1. Anthony W. Thomas CSSM Asia-Pacific Mini-Workshop on Lattice QCD Tsukuba – January 2003 Model Independent Chiral Extrapolation of Baryon Masses

2. Outline PROBLEM: Lack of computer power ) need chiral extrapolation  PT : Radius of convergence TOO SMALL STOP?? NO! Just impose a little physics insight ! ) MODEL INDEPENDENT RESULTS! Then: 10 Teraflops (next generation) machines will allow 1% accuracy

3. Collaborators Derek LeinweberRoss Young (Stewart Wright)

4. Problem of Chiral Extrapolation Time to decrease m  by factor of 2 : 2 7 » 100 Furthermore EFT implies ALL hadronic properties are non-analytic functions of m q …….. HENCE: NO simple power series expansion about m q = 0 : NO simple chiral extrapolation ♦ Currently limited to m q > 30-50 MeV ♦ NEED perhaps 500 Teraflops to get to 5 MeV !

5. Major Recent Advance Study of current lattice data for hadron properties as a function of quark mass (c.f. vs N c ) Implies: Major surprise plus solution Can ALREADY make accurate, controlled chiral extrapolation, respecting model independent constraints of χ P T

6. Formal Chiral Expansion Formal expansion of Hadron mass: M N = c 0 + c 2 m  2 + c LNA m  3 + c 4 m  4 + c NLNA m  4 ln m  + c 6 m  6 + ….. Mass in chiral limit No term linear in m  (in FULL QCD…… there is in QQCD) First (hence “leading”) non-analytic term ~ m q 3/2 ( LNA) Source: N ! N  ! N Another branch cut from N !   ! N - higher order in m  - hence “next-to-leading” non-analytic (NLNA) Convergence?

7. Failure of Convergence of χ P T  +  m π 2 +  Nπ c.f. series expansion…. (From thesis: S. V. Wright ) For NJL model, see also: T. Hatsuda, Phys. Rev. Lett., 27 (1991) 455.

8. Relevance for Lattice data Knowing χ PT, fit with: α + β m  2 + γ m  3 (dashed curve) Problem: γ = - 0.76 c.f. model independent value -5.6 !! Best fit with γ as in χ P T ( From: Leinweber et al., Phys. Rev., D61 (2000) 074502 )

9. The Solution There is another SCALE in the problem (not natural in dim-regulated χPT ) Λ ~ 1 / Size of Source of Goldstone Boson ~ 400 – 500 MeV IF Pion Compton wavelength is smaller than source….. ( m  ≥ 0.4 – 0.5 GeV ; m q ≥ 50-60 MeV) ALL hadron properties are smooth, slowly varying (with m q ) and Constituent Quark like ! (Pion loops suppressed like (Λ / m  ) n ) WHERE EXPANSION FAILS: NEW, EFFECTIVE DEGREE OF FREEDOM TAKES OVER

10. Model Independent Non-Analytic Behaviour When Pion Compton wavelength larger than pion source: ( i.e. m  < 0.4 -0.5 GeV : m q ≤ 50 MeV ) All hadron properties have rapid, model independent, non-analytic behaviour as function of m q : m H (Full QCD) ~ m q 3/2 ; m H (QQCD) ~ m q 1/2 ; ch ~ ln m q ; μ H ~ m q 1/2 ; mag ~ 1 / m q 1/2

11. Acceptable Extrapolation Procedure 1. Must respect non-analytic behaviour of χ P T in region m  < 400 – 500 MeV (m q ≤ 50 MeV) ………..with correct coefficients! 2. Must suppress chiral behaviour as inverse power of m  in region m  > 400 – 500 MeV 3. No more parameters than lattice data can determine

12. Extrapolation of Masses At “large m  ” preserve observed linear (constituent-quark-like) behaviour: M H ~ m  2 As m  ~ 0 : ensure LNA & NLNA behaviour: ( BUT must die as (Λ / m  ) 2 for m  > Λ) Hence use: M H = α + β m  2 +  LNA (m ,Λ) +  NLNA (m ,Λ) Evaluate self-energies with form factor with  » 1/Size of Hadron

13. Behaviour of Hadron Masses with m π Lattice data from CP-PACS & UKQCD From: Leinweber et al., Phys. Rev., D61 (2000) 074502 Point of inflection at opening of N  channel BUT how model dependent is the extrapolation to the physical point? » 3M » 2M

14. Evaluation of Self-Energies: Finite Range Regulator Evaluation of Self-Energies: Finite Range Regulator Infrared behaviour is NOT affected by UV cut-off on Goldstone boson loops. e.g. for N ! N  ! N use sharp cut-off (SC),  (  – k) :  ( ,m  ) = - 3g A 2 / (16  2 f  2 ) [ m  3 arctan (  / m  ) +  3 /3 -  m  2 ] Coefficient of LNA term !  /2 as m  ! 0 But coefficient gets rapidly smaller for m  ~  (hence hard to determine by naïve fitting procedure …….. explains “  problem” discussed earlier)  ~ 1/m  2 for m  >  (/// to “Long Distance Regularization”: Donoghue et al., Phys Rev D59 (1999) 036002)

