probing the chiral limit with clover fermions ii: the ... fileintroduction masses ga hxi conclusions...

Introduction Masses gA 〈x〉 Conclusions

Probing the chiral limit with clover fermions II:The baryon sector

M. Göckeler, P. Hägler, R. Horsley, Y. Nakamura, M. Othani,D. Pleiter, P.E.L. Rakow, A. Schäfer, G. Schierholz,

W. Schroers, H. Stüben, J.M. Zanotti

(QCDSF collaboration)

DESY, Zeuthen

Lattice 2007, Regensburg3 August 2007

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Outline

Introduction

Masses

Nucleon axial coupling

Moments of unpolarised structure functions

Conclusions

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Content

Introduction

Masses



Conclusions

3 / 34


Introduction

I Today simulations with light dynamical Wilson fermions feasible• Recent algorithmic improvements• Availability of new generation of computers

I Simulations sufficiently close to chiral limit?I Control on systematic errors?

• Finite size corrections• Discretisation errors• Chiral extrapolations

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Sommer scale r0

0 0.1 0.2 0.3

(a mPS

)2

4

5

6

7

r 0 / a

I Global fit ansatz:

lnr0

a= A0(β) + A2(β) am2

PS

where

Ai(β) = Ai0+Ai1(β−β0)+Ai2(β−β0)2

β0 = 5.29, amPS < 0.5, χ2/Nd.o.f. = 6.7/12

I Here we use r c0 = r0(mPS = 0) and set r0 = 0.467 fm

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Content

Introduction

Masses



Conclusions

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Introduction

I mN and m∆ are “golden plate” quantitiesI Test case for control on systematic errorsI BχPT ⇒ non-trivial quark mass dependence

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Quark mass dependence of the N mass

I p-expansion at O(p4) gives (in infinite volume):

[A. Ali Khan et al. (QCDSF), 2003]

mN(mPS) = M0 − 4c1m2PS −

3gA,02

32πF 20

m3PS

+

[er

1(λ)− 364π2F0

2

(gA,0

2

M0− c2

2

)−

332π2F0

2

(gA,0

2

M0− 8c1 + c2 + 4c3

)ln

mPS

λ

]m4

PS

+3gA,0

2

256πF02M0

2 m5PS + O(m6

PS)

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Comparison with lattice results

I Not sufficient lattice results for 0 < mPS . 350 MeV available tofit all parameters

I Use phenomenological input for• Leading order LECs:

• Pion decay constant F0: O(0.5%) uncertainty• Nucleon axial coupling gA,0: O(5%) uncertainty

• Higher order LECs:• c2: O(6%) uncertainty• c3: O(25%) uncertainty

I Free fit parameters: M0, c1, er1(λ = 1GeV)

I Restrict fit interval to 0 ≤ mPS ≤ 650 MeV

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0 0.2 0.4 0.6 0.8

m2

PS [GeV

2]

0.8

1

1.2

1.4

1.6

1.8

2

mN

[G

eV]

I Fit to results withmPS < 650MeV

I Error analysis: only statisticalerrors of lattice results

I Fit parameters consistent withexpectations, e.g.

• c1 = −1.02(7) GeV−1 vs.χPT analysis:c1 = −0.9+0.2

−0.5 GeV−1

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Finite size effects

I χPT allows for calculation of finite size effects

mN(L)−mN(∞) = ∆a(L) + ∆b(L) + O(p5)

I O(p3) result for volume dependence:

3g2A,0M0m2

PS

16π2F 20

∫ ∞

0dx∑~n

′ K0

(L|~n|

√M2

0 x2 + m2PS(1− x)

)

I O(p4) corrections:

∆b(L) =3m4

PS

4π2F 20

∑~n

′[(2c1 − c3)

K1(L|~n|mPS)

L|~n|mPSi + c2

K2(L|~n|mPS)

(L|~n|mPS)2

]

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Comparison with lattice data

I In formulae describing volume dependence no new coefficientshave been introduced

