modern determination of this ratio by JLQCD [24] hasfound Rd/c = 0.41(5) (taking PQ-b as the “best estimate”) — suggesting even further re-duction of this ratio as results better describe the physical regime. This study constrainedthe nucleon mass using light-quark masses taking values over the range ms/6–ms (or pionmasses ∼ 290–750MeV).In a SU(3) symmetric (mu = md = ms) 3-flavour dynamical calculation, the discon-nected contributions of each of the three quarks are identical, satisfying the identity2〈N |s¯s|N〉discSU(3)= 〈N |u¯u+ d¯d|N〉disc . (6)In a 2-flavour simulation, the literal s¯s contribution is zero, yet it could be reasonableto adopt the above relation to infer the s¯s from a 2-flavour calculation. Indeed this isthe approach taken by [24], where at the SU(3) symmetric point one can infer y ∼ 0.09.Further taking the 2-flavour result for the increase in both the connected and disconnectedlight-quark terms as the light-quarks are reduced to the physical point reduces the estimateof y to ∼ 0.03 [24] (assuming that the strangeness term is only weakly dependent on thelight-quark masses). Indeed, when combined with their extraction for the σ`, this leadsto a significantly reduced value of the strangeness sigma term σs = 20± 12MeV.The major cause of this reduction compared with the previous estimates comes fromthe observation that the derivative of the nucleon mass ( ∂MN∂msea) is quite strongly dependenton the sea-quark mass (msea). Indeed, this derivative was observed to be enhanced bya factor of ∼10 in going from the strange-quark mass to the light-quark mass. Beyondthe numerical improvements in the underlying lattice simulations, this study has alsoincorporated the consequences of chiral symmetry breaking, where there are known to besubstantial chiral logarithms in these derivatives — leading to the observed enhancementin the light-quark domain.These authors have recently reported results on testing their method against a directcalculation of the disconnected strangeness component, finding compatible results [27].Further, the analysis of the JLQCD Collaboration is being extended to 2+1 flavours ofdynamical quarks, where preliminary results also support their 2-flavour findings [28].Another opportunity to study the strange-quark mass dependence came with a series ofcalculations of the baryon spectrum in 2+1-flavour dynamical simulations which emergedin 2008 [29,30,31,32] (after an early venture in 2001 [33]). These new 3-flavour resultsallowed tests of the SU(3) chiral expansion at the lattice quark masses [29,25,26]. Indeedthe poor convergence of the SU(3) expansion, already recognised at the physical quarkmasses [12,11], was rediscovered on the lattice. These three chiral-lattice papers tookthree different approaches to deal with this problem. PACS-CS [26] simply abondonedRecent results on nucleon sigma terms in lattice QCD 5the SU(3) formulation; LHPC [29] sacrificed the coefficients of the chiral logarithms inorder to stabilise the fits; we [25] introduced a single new fit parameter that acts toseparate the low- and high-energy contributions to the chiral loop integrals.The results of Ref. [25] built on the established benefits of using finite-range regulari-sation, which has previously been shown to dramatically improve the SU(3) convergence[11,34] and offer robust chiral extrapolation for lattice QCD [20,35]. The analysis ofRef. [25] demonstrated the ability to extrapolate in both the light- and strange-quarkmasses to accurately reproduce the physical octet baryon spectrum, and also extrapolateto the heavier quark masses (not used in the fits) as calculated on the lattice. The resultswere found to be consistent with two different lattice discretizations, which also allowedan estimate of potential discretization effects.Using the Feynman-Hellman relation, we found σ` to be compatible with phenomenolog-ical and lattice estimates. And the strangeness sigma term was identified to be relativelysmall, supporting the 2-flavour calculation of JLQCD [24].Another recent 2+1-flavour dynamical result for σs has been reported by Toussaintand Freeman [36], using an application of the Feynman-Hellman relation on the nucleoncorrelator. This technique substantially differs from the approaches based on fitting thethe nucleon mass and therefore provides a strong independent test. The result of Toussaintand Freeman [36] is