15. HOW MODEL DEPENDENT? Young, Leinweber & Thomas, hep-lat/0212031 Quantitative comparison of 6 different regulator schemes Ask over what range in m  2 is there consistency at 1% level? Use new CP-PACS data from >1 year dedicated running: - Iwasaki gluon action - perturbatively improved clover fermions - use data on two finest lattices (largest  ): a = 0.09 & 0.13 fm - follow UKQCD (Phys. Rev. D60 (’99) 034507) in setting physical scale with Sommer scale, r 0 = 0.5 fm (because static quark potential insensitive to χ’al physics) - restricted to m  2 > 0.3 GeV 2 Also use measured nucleon mass in study of model dependence

16. Regularization Schemes M N = c 0 + c 2 m  2 + c LNA m  3 + c 4 m  4 + c NLNA m  4 ln m  + c 6 m  6 + ….. DR: Naïve dimensional regularization BP: Improved dim-reg.... for  -  loop use correct branch point at m  =  m  4 ln m  ! (  2 – m  2 ) 3/2 ln (  + m  - [  2 – m  2 ] 1/2 ) -  /2 (2  2 -3m  2 ) ln m  - for m   FRR: Finite Range Regulator – use MONopole, DIPole, GAUssian and Sharp Cutoff in  -N and  -  self-energy integrals: M N = a 0 + a 2 m  2 + a 4 m  4 +  LNA ( m ,  ) +  NLNA (m ,  ) ? Is this more convergent with good choice of FRR ? Fit parameters shown in black {

17. Constraint Curves BP DR All 4 FRR curves indistinguishable. CP-PACS data Physical nucleon mass Dim-Reg curves wrong above last data point

18. Best fit (bare) parameters Regulator a 0 a 0 a 2 a 2 a 4 a 4 a 6 a 6  (GeV) DR DR 0.882 0.882 3.82 3.82 6.65 6.65-4.24 - BP BP 0.825 0.825 4.37 4.37 9.72 9.72-2.77 - SC SC 1.03 1.03 1.12 1.12-0.292 - 0.418 0.418 MON MON 1.56 1.56 0.88 0.88-0.204 - 0.496 0.496 DIP DIP 1.20 1.20 0.97 0.97-0.229 - 0.785 0.785 GAU GAU 1.12 1.12 1.01 1.01-0.247 - 0.616 0.616 N.B. Improved convergence of residual series is dramatic!

19. Finite Renormalization ) Physical Low Energy Constants True low energy constants, c 0, 2, 4, 6 should not depend on either regularization or renormalization procedure To check we need formal expansion of  LNA and  NLNA * :  LNA (m ,  ) = b 0 + b 2 m  2 + c LNA m  3 + b 4 m  4 + + …  NLNA (m ,  ) = d 0 + d 2 m  2 + d 4 m  4 + c NLNA m  4 ln m  4 + … * Complete algebraic expressions for d i, b i given in hep-lat/0212031: for all regulators Hence: c i ´ a i + b i + d i and even though each of a i, b i and d i is scheme dependent c i should be scheme independent!

20. Low Energy Parameters are Model Independent for FRR Low Energy Parameters are Model Independent for FRRRegulator c 0 c 0 c 2 c 2 c 4 c 4 DR DR 0.882 0.882 3.82 3.82 6.65 6.65 BP BP 0.885 0.885 3.64 3.64 8.50 8.50 SC SC 0.894 0.894 3.09 3.09 13.5 13.5 MON MON 0.898 0.898 2.80 2.80 23.6 23.6 DIP DIP 0.897 0.897 2.84 2.84 22.0 22.0 GAU GAU 0.897 0.897 2.87 2.87 20.7 20.7 } Note consistency of low energy parameters! Note terrible convergence properties of series….. SC not as good as other FRR } Inaccurate higher order coefficients

21. Unbiased Assessment of Schemes Take each fit, in turn, as the “exact” data Then see how well each of the other regulators can reproduce this “data” Define “goodness of fit”, χ, as area between exact “data” and best fit for given regulator over window m  2  (0, m W 2 ) e.g. with m W 2 = 0.5 GeV 2,  = 700 MeV 2 on average fit is within 1 MeV of “data”

22. Fits - 1 DR constraint curve - all regulators fit well up to 0.4 GeV 2 DIP constraint curve - DR & BP (dim-reg) fail above 0.3 GeV 2 - all FRR work over entire range!

