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Anthony W. Thomas CSSM

Model Independent Chiral Extrapolation of Baryon Masses. Anthony W. Thomas CSSM. Asia-Pacific Mini-Workshop on Lattice QCD Tsukuba – January 2003. Outline. PROBLEM: Lack of computer power. ) need chiral extrapolation.

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Anthony W. Thomas CSSM

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

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

  3. Collaborators (Stewart Wright) Derek Leinweber Ross Young

  4. Problem of Chiral Extrapolation ♦Currently limited to mq > 30-50 MeV Time to decrease m by factor of 2 : 27 » 100 ♦NEED perhaps 500 Teraflops to get to 5 MeV ! Furthermore EFT implies ALL hadronic properties are non-analytic functionsof mq …….. HENCE:NO simple power series expansion about mq = 0 : NO simple chiral extrapolation

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

  6. Formal Chiral Expansion Formal expansion of Hadron mass: MN = c0 + c2 m2 + cLNA m3+ c4 m4 + cNLNA m4 ln m + c6 m6 + ….. Mass in chiral limit • Another branch cut • from N!! N • higher order in m • hence “next-to-leading” • non-analytic (NLNA) First (hence “leading”) non-analytic term ~ mq3/2 (LNA) Source: N ! N ! N No term linear in m(in FULL QCD…… there is in QQCD) Convergence?

  7. Failure of Convergence of χ P T  +  mπ2 + 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) Best fit with γ as in χ P T Problem: γ = - 0.76 c.f. model independent value -5.6 !! ( From: Leinweber et al., Phys. Rev., D61 (2000) 074502 )

  9. The Solution There is another SCALE in the problem (not natural in dim-regulatedχPT )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 ; mq ≥ 50-60 MeV)ALL hadron properties are smooth, slowly varying (with mq )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 : mq ≤ 50 MeV ) All hadron properties have rapid, model independent, non-analytic behaviour as function of mq : mH (Full QCD) ~ mq3/2 ; mH (QQCD) ~ mq1/2 ; <r2 >ch~ ln mq ; μH ~ mq1/2 ; <r2 >mag ~ 1 / mq1/2

  11. Acceptable Extrapolation Procedure 1. Must respect non-analytic behaviour of χ P T in region m< 400 – 500 MeV (mq ≤ 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: MH ~ m2 As m~ 0 : ensure LNA & NLNA behaviour: (BUTmust die as (Λ / m )2 for m> Λ) Hence use: MH = 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 mπ From: Leinweber et al., Phys. Rev., D61 (2000) 074502 » 3M Point of inflection at opening of N channel Lattice data fromCP-PACS & UKQCD » 2M BUT how model dependent is the extrapolation to the physical point?

  14. 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 ) = - 3gA2 / (16 2 f2) [ m3 arctan ( / m) + 3/3 -  m2 ] (/// to “Long Distance Regularization”: Donoghue et al., Phys Rev D59 (1999) 036002) • 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 > 

  15. HOW MODEL DEPENDENT? • 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, r0 = 0.5 fm (because static quark potential insensitive to χ’al physics) - restricted to m2 > 0.3 GeV2 • Also use measured nucleon mass in study of model dependence Young, Leinweber & Thomas, hep-lat/0212031

  16. { Fit parameters shown in black Regularization Schemes DR: Naïve dimensional regularization MN = c0 + c2m2 + cLNA m3 + c4 m4 + cNLNA m4 ln m+ c6 m6 + ….. 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 <  ; ln ! arctan for m >  FRR: Finite Range Regulator – use MONopole, DIPole, GAUssian and Sharp Cutoff in -N and - self-energy integrals: MN = a0 + a2 m2 + a4 m4 + LNA( m, ) + NLNA(m, ) ? Is this more convergent with good choice of FRR ?

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

  18. Best fit (bare) parameters 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,) = b0 + b2 m2 + cLNA m3 + b4 m4 + + … NLNA(m,) = d0 + d2 m2 + d4 m4 + cNLNAm4 ln m4 + … • 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! * Complete algebraic expressions for di , bi given in hep-lat/0212031: for all regulators

  20. Low Energy Parameters are Model Independent for FRR • Inaccurate higher order coefficients } • SC not as good as other FRR • Note consistency of low energy parameters! } • Note terrible convergence properties of series…..

