Evidence for chiral logarithms in the baryon spectrum
EEvidence for chiral logarithms in the baryonspectrum
André Walker-Loud ∗ Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720E-mail: [email protected]
Using precise lattice QCD computations of the baryon spectrum, we present the first direct evi-dence for the presence of contributions to the baryon masses which are non-analytic in the lightquark masses; contributions which are often denoted chiral logarithms . We isolate the poor con-vergence of SU ( ) baryon chiral perturbation theory to the flavor-singlet mass combination. Theflavor-octet baryon mass splittings, which are corrected by chiral logarithms at next to leadingorder in SU ( ) chiral perturbation theory, yield baryon-pion axial coupling constants D , F , C and H consistent with QCD values; the first evidence of chiral logarithms in the baryon spectrum.The Gell-Mann–Okubo relation, a flavor- baryon mass splitting, which is dominated by chiralcorrections from light quark masses, provides further evidence for the presence of non-analyticlight quark mass dependence in the baryon spectrum; we simultaneously find the GMO relation tobe inconsistent with the first few terms in a taylor expansion in m s − m l , which must be valid forsmall values of this SU ( ) breaking parameter. Additional, more definitive tests of SU ( ) chiralperturbation theory will become possible with future, more precise, lattice calculations. The XXIX International Symposium on Lattice Field Theory - Lattice 2011July 10-16, 2011Squaw Valley, Lake Tahoe, California ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] D ec aryon chiral logs André Walker-Loud
1. Introduction
Lattice QCD calculations are now performed with light quark masses at or near their physi-cal values [1], opening a new era for detailed comparisons with chiral perturbation theory ( χ PT).While this program has been very successful for mesons [2], the application to baryon propertieshas been wrought with significant challenges mainly from issues of convergence of the perturbativeexpansion. Recent analysis suggests the convergence of the two-flavor expansion for the nucleonmass is limited to m π (cid:46)
300 MeV [3, 4]. The SU ( ) chiral expansion has similar but more se-vere problems. In heavy baryon χ PT [5] (HB χ PT), the small expansion parameter is given by ε ∼ m K / Λ χ , whereas for the pion-octet χ PT, the small expansion parameter is ε φ ∼ ε . Severaloffshoots of HB χ PT have been developed in an effort to improve the convergence of the theory [6].We review a new application of an old idea: combining the large N c expansion with the SU ( ) chi-ral expansion [7]. This approach has a few formal advantages over the other methods. In the large N c limit, there is an extra symmetry, the contracted spin-flavor symmetry allowing for an unam-biguous field-theoretic method to include the low lying decuplet baryon resonances in the theory;in the large N c limit, the spin-1 / / χ PT are the contributions to hadronic observables which are non-analytic inthe light quark masses, arising from pion-octet loops, which often contribute ln ( m K , π , η ) terms tohadronic observables, and are commonly referred to as chiral logs . These contributions can notarise from a finite number of local counterterms but only from the long range contributions fromthe light pion octet degrees of freedom, the pion cloud . Isolating this predicted light quark massdependence in lattice QCD results has been a major challenge for many years. The definitive iden-tification of these contributions is hailed as a signal that the up and down (and strange ) quarks aresufficiently light that the lattice results can be described accurately by χ PT. This task has provedto be very challenging, as often, these non-analytic light quark mass contributions are subleading,or masked by other systematics.We report on the first substantial and direct evidence of the presence of non-analytic lightquark mass dependence in the baryon spectrum, work which was performed in Ref. [8].
