Anthony W. ThomasAnthony W. Thomas

CSSMCSSM

Asia-Pacific Mini-Workshop on Lattice QCD Tsukuba – January 2003

Model Independent Chiral Extrapolation of Baryon Masses

OutlineOutline

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

CollaboratorsCollaborators

Derek Leinweber Ross Young (Stewart Wright)

Problem of Chiral ExtrapolationProblem of Chiral Extrapolation

Time to decrease m by factor of 2 : 27 » 100

Furthermore EFT implies ALL hadronic properties are

non-analytic functions of mq ……..

HENCE: NO simple power series expansion about mq = 0 : NO simple chiral extrapolation

♦ Currently limited to mq > 30-50 MeV

♦ NEED perhaps 500 Teraflops to get to 5 MeV !

Major Recent AdvanceMajor Recent Advance

Study of current lattice data for hadron properties as a function of quark mass (c.f. vs Nc )

Implies: Major surprise plus solution

Can ALREADY make accurate, controlled chiral

extrapolation, respecting model independent constraints

of χ P T

Formal Chiral ExpansionFormal Chiral Expansion

Formal expansion of Hadron mass:

MN = c0 + c2 m2 + cLNA m

3 + c4 m4

+ cNLNA m4 ln m + c6 m

6 + …..

Mass in chiral limit

No term linear in m

(in FULL QCD…… there is in QQCD)

First (hence “leading”)non-analytic term ~ mq

3/2

( LNA)Source: N ! N ! N

Another branch cutfrom N ! ! N- higher order in m

- hence “next-to-leading” non-analytic (NLNA)

Convergence?

Failure of Convergence of Failure of Convergence of χχ P T 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.

Relevance for Lattice dataRelevance for Lattice dataKnowing χ PT , fit with: α + β m

2 + γ m3 (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 )

The SolutionThe 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 ; 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

Model Independent Model Independent Non-Analytic BehaviourNon-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) ~ mq

1/2 ;

<r2 >ch ~ ln mq ; μH ~ mq

1/2 ; <r2 >mag ~ 1 / mq1/2

Acceptable Extrapolation ProcedureAcceptable 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

Extrapolation of MassesExtrapolation of Masses

At “large m” preserve observed linear (constituent-quark-like) behaviour: MH ~ m

2

As m~ 0 : ensure LNA & NLNA behaviour:

( BUT must die as (Λ / m )2 for m > Λ)

Hence use:

MH = α + β m2 + LNA(m ,Λ) + NLNA(m ,Λ)

• Evaluate self-energies with form factor with » 1/Size of Hadron

Behaviour of Hadron Masses with mBehaviour of Hadron Masses with mππ

Lattice data fromCP-PACS & UKQCD

From: Leinweber et al., Phys. Rev., D61 (2000) 074502

Point ofinflection at opening of N channel

BUT how model dependent is the extrapolation to the physical point?

» 3M

» 2M

Evaluation of Self-Energies: Finite Range RegulatorEvaluation 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) [ m3 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/m2 for m >

(/// to “Long Distance Regularization”: Donoghue et al., Phys Rev D59 (1999) 036002)

HOW MODEL DEPENDENT?HOW MODEL DEPENDENT?

Young, Leinweber & Thomas, hep-lat/0212031

• Quantitative comparison of 6 different regulator schemes

• Ask over what range in m2 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

Regularization SchemesRegularization Schemes

MN = c0 + c2 m2 + cLNA m

3 + c4 m4 + cNLNA m

4 ln m + c6 m6 + …..

DR: Naïve dimensional regularization

BP: Improved dim-reg.... for - loop use correct branch point at m=

m4 ln m ! (2 – m

2)3/2 ln ( + m- [2 – m2]1/2) - /2 (22 -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 m2 + a4 m

4 + LNA( m, ) + NLNA(m, )

? Is this more convergent with good choice of FRR ?

Fit parameters shown in black{

Constraint CurvesConstraint Curves

BP

DR

All 4 FRR curves indistinguishable.

CP-PACS data

Physicalnucleon mass

Dim-Reg curves wrong above last data point

Best fit (bare) parametersBest fit (bare) parameters

RegulatorRegulator aa00 aa22 aa44 aa66 (GeV)(GeV)

DRDR 0.8820.882 3.823.82 6.656.65 -4.24-4.24 --

BPBP 0.8250.825 4.374.37 9.729.72 -2.77-2.77 --

SCSC 1.031.03 1.121.12 -0.292-0.292 -- 0.4180.418

MONMON 1.561.56 0.880.88 -0.204-0.204 -- 0.4960.496

DIPDIP 1.201.20 0.970.97 -0.229-0.229 -- 0.7850.785

GAUGAU 1.121.12 1.011.01 -0.247-0.247 -- 0.6160.616

N.B. Improved convergence of residual series is dramatic!