23. Fits - 2 BP constraint curve - all regulators fit well up to 0.4 GeV 2 GAU constraint curve - DR & BP (dim-reg) fail above 0.3 GeV 2 - all FRR work over entire range!

24. Recovery of Chiral Coefficients Region leading to 1% error in M N Recovery of c 0 from fit to BP “data” over window (0,m W 2 ) All regulators work up to 0.7 GeV 2 – where BP “data” wrong anyway

25. Recovery of Chiral Coefficients - 2 Region leading to 1% error in M N Recovery of c 0 from fit to DIP “data” over window (0,m W 2 ) DR & BP fail above 0.4 GeV 2 : all FRR work over entire range!

26. Recovery of Chiral Coefficients - 3 Region leading to 1% error in M N Recovery of c 2 from fit to BP “data” over window (0,m W 2 ) All regulators work up to 0.7 GeV 2 – where BP “data” wrong anyway

27. Recovery of Chiral Coefficients - 4 Region leading to 1% error in M N Recovery of c 2 from fit to DIP “data” over window (0,m W 2 ) DR & BP fail above 0.4 GeV 2 : all FRR work over entire range!

28. Summary : Extraction of Low Energy Constants For “data” based on DR &BP: all schemes give c 0 to <1% for window up to 0.7 GeV 2 BUT dim-reg schemes FAIL at 1% level for FRR once window exceeds 0.5 GeV 2 C0:C0: C2:C2: Similar story for c 2 : - for “data” based on DR & BP all schemes good enough to give M N phys to 1% for window up to 0.7 GeV 2 - for DIP “data” DR & BP fail above 0.4 GeV 2 while all other FRR’s work over entire range (and all except SC give c 2 itself to 1% !)

29. Two-loops : Why SC is not so good Birse and McGovern * show that at two-loops (in dim-reg)  PT one obtains a term » m  5 : * J. McGovern and M. Birse, Phys Lett B446 (’99) 300 _ _ C 5 = 3g 2 2l 4 - 4 (2 d 16 –d 18 ) + g 2 + 1 --------- ---- --------------- ------ --- 32  f 2 f 2 g 8  2 f 2 8m  2 BUT (apart from tiny term) they show that its origin is the Goldberger-Treiman discrepancy i.e. same as difference: g  NN (q 2 =m  2 ) – g  NN (0) FRR : form factor (other than SC, where vanishes) already accounts for the GT discrepancy at 1-loop – e.g. for MONopole correction goes like g  NN (1 – m  2 /  2) _ _ [Savage & Beane (UW-02-023): -2.6<d 16 < - 0.17 ; -1.54 < d 18 < -0.51]

30. Application to lattice extrapolation problem Use DIP curve as the “lattice data” Take points at 0.05 GeV 2 intervals from 0.8 GeV 2 down to m L 2 Then, for each regulator, fit 4 free parameters Ask how low must m L 2 be to determine c 0,2,4 ? How low do we need to go?

31. Finite Range Regulators Fix c 0 Accurately Dimensional regularization fails at 10% level ! All FRR work at 2% level; omitting SC they work to better than 1% for m L 2 below 0.4 GeV 2 !

32. FRR Permit Accurate Determination of c 2 Too DR & BP fail miserably SC not so good MON & GAU yield 3% accuracy even for m L 2 = 0.4

33. Does it help dim-reg to lower upper limit? Instead set upper limit at 0.5 GeV 2 and use “perfect data” at intervals of 0.05 GeV 2 : Dim-reg fails even with data down to m u,d » m s /5 (m  2 » 0.1 GeV 2 ) FRR still works to fraction of a percent

34. Extraction of c 2 using smaller interval DR and BP both introduce error of 40% even with data down to m u,d » m s /5 FRR show no significant systematic error

35. Analysis of the Best Available Data Data : CP-PACS Phys Rev D65 ( 2002) Accuracy better than 1% ! Sufficient to determine 4 parameters ……..

36. Analysis of the Best Available Data

41. Scheme Dependent Fit Parameters Regulator a 0 a 0 a 2 a 2 a 4 a 4 a 6 a 6 D-R 0.700 0.700 5.10 5.10 3.87 3.87 -2.35 -2.35 - Sharp 1.06 1.06 - 0.249 - 0.249 - 0.440 0.440 Monopole 1.38 1.38 0.950 0.950 - 0.217 - 0.217 - 0.443 0.443 Dipole 1.15 1.15 0.998 0.998 - 0.227 - 0.227 - 0.732 0.732 Gaussian 1.11 1.11 1.02 1.02 - 0.234 - 0.234 - 0.593 0.593 ( Young, Leinweber & Thomas )