  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, mW2) • e.g. with mW2 = 0.5 GeV2,  = 700 MeV2 on average fit is within 1 MeV of “data”

  22. Fits - 1 DIP constraint curve - DR & BP (dim-reg) fail above 0.3 GeV2- all FRR work over entire range! DR constraint curve - all regulators fit well up to 0.4 GeV2

  23. Fits - 2 GAU constraint curve - DR & BP (dim-reg) fail above 0.3 GeV2- all FRR work over entire range! BP constraint curve - all regulators fit well up to 0.4 GeV2

  24. Recovery of Chiral Coefficients Recovery of c0 from fit to BP “data” over window (0,mW2) Region leading to 1% error in MN All regulators work up to 0.7 GeV2 – where BP “data” wrong anyway

  25. Recovery of Chiral Coefficients - 2 Recovery of c0 from fit to DIP “data” over window (0,mW2) Region leading to 1% error in MN DR & BP fail above 0.4 GeV2 : all FRR work over entire range!

  26. Recovery of Chiral Coefficients - 3 Recovery of c2 from fit to BP “data” over window (0,mW2) Region leading to 1% error in MN All regulators work up to 0.7 GeV2 – where BP “data” wrong anyway

  27. Recovery of Chiral Coefficients - 4 Recovery of c2 from fit to DIP “data” over window (0,mW2) Region leading to 1% error in MN DR & BP fail above 0.4 GeV2 : all FRR work over entire range!

  28. Summary : Extraction of Low Energy Constants • For “data” based on DR &BP:all schemes give c0 to <1% for window up to 0.7 GeV2 • BUT dim-reg schemes FAIL at 1% level for FRR once window exceeds 0.5 GeV2 C0: • Similar story for c2:- for “data” based on DR & BP all schemes good enough to • give MNphys to 1% for window up to 0.7 GeV2 - for DIP “data” DR & BP fail above 0.4 GeV2 while all other FRR’s work over entire range (and all except SC give c2 itself to 1% !) C2:

  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 : _ _ C5 = 3g2 2l4 - 4 (2 d16 –d18) + g2 + 1 --------- ---- --------------- ------ --- 32  f2 f2 g 8 2 f2 8m2 _ _ [Savage & Beane (UW-02-023): -2.6<d16 < - 0.17 ; -1.54 < d18 < -0.51] • BUT (apart from tiny term) they show that its • origin is the Goldberger-Treiman discrepancy i.e. same as difference: g NN (q2=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) * J. McGovern and M. Birse, Phys Lett B446 (’99) 300

  30. Application to lattice extrapolation problem How low do we need to go? • Use DIP curve as the “lattice data” • Take points at 0.05 GeV2intervals from 0.8 GeV2down to mL2 • Then, for each regulator, fit 4 free parameters • Ask how low must mL2 be to determine c0,2,4 ?

  31. Finite Range Regulators Fix c0 Accurately • All FRR work at 2% level; omitting SC they work to better than 1% for mL2 below 0.4 GeV2 ! Dimensional regularization fails at 10% level !

  32. FRR Permit Accurate Determination of c2 Too • DR & BP fail miserably • SC not so good • MON & GAU yield 3% accuracy even for mL2 = 0.4

  33. Does it help dim-reg to lower upper limit? Instead set upper limit at 0.5 GeV2and use “perfect data” at intervals of 0.05 GeV2 : • Dim-reg fails even with data down to mu,d » ms /5 (m2» 0.1 GeV2) • FRR still works to fraction of a percent

  34. Extraction of c2 using smaller interval • DR and BP both introduce error of 40% even with data down to mu,d » ms /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

  37. Analysis of the Best Available Data

  38. Analysis of the Best Available Data

  39. Analysis of the Best Available Data

  40. Analysis of the Best Available Data

  41. Scheme Dependent Fit Parameters ( Young, Leinweber & Thomas )

  42. Output: Chiral Coefficients, Nucleon Mass and Sigma Commutator ( 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 mq 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 GeV2 • Accurate data point needed at 0.1 GeV2 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 +c1 m + c2 m2 + cLNA m3 + c4 m4 + cNLNA m4 ln m +..… Contribution from ´ and  LNA term now ~ mq1/2 origin is ´ double pole

  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. Lattice data (from MILC Collaboration) : red triangles • Green boxes: fit evaluating ’s on same finite grid as lattice • Lines are exact, continuum results  (QQCD) • Bullet points  N (QQCD) N From: Young et al., hep-lat/0111041; Phys. Rev. D66 (2002) 094507

  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 & NLNAbehaviour in QQCD by corresponding terms in full QCD Quenched QCD would then no longer be an “uncontrolled approximation” !

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