Finite Renormalization Finite Renormalization )) Physical Low Energy Constants 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 m2 + cLNA m

3 + b4 m4 + + …

NLNA(m,) = d0 + d2 m2 + d4 m

4 + cNLNAm4 ln m

4 + …

* Complete algebraic expressions for di , bi 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!

Low Energy Parameters Low Energy Parameters areare Model Model Independent for FRRIndependent for FRR

RegulatorRegulator cc00 cc22 cc44

DRDR 0.8820.882 3.823.82 6.656.65

BPBP 0.8850.885 3.643.64 8.508.50

SCSC 0.8940.894 3.093.09 13.513.5

MONMON 0.8980.898 2.802.80 23.623.6

DIPDIP 0.8970.897 2.842.84 22.022.0

GAUGAU 0.8970.897 2.872.87 20.720.7

}• Note consistency of low energy parameters!

• Note terrible convergence properties of series…..

• SC not as good as other FRR

}

• Inaccurate higher order coefficients

Unbiased Assessment of SchemesUnbiased 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”

Fits - 1Fits - 1

DR constraint curve - all regulators fit well up to 0.4 GeV2

DIP constraint curve - DR & BP (dim-reg) fail above 0.3 GeV2

- all FRR work over entire range!

Fits - 2Fits - 2

BP constraint curve - all regulators fit well up to 0.4 GeV2

GAU constraint curve - DR & BP (dim-reg) fail above 0.3 GeV2

- all FRR work over entire range!

Recovery of Chiral CoefficientsRecovery of Chiral Coefficients

Region leading to 1%error in MN

Recovery of c0 from fit to BP “data” over window (0,mW2)

All regulators work up to 0.7 GeV2 – where BP “data” wrong anyway

Recovery of Chiral Coefficients - 2Recovery of Chiral Coefficients - 2

Region leading to 1%error in MN

Recovery of c0 from fit to DIP “data” over window (0,mW2)

DR & BP fail above 0.4 GeV2 : all FRR work over entire range!

Recovery of Chiral Coefficients - 3Recovery of Chiral Coefficients - 3

Region leading to 1%error in MN

Recovery of c2 from fit to BP “data” over window (0,mW2)

All regulators work up to 0.7 GeV2 – where BP “data” wrong anyway

Recovery of Chiral Coefficients - 4Recovery of Chiral Coefficients - 4

Region leading to 1%error in MN

Recovery of c2 from fit to DIP “data” over window (0,mW2)

DR & BP fail above 0.4 GeV2 : all FRR work over entire range!

Summary : Extraction of Low Energy ConstantsSummary : 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:

C2:

• Similar story for c2:

- for “data” based on DR & BP all schemes good enough to give MN

phys 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% !)

Two-loops : Why SC is not so goodTwo-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

_ _C5 = 3g2 2l4 - 4 (2 d16 –d18) + g2 + 1 --------- ---- --------------- ------ --- 32 f2 f2 g 8 2 f2 8m

2

• 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)

_ _[Savage & Beane (UW-02-023): -2.6<d16 < - 0.17 ; -1.54 < d18 < -0.51]

Application to latticeApplication to lattice extrapolation extrapolation problemproblem

•Use DIP curve as the “lattice data”

• Take points at 0.05 GeV2 intervals from 0.8 GeV2 down to mL2

• Then, for each regulator, fit 4 free parameters

• Ask how low must mL2 be to determine c0,2,4 ?

How low do we need to go?How low do we need to go?

Finite Range Regulators Fix cFinite Range Regulators Fix c0 0 AccuratelyAccurately

Dimensional regularization fails at 10% level !

• All FRR work at 2% level; omitting SC they work to better than 1% for mL

2 below 0.4 GeV2 !

FRR Permit Accurate Determination of cFRR Permit Accurate Determination of c2 2 TooToo

• DR & BP fail miserably

• SC not so good

• MON & GAU yield 3% accuracy even for mL

2 = 0.4

Does it help dim-reg to lower upper limit?Does it help dim-reg to lower upper limit?

Instead set upper limit at 0.5 GeV2 and 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

Extraction of cExtraction of c2 2 using smaller intervalusing 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

Analysis of the Best Available DataAnalysis of the Best Available Data

Data :CP-PACS Phys Rev D65 ( 2002)

Accuracybetter than1% !