42. Output: Chiral Coefficients, Nucleon Mass and Sigma Commutator Regulator c 0 c 0 c 2 c 2 c 4 c 4 m N m N (MeV) (MeV)  N  N(MeV)DR 0.700 0.700 5.10 5.10 3.87 3.87 782 782 72.8 72.8 Sharp 0.892 0.892 3.16 3.16 939 939 40.9 40.9 Monopole 0.914 0.914 2.63 2.63 953 953 35.1 35.1 Dipole 0.910 0.910 2.72 2.72 951 951 36.1 36.1 Gaussian 0.906 0.906 2.79 2.79 948 948 36.9 36.9 ( Young, Leinweber & Thomas ) N.B. Full error analysis to be done and no finite volume corrections applied

43. Conclusions ♦ Study of hadron properties as function of m q in lattice QCD is extremely valuable….. Has given major qualitative & quantitative advance in understanding ! ♦ Inclusion of model independent constraints of chiral perturbation theory to get to physical quark mass is essential ♦ But radius of convergence of dim-reg χ P T is too small

44. Conclusions - 2 Chiral extrapolation problem is solved ! Use of FRR yields accurate, model independent: - physical nucleon mass - low energy constants Can use lattice QCD data over entire range m  2 up to 0.8 GeV 2 Accurate data point needed at 0.1 GeV 2 in order to reduce statistical errors

45. Conclusions - 3 Expect similar conclusion will apply to other cases where chiral extrapolation has been successfully applied - Octet magnetic moments - Octet charge radii - Moments of parton distribution functions Findings suggest new approach to quark models i.e. build Constituent Quark Model where chiral loops are suppressed and make chiral extrapolation to compare with experimental data (see Cloet et al., Phys Rev C65 (2002) 062201)

46. Baryon Masses in Quenched QCD: An Extreme Test of Our Understanding! Chiral behaviour in QQCD quite different from full QCD  ´ is an additional Goldstone Boson, so that: m N = m 0 +c 1 m  + c 2 m  2 + c LNA m  3 + c 4 m  4 + c NLNA m  4 ln m  +..… LNA term now ~ m q 1/2 origin is  ´ double pole Contribution from  ´ and 

47. Extrapolation Procedure for Nucleon in QQCD Coefficients of non-analytic terms again model independent (Given by: Labrenz & Sharpe, Phys. Rev., D64 (1996) 4595) Let: m N =  ´ +  ´ m  2 +  QQCD with same  as full QCD

48.  in QQCD LNA term linear in m   ! N  contribution has opposite sign in QQCD (repulsive) Overall  QQCD is repulsive !

49.  Bullet points  (QQCD)  N (QQCD) N Green boxes: fit evaluating  ’s on same finite grid as lattice Lines are exact, continuum results Lattice data (from MILC Collaboration) : red triangles From: Young et al., hep-lat/0111041; Phys. Rev. D66 (2002) 094507 N N N N N N N N         FULL FULL 1.24 (2) 1.24 (2) 0.92 (5) 0.92 (5) 1.43 (3) 1.43 (3) 0.75 (8) 0.75 (8) QQCD QQCD 1.23 (2) 1.23 (2) 0.85 (8) 0.85 (8) 1.45 (4) 1.45 (4) 0.71 (11)

50. Suggests Connection Between QQCD & QCD IF lattice scale is set using static quark potential (e.g. Sommer scale) (insensitive to chiral physics) Suppression of Goldstone loops for m  > Λ implies: can determine linear term ( α + β m  2 ) representing “hadronic core” : very similar in QQCD & QCD Can then correct QQCD results by replacing LNA & NLNA behaviour in QQCD by corresponding terms in full QCD Quenched QCD would then no longer be an “uncontrolled approximation” !

51. Octet Masses Octet Masses (Preliminary results from CSSM group: Young, Zanotti et al.) Fit quenched data with : α + β m  2 +  QQCD ; then let  QQCD !  QCD Errors for: Stats a ! 0 finite L NOT SHOWN

52. Title

53. Radius of Convergence of  PT is very small! (From : Hatsuda, Phys. Rev. Lett., 65 (1990) 543 : NJL c.f. χPT)

54. Output: Chiral Coefficients, Nucleon Mass and Sigma Commutator Regulator c 0 c 0 c 2 c 2 c 4 c 4 m N m N (MeV) (MeV)  N  N(MeV)DR 0.700 0.700 5.10 5.10 3.87 3.87 782 782 72.8 72.8 Sharp 0.892 0.892 3.16 3.16 12.9 12.9 939 939 40.9 40.9 Monopole 0.914 0.914 2.63 2.63 26.1 26.1 953 953 35.1 35.1 Dipole 0.910 0.910 2.72 2.72 23.4 23.4 951 951 36.1 36.1 Gaussian 0.906 0.906 2.79 2.79 21.4 21.4 948 948 36.9 36.9 ( Young, Leinweber & Thomas : to be published) N.B. Full error analysis to be done and no finite volume corrections applied

55. Title