2. Evidence for non-analytic light quark mass dependence
In Ref. [9], linear combinations of the ground state baryon spectrum were constructed to isolatevarious operators in the combined SU ( ) and large N c expansions. These mass relations werecompared with lattice calculations and it was demonstrated the predicted mass hierarchy persistedover a large range of quark masses [10]. Here, we focus on three of these mass relations in additionto the Gell-Mann–Okubo relation, and provide evidence for the presence of non-analytic lightquark mass dependence in the baryon spectrum. The heavy baryon Lagrangian was formulated inthe 1 / N c expansion in Ref. [11], providing relations amongst the various LECs. In particular, theleading quark mass dependent operators satisfy the following relations at subleading order in 1 / N c b D = b , b F = b + b , b T = − b − b , σ B = b + b , σ T = b + b , (2.1)2 aryon chiral logs André Walker-Loud m lattl [MeV] R [ M e V ] R ( m lattl , m latts )[ LO ] R ( m lattl , m latts,phy )[ LO ] R phy m lattl [MeV] R [ M e V ] R ( m lattl , m latts )[ NLO ] R ( m lattl , m latts,phy )[ NLO ] R phy Figure 1:
Representative fits to R from LO (left) and NLO (right) HB χ PT analysis. The blue star is thephysical value, not used in the analysis. while the axial couplings satisfy the relations at leading order in 1 / N c D = a , F = a , C = − a , H = − a , (2.2)significantly reducing the number of LECs to be determined in the analysis.The numerical data is take from Ref. [3], which is a mixed-action lattice calculation withdomain-wall valence fermions on the dynamical MILC configurations. While the relevant mixed-action EFT is known [12], the lattice results exist at only a single lattice spacing. We thereforerestrict our analysis to that of the continuum HB χ PT. R We begin with the flavor singlet mass relation R : R = ( M N + M Λ + M Σ + M Ξ ) − ( M ∆ + M Σ ∗ + M Ξ ∗ + M Ω ) . (2.3)To NLO in the chiral expansion and using the large N c operator relations (2.1), (2.2),32 R ( m l , m s ) = M − (cid:18) b + b (cid:19) ( m l + m s ) − a ( π f ) (cid:20) (cid:0) F π + F K + F η (cid:1) + (cid:0) F ∆ π + F ∆ K + F ∆ η (cid:1) − (cid:0) F − ∆ π + F − ∆ K + F − ∆ η (cid:1) (cid:21) , (2.4)with F ∆ φ = F ( m φ , ∆ , µ ) defined in Ref. [8], encoding the leading non-analytic light quark massdependence in the baryon spectrum. Both LO ( a =
0) and NLO fits were performed to the latticedata, for a variety of ranges of the light quark masses. The NLO analysis yielded the LECs M [ NLO ] = ( ) MeV , (cid:20) b + b (cid:21) [ NLO ] = − . ( ) , a [ NLO ] = . ( ) , (2.5)Figure 1 displays representative fits. The lower error band is obtained by setting m latts → m latts , phy ,determined from an NLO χ PT [13] analysis of the pion and kaon spectrum. The small value forthe axial coupling, a signals a lack of contributions from the non-analytic light quark mass effects,consistent with the SU ( ) chiral extrapolation analysis of the nucleon mass [3, 4], but inconsistentwith their phenomenological determination [14] or direct computation from lattice QCD [15]. Oneis left to conclude that SU ( ) HB χ PT does not provide a controlled perturbative expansion for R over the range of quark masses explored in this work.3 aryon chiral logs André Walker-Loud R and R We next examine the flavor-octet mass relations R and R R = ( M N + M Λ − M Σ − M Ξ ) − ( M ∆ − M Ξ ∗ − M Ω ) R = M N + M Λ − M Σ + M Ξ SU ( ) chiral and vector limits, making them more sensitiveto the non-analytic light quark mass dependence appearing at NLO in the chiral expansion. To NLOin the chiral expansion and using the large N c operator relations (2.1), (2.2), R ( m l , m s ) = b ( m s − m l ) − a ( π f ) (cid:20) (cid:0) F π − F K − F η (cid:1) + (cid:0) F ∆ π − F ∆ K − F ∆ η (cid:1) − (cid:0) F − ∆ π − F − ∆ K − F − ∆ η (cid:1) (cid:21) , (2.7) R ( m l , m s ) = − b ( m s − m l ) + a (cid:0) F π − F K − F η (cid:1) − (cid:0) F ∆ π − F ∆ K − F ∆ η (cid:1) ( π f ) . (2.8)The LO expressions ( a =
0) fail to describe the numerical results; it is clear higher order contribu-tions are necessary for the extrapolations of these mass relations. At NLO, the analysis of R and R becomes correlated. The full covariance matrix is constructed as described in Ref. [10]. TheNLO analysis, considering several possible ranges of m lattl yields values of the LECs b [ NLO ] = − . ( ) , b [ NLO ] = . ( ) , a [ NLO ] = . ( ) . (2.9)Using the leading large N c relations, Eq. (2.2), this corresponds to D = . ( ) , F = . ( ) , C = − . ( ) , H = − . ( ) . (2.10)The significance of this is prominent; the large value of the axial coupling is strong evidence for thepresence of the non-analytic light quark mass dependence in these mass relations. Further, this isthe first time an analysis of the baryon spectrum has returned values of the axial couplings consis-tent with phenomenology. However, caution is in order. Examining the resulting contributions to R and R from LO and NLO separately, one observes a delicate cancellation between the differentcontributions, see Fig. 2. Further studies are needed with more numerical data sufficient to alsoconstrain the sub-leading large N c axial coefficient a as well as the NNLO contributions. The last mass relation we study is the flavor- Gell-Mann–Okubo relation ∆ GMO = M Λ + M Σ − M N − M Ξ . (2.11) It is interesting to note that while the SU ( ) chiral expansion for the baryon spectrum is not convergent, it wasfound that the volume dependence of the octet baryon masses is consistent with SU ( ) HB χ PT. Analysis of the volumedependence yielded a large value of g π N ∆ ( C ) with g A fixed to its physical value [16]. aryon chiral logs André Walker-Loud m lattl [MeV] − − − R [ M e V ] R [ LO ] R [ NLO ] R phy m lattl [MeV] − − − − R [ M e V ] R [ LO ] R [ NLO ] R phy Figure 2:
The LO and NLO contributions to R (left) and R (right) from a sample analysis. A (blue) staris used to denote the physical values, not included in the analysis. The first non-vanishing contribution to ∆ GMO comes from the NLO chiral loops, which are non-analytic in the light quark masses. For this reason, the GMO relation is of particular interest tostudy with lattice QCD. We extend the previous analysis [17, 3] in a few important ways. Close tothe SU ( ) vector limit, the GMO relation can be described by a taylor expansion in m s − m l , ∆ V GMO = d ( m s − m l ) + d ( m s − m l ) + · · · (2.12)The leading term proportional to ( m s − m l ) must vanish as it transforms as a flavor- . The firstnon-vanishing contribution is equivalent to a next-to-next-to-leading order (NNLO) contributionfrom HB χ PT and the ( m s − m l ) contribution is equivalent to an NNNNLO HB χ PT contribution.We demonstrate these first few terms in the Taylor expansion about the SU ( ) -vector limit areinconsistent with the lattice data as m lattl →
0. We extend the previous analysis to include the NNLOHB χ PT contributions, with the axial couplings constrained by the analysis of R and R . It is foundonly NNLO HB χ PT, which is dominated by the non-analytic light quark mass contributions, cannaturally accommodate the strong light quark mass dependence observed in the numerical results.At NLO in the chiral expansion and using the large N c operator relations (2.1), ∆ GMO [ NLO ] = a ( π f ) (cid:2) F π − F K + F η + (cid:0) F ∆ π − F ∆ K + F ∆ η (cid:1)(cid:3) . (2.13)The full NNLO formula, determined from Ref. [18] can be found in Ref. [8].In Fig. 3, four plots are displayed. The first plot (upper left) is the result of an NLO analysisof the GMO formula, allowing the axial coupling to be determined from the data, resulting in asmall, but non-zero value for a . The second plot (upper right) displays the predicted value ofthe GMO relation from NLO taking the determination of a from the analysis of R and R . Thethird plot (bottom left) shows the result of a taylor expansion about the SU ( ) vector limit fittingthe first two non-vanishing terms. Finally, the NNLO analysis is displayed, using the value of a determined from R and R (bottom right). Only the NNLO analysis is consistent with the valuesof the numerical data over the full range of light quark masses, in particular, the steep rise observedas m lattl →
0, as well as the value of the axial coupling a determined from phenomenology. Thisis further evidence for non-analytic light quark mass dependence in the baryon spectrum.5 aryon chiral logs André Walker-Loud − ∆ G M O [ M e V ] SU (3) χ PTNLO Fit a = 0 . − SU (3) χ PTNLO Fixed a = 1 . m lattl [MeV] − ∆ G M O [ M e V ] SU (3) Vector d ( m s − m l ) + d ( m s − m l ) m lattl [MeV] − SU (3) χ PTNNLO Fit
Figure 3:
GMO mass splitting plotted as a function of m latt l . The ∗ is the PDG point, not included in theanalysis. The various fits are described in the text. In a given plot, the filled (blue) circles denote resultsincluded in the analysis while the open (gray) boxes are excluded.
3. Conclusions
We have presented the first substantial evidence for the presence of non-analytic light quarkmass dependence in the baryon spectrum, with further analysis details in Ref. [8]. This wasachieved by comparing the predictions from HB χ PT combined with the large N c expansion torelatively high statistics lattice computations of the octet and decuplet baryon spectrum. An anal-ysis of mass relations R and R provided for the first time, values of the axial couplings whichare consistent with the phenomenological determination, signaling significant contributions fromnon-analytic light quark mass dependence in R and R : utilizing the leading large N c expansion, D = . ( ) , F = . ( ) , C = − . ( ) , H = − . ( ) . It was further demonstrated that the Gell-Mann–Okubo relation is inconsistent with the first twonon-vanishing terms in a taylor expansion about the SU ( ) vector limit, and that the steep risein the numerical data, observed as m lattl →
0, can only be described by the NNLO heavy baryon χ PT formula which is dominated by chiral loop contributions. Taken together, these observationsindicate the first significant evidence for the presence of non-analytic light quark mass dependencein the baryon spectrum.However, there are several known systematics which were not addressed in the present article,and require future, more precise lattice results: the numerical data used [3] exist at only a singlelattice spacing: continuum χ PT analysis was performed: there may be contamination from finitevolume effects [16]: the convergence issues need further examination: more precise numericalresults are needed to explore mass relations R – R which are more sensitive to non-analytic light6 aryon chiral logs André Walker-Loud quark mass dependence: results with smaller values of the light quark mass are desirable: thestrange quark mass used in this work is known to be ∼
25% to large [19].