marginally higher than the values reported by JLQCD [24] and byus [25]. Nevertheless, as seen in Figure 1(b), the modern lattice results for σs are inagreement that the size of the strangeness sigma term is substantially smaller than hasbeen previously thought.3. IMPACTAs described above, the dominant uncertainty in knowledge of the scalar quark cou-plings has lies in the strangeness component. The current generation of lattice QCDcalculations has primarily resolved this limitation by providing stringent new limits onthe strange quark sigma term. As a result of this improvement, the predicted cross sec-tions for models of dark matter can now be revised. In a recent paper, Giedt et al. [37]have investigated the impact of the new lattice QCD results on the cross sections for aclass of supersymmetric models. The poorly known strangeness σ-term had previouslyresulted in cross sections which varied by roughly an order of magnitude [3]. The uncer-tainties have been dramatically reduced by the lattice calculations and importantly, onenow has significant discrimination power between the ensemble of models considered.This work was supported by the U.S. Department of Energy, Office of Nuclear Physics,under Contract No. DE-AC02-06CH11357; and the Australian Research Council.REFERENCES1. A. E. Nelson and D. B. Kaplan, Phys. Lett. B 192, 193 (1987).2. A. Bottino et al., Astropart. Phys. 13, 215 (2000) [arXiv:hep-ph/9909228]; Phys. Rev.D 78, 083520 (2008) [arXiv:0806.4099 [hep-ph]].3. J. R. Ellis et al., Phys. Rev. D 77, 065026 (2008) [arXiv:0801.3656 [hep-ph]].4. T. P. Cheng and R. F. Dashen, Phys. Rev. Lett. 26, 594 (1971).5. L. S. Brown, W. J. Pardee and R. D. Peccei, Phys. Rev. D 4, 2801 (1971).6 R. D. Young and A. W. Thomas6. J. Gasser, H. Leutwyler and M. E. Sainio, Phys. Lett. B 253, 252 (1991).7. V. Bernard et al., Phys. Lett. B 389, 144 (1996) [arXiv:hep-ph/9607245].8. J. Gasser, H. Leutwyler and M. E. Sainio, Phys. Lett. B 253, 260 (1991).9. R. Koch, Z. Phys. C 15, 161 (1982).10. M. M. Pavan et al., PiN Newslett. 16, 110 (2002) [arXiv:hep-ph/0111066].11. J. Gasser, Annals Phys. 136, 62 (1981).12. B. Borasoy and U.-G. Meißner, Annals Phys. 254, 192 (1997) [arXiv:hep-ph/9607432].13. L. Maiani et al., Nucl. Phys. B 293, 420 (1987).14. S. Gu¨sken et al. [SESAM], Phys. Lett. B 212, 216 (1988).15. M. Fukugita et al., Phys. Rev. D 51, 5319 (1995) [arXiv:hep-lat/9408002].16. S. J. Dong et al., Phys. Rev. D 54, 5496 (1996) [arXiv:hep-ph/9602259].17. S. Gu¨sken et al. [SESAM], Phys. Rev. D 59, 054504 (1999) [arXiv:hep-lat/9809066].18. C. Michael et al. [UKQCD], Nucl. Phys. Proc. Suppl. 106, 293 (2002)[arXiv:hep-lat/0109028].19. D. B. Leinweber et el., Phys. Lett. B 482, 109 (2000) [arXiv:hep-lat/0001007].20. D. B. Leinweber et al., Phys. Rev. Lett. 92, 242002 (2004) [arXiv:hep-lat/0302020].21. M. Procura et al., Phys. Rev. D 69, 034505 (2004) [arXiv:hep-lat/0309020].22. M. Procura et al., Phys. Rev. D 73, 114510 (2006) [arXiv:hep-lat/0603001].23. C. Alexandrou et al. [ETM], Phys. Rev. D 78, 014509 (2008) [arXiv:0803.3190 [hep-lat]].24. H. Ohki et al. [JLQCD], Phys. Rev. D 78, 054502 (2008) [arXiv:0806.4744 [hep-lat]].25. R. D. Young and A. W. Thomas, arXiv:0901.3310 [hep-lat].26. K. I. Ishikawa et al. [PACS-CS], Phys. Rev. D 80, 054502 (2009) [arXiv:0905.0962[hep-lat]].27. K. Takeda et al. [JLQCD], arXiv:0910.5036 [hep-lat].28. H. Ohki et al. [JLQCD], arXiv:0910.3271 [hep-lat].29. A. Walker-Loud et al. [LHPC], Phys. Rev. D 79, 054502 (2009) [arXiv:0806.4549[hep-lat]].30. S. Aoki et al. [PACS-CS], Phys. Rev. D 79, 034503 (2009) [arXiv:0807.1661 [hep-lat]].31. H. W. Lin et al. [HSC], Phys. Rev. D 79, 034502 (2009) [arXiv:0810.3588 [hep-lat]].32. S. Du¨rr et al., Science 322, 1224 (2008) [arXiv:0906.3599 [hep-lat]].33. C. W. Bernard et al., Phys. Rev. D 64, 054506 (2001) [arXiv:hep-lat/0104002].34. J. F. Donoghue et al., Phys. Rev. D 59, 036002 (1999) [arXiv:hep-ph/9804281].35. D. B. Leinweber et al., Phys. Rev. D 61, 074502 (2000) [arXiv:hep-lat/9906027].36. D. Toussaint and W. Freeman [MILC], Phys. Rev. Lett. 103, 122002 (2009)[arXiv:0905.2432 [hep-lat]].37. J. Giedt et al., to appear Phys. Rev. Lett., arXiv:0907.4177 [hep-ph].

Recent results on nucleon sigma terms in lattice QCD

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Recent results on nucleon sigma terms in lattice QCD

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