Sufficient to determine 4 parameters ……..

Analysis of the Best Available DataAnalysis of the Best Available Data

Analysis of the Best Available DataAnalysis of the Best Available Data

Analysis of the Best Available DataAnalysis of the Best Available Data

Analysis of the Best Available DataAnalysis of the Best Available Data

Analysis of the Best Available DataAnalysis of the Best Available Data

Scheme Dependent Fit ParametersScheme Dependent Fit Parameters

RegulatorRegulator aa00 aa22 aa44 aa66

D-RD-R 0.7000.700 5.105.10 3.873.87 -2.35-2.35 --

SharpSharp 1.061.06 1.061.06 - 0.249- 0.249 -- 0.4400.440

MonopoleMonopole 1.381.38 0.9500.950 - 0.217- 0.217 -- 0.4430.443

DipoleDipole 1.151.15 0.9980.998 - 0.227- 0.227 -- 0.7320.732

GaussianGaussian 1.111.11 1.021.02 - 0.234- 0.234 -- 0.5930.593

( Young, Leinweber & Thomas )

Output: Chiral Coefficients, Nucleon Output: Chiral Coefficients, Nucleon Mass and Sigma CommutatorMass and Sigma Commutator

RegulatorRegulator cc00 cc22 cc44 mmNN

(MeV)(MeV)

NN

(MeV)(MeV)

DRDR 0.7000.700 5.105.10 3.873.87 782782 72.872.8

SharpSharp 0.8920.892 3.163.16 939939 40.940.9

MonopoleMonopole 0.9140.914 2.632.63 953953 35.135.1

DipoleDipole 0.9100.910 2.722.72 951951 36.136.1

GaussianGaussian 0.9060.906 2.792.79 948948 36.936.9

( Young, Leinweber & Thomas )

N.B. Full error analysis to be done and no finite volume corrections applied

ConclusionsConclusions

♦ 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

Conclusions - 2Conclusions - 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

Conclusions - 3Conclusions - 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)

Baryon Masses in Quenched QCD:Baryon Masses in Quenched QCD:An Extreme Test of Our Understanding!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 m3 + c4 m

4 + cNLNA m4 ln m +..…

LNA term now ~ mq1/2

origin is ´ double pole

Contribution from ´ and

Extrapolation Procedure for Nucleon in Extrapolation Procedure for Nucleon in QQCDQQCD

Coefficients of non-analytic terms again model independent

(Given by: Labrenz & Sharpe, Phys. Rev., D64 (1996) 4595)

Let:

m N = ´ + ´ m2+QQCD

with same as

full QCD

in QQCDin QQCD

LNA term linear in m

! N contributionhas opposite sign in QQCD (repulsive)

Overall QQCD

is repulsive !

Bullet pointsBullet 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

NN NN

FULLFULL 1.24 (2)1.24 (2) 0.92 (5)0.92 (5) 1.43 (3)1.43 (3) 0.75 (8)0.75 (8)

QQCDQQCD 1.23 (2)1.23 (2) 0.85 (8)0.85 (8) 1.45 (4)1.45 (4) 0.71 (11)0.71 (11)

Suggests Connection BetweenSuggests Connection Between QQCD & QCD 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” !

Octet MassesOctet Masses

(Preliminary results from CSSM group: Young, Zanotti et al.)

Fit quenched data with : α + β m2 + QQCD ; then let QQCD ! QCD

Errors for:•Stats•a ! 0•finite LNOT SHOWN

TitleTitle

Radius of Convergence of Radius of Convergence of PT is very PT is very small!small!

(From : Hatsuda, Phys. Rev. Lett., 65 (1990) 543 : NJL c.f. χPT)

Output: Chiral Coefficients, Nucleon Output: Chiral Coefficients, Nucleon Mass and Sigma CommutatorMass and Sigma Commutator

RegulatorRegulator cc00 cc22 cc44 mmNN

(MeV)(MeV)

NN

(MeV)(MeV)

DRDR 0.7000.700 5.105.10 3.873.87 782782 72.872.8

SharpSharp 0.8920.892 3.163.16 12.912.9 939939 40.940.9

MonopoleMonopole 0.9140.914 2.632.63 26.126.1 953953 35.135.1

DipoleDipole 0.9100.910 2.722.72 23.423.4 951951 36.136.1

GaussianGaussian 0.9060.906 2.792.79 21.421.4 948948 36.936.9

( Young, Leinweber & Thomas : to be published)

N.B. Full error analysis to be done and no finite volume corrections applied

TitleTitle