References [1] S. Durr, et al. , Science (2008) 1224; S. Aoki et al. [PACS-CS Collaboration], Phys. Rev. D (2009) 034503; A. Bazavov, et al. , Rev. Mod. Phys. (2010) 1349; S. Aoki et al. [PACS-CSCollaboration], Phys. Rev. D (2010) 074503; S. Durr, et al. , Phys. Lett. B (2011) 265; S. Durr, et al. , JHEP (2011) 148.[2] G. Colangelo et al. , Eur. Phys. J. C (2011) 1695 [arXiv:1011.4408 [hep-lat]].[3] A. Walker-Loud et al. , Phys. Rev. D (2009) 054502 [arXiv:0806.4549 [hep-lat]].[4] A. Walker-Loud, PoS LATTICE (2008) 005 [arXiv:0810.0663 [hep-lat]].[5] E. E. Jenkins and A. V. Manohar, Phys. Lett. B (1991) 558; Phys. Lett. B (1991) 353.[6] J. F. Donoghue, B. R. Holstein and B. Borasoy, Phys. Rev. D (1999) 036002; D. B. Leinweber,A. W. Thomas, K. Tsushima and S. V. Wright, Phys. Rev. D (2000) 074502; D. B. Leinweber,A. W. Thomas and R. D. Young, Phys. Rev. Lett. (2004) 242002; T. Becher and H. Leutwyler, Eur.Phys. J. C (1999) 643; T. Fuchs, J. Gegelia, G. Japaridze and S. Scherer, Phys. Rev. D (2003)056005; J. Martin Camalich, L. S. Geng and M. J. Vicente Vacas, Phys. Rev. D (2010) 074504;B. C. Tiburzi and A. Walker-Loud, Phys. Lett. B (2008) 246; F. -J. Jiang, B. C. Tiburzi andA. Walker-Loud, Phys. Lett. B (2011) 329.[7] G. ’t Hooft, Nucl. Phys. B (1974) 461; E. Witten, Nucl. Phys. B (1979) 57; R. F. Dashen andA. V. Manohar, Phys. Lett. B (1993) 425; Phys. Lett. B (1993) 438; E. E. Jenkins, Phys. Lett.B (1993) 441; R. F. Dashen, E. E. Jenkins and A. V. Manohar, Phys. Rev. D (1994) 4713;R. Flores-Mendieta, C. P. Hofmann, E. E. Jenkins and A. V. Manohar, Phys. Rev. D (2000) 034001.[8] A. Walker-Loud, arXiv:1112.2658 [hep-lat].[9] E. E. Jenkins and R. F. Lebed, Phys. Rev. D (1995) 282 [hep-ph/9502227].[10] E. E. Jenkins, A. V. Manohar, J. W. Negele and A. Walker-Loud, Phys. Rev. D (2010) 014502.[11] E. E. Jenkins, Phys. Rev. D (1996) 2625 [hep-ph/9509433].[12] O. Bar, C. Bernard, G. Rupak and N. Shoresh, Phys. Rev. D (2005) 054502; B. C. Tiburzi, Phys.Rev. D (2005) 094501; J. -W. Chen, D. O’Connell and A. Walker-Loud, Phys. Rev. D (2007)054501; K. Orginos and A. Walker-Loud, Phys. Rev. D (2008) 094505; J. -W. Chen, D. O’Connelland A. Walker-Loud, JHEP (2009) 090; J. -W. Chen, M. Golterman, D. O’Connell andA. Walker-Loud, Phys. Rev. D (2009) 117502.[13] J. Gasser and H. Leutwyler, Nucl. Phys. B (1985) 465.[14] R. Flores-Mendieta, E. E. Jenkins and A. V. Manohar, Phys. Rev. D (1998) 094028.[15] H. -W. Lin and K. Orginos, Phys. Rev. D (2009) 034507 [arXiv:0712.1214 [hep-lat]].[16] S. R. Beane et al. , Phys. Rev. D (2011) 014507 [Phys. Rev. D (2011) 039903.[17] S. R. Beane, K. Orginos and M. J. Savage, Phys. Lett. B (2007) 20 [hep-lat/0604013].[18] A. Walker-Loud, Nucl. Phys. A (2005) 476 [hep-lat/0405007]; hep-lat/0608010.[19] C. Aubin et al. [HPQCD and MILC and UKQCD Collaborations], Phys. Rev. D (2004) 031504.(2004) 031504.