I Consider δ(L) = mN(L)−mN(∞)mN(∞)

I Compare “predicted” volume dependence with lattice results:(for mPS ' 590 MeV)

1 1.5 2 2.5 3L [fm]

0

0.2

0.4

δ mN

O(p3)

O(p4)

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Scale dependence

I Repeat analysis for different values of r0:

0.38 0.39 0.4 0.41 0.42 0.43 0.44

r0

-1 [GeV

-1]

0.85

0.9

0.95

mN

[G

eV]

mN

= mN

phys

r0 = 0.5 fm

I Lattice and experimental results agree for r0 = 0.457(3) fm

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Quark mass dependence of the ∆(1232) mass

0 0.1 0.2 0.3 0.4 0.5

m2

PS [GeV

2]

1

1.2

1.4

1.6

1.8

2

m∆ [

GeV

]

I Lower orders of χPT:

m∆(mPS) = M∆,0−

4a1m2PS−

332πF 2

0

25h2A

81m3

PS

I Fit data for0 ≤ mPS ≤ 650 MeV

I Fits parameters consistentwith expectations:

• a1 = −0.8(3) GeV−1 ' c1• hA = 1.5(4) . 9gA/5

I No investigation of FSE, yet

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Discussion

I Finite size correction of mN start to become relevant for oursimulation parameters

• Finite size corrections of m∆ still to be investigatedI Discretisation errors seem to be smallI χPT not expected to converge in considered quark mass region

• Nevertheless good agreement foundI From mN(mPS = mπ) = mexp

N Ô r0 = 0.457(3) fm

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Content

Introduction

Masses



Conclusions

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Introduction

I Form factor of the nucleon axial current:

〈p′, s′|A(u)µ |p, s〉 =

u(p′, s′)[γµγ5GA(Q2) + γ5

qµ

2mNGP(Q2)

]u(p, s)

I Axial coupling constant: gA = GA(0)

I Consider non-singlet case (no disconnected contributions):

〈p, s|Au−dµ |p, s〉 = 2gAsµ = 2(∆u −∆d)sµ

I Non-perturbative renormalisation (using perturbative bA)

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Quark mass dependence

[QCDSF, Phys.Rev. D74 (2006)]

I Evaluation of matrix element of iso-vector axial current withinHBχPT feasible:

gA(mPS)

I Calculation for finite spatial cubic box of length L:

gA(mPS, L) = gA(mPS) + ∆gA(mPS, L)

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Infinite volume limit

gA(mPS) = gA,0 −(gA,0)

3m2PS

16π2F02

+ 4

{CSSE(λ) +

cA2

4π2F02

[155972

g1 −1736

gA,0

]+ γSSE ln

mPS

λ

}m2

PS

+4cA

2gA,0

27πF02∆0

m3PS +

827π2F0

2 cA2gA,0m2

PS

√1−

m2PS

∆20

ln R

+cA

2∆20

81π2F02

(25g1 − 57gA,0

){ln[

2∆0

mPS

]−

√1−

m2PS

∆20

ln R

}+O(ε4)

where

γSSE =1

16π2F02

»5081

cA2g1 −

12

gA,0 −29

cA2gA,0 − gA,0

3–

,

R = ∆0/mPS +q

∆02/m2

PS − 1 .

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Fit ansatz

I Lattice results not sufficient to fix all parameters→ phenomenological input required:

• Pion decay constant F0: O(0.5%) uncertainty• N∆ mass splitting ∆0: rather large uncertainty (271 · · ·330 MeV)• Axial N∆ coupling cA: O(5%) uncertainty• Coupling Br

20(λ)

I Free fit parameters: gA,0, B9(λ = mπ), g1

I Finite size correction can not be ignored→ fit to finite size corrected results

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0 0.1 0.2 0.3 0.4 0.5

m2

PS [GeV

2]

0.9

1

1.1

1.2

1.3

g A

I Strong quark massdependence0 ≤ mPS ≤ 300 MeV

I Fits parameters consistentwith expectations:

Phen. LatticegA,0 1.2(1) 1.20(8)B9 [GeV−2] -1.4(12) -1.1(2)g1 O(2) 3.6(10)

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Finite size corrections

0 0.1 0.2 0.3 0.4 0.5

m2

PS [GeV

2]

0.9

1

1.1

1.2

1.3

g A

L=1.61 fmL=1.91 fmL=2.41 fmL=∞

I Results for lightestquark mass atβ = 5.25, 5.29, 5.40

I Consistent descriptionof data assuming finitesize effects

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Discussion

I Lattice results for gA smaller then phenomenological valueI Indications for this to be due to

• Strong quark mass dependence for mPS . 400 MeV• Large finite size corrections

I Little control on discretization effectsI Need L & 2.5 fm to explore mPS . 300 MeV region

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Content

Introduction

Masses



Conclusions

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Introduction

I First moment of PDF Hq(x , ξ, Q2) at ξ = 0:∫dx x Hq(x , 0, Q2) = Aq

2,0(Q2)

I Here we consider only Q2 = 0 case:

〈x〉 = Aq2,0(0)

I Aq2,0 obtained from matrix element

〈N(~p)|[u γ{µ1

↔D

µ2}u]|N(~p)〉 := 2Aq

2,0 [pµ1pµ2 ]

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RenormalisationI Results have to be renormalised using a scheme (e.g. MS) and

scale (e.g. 2 GeV):

OMS(µ = 2 GeV) =(∆Z MS

O (µ = 2GeV))−1

∆Z latO (a) Olat(a)

(lattice, a)Lattice to RGI ⇓ ∆Z lat

O (a)

RGI

RGI to MS ⇓(∆Z MS

O (µ = 2GeV))−1

(MS, 2GeV)

• Non-perturbative determination of ∆Z latO (a) (RI’-MOM)

• Perturbative determination of(∆Z MS

O (µ = 2GeV))−1

(Using non-perturbatively determined ΛMS)

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Quark mass dependence from BχPT: isovector case[Dorati et al., nucl-th/0703073v1]

〈x〉(u−d)(mPS) = av2,0+

av2,0m2

PS

(4πFπ)2

{−(3gA,0

2+1) logm2

PSλ2 −2gA,0

2+gA,02 m2

PSM0

2

(1+3 log

m2PS

M02

)− 1

2gA,0

2 m4PS

M04 log

m2PS

M02

+ gA,02 mPS√

4M02 −m2

PS

(14− 8

m2PS

M02 +

m4PS

M04

)arccos

(mPS

2M0

)}

+∆av

2,0gA,0m2PS

3(4πF0)2

{2

m2PS

M02

(1 + 3 log

m2PS

M02

)−

m4PS

M04 log

m2PS

M02

+2mPS(4M0

2 −m2PS)

32

M04 arccos

(mPS

2M0

)}+ 4m2

PSc(r)

8 (λ)

M02 +O(p3)

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Quark mass dependence: isoscalar case

[Dorati et al., nucl-th/0703073v1]

〈x〉(u+d)(mPS) = a(s)2,0 + 4

c9

M02 m2

PS−

−3a2,0

(s)gA,02m2

PS

16π2F02

[m2

PSM0

2 +m2

PSM0

2

(2−

m2PS

M02

)log(

mPS

M0

)

+mPS√

4M02 −m2

PS

(2− 4

m2PS

M02 +

m4PS

M04

)arccos

(mPS

2M0

)]+O(p3).

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Parameters

I Phenomenological input parameters• F0, M0, gA• ∆av

2,0

I Free fit parameters: av2,0, as

2,0, c8(λ = 1 GeV), c9(λ = 1 GeV)

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Comparison with lattice results: isovector case

0 0.25 0.5 0.75 1

mPS

2 [GeV

2]

0.1

0.2

0.3

0.4

v 2bMS (2

GeV

)

β = 5.29, u-d

0 0.02 0.04

(a / r0

c )2

u-d

I Small number of parameters → fit results for same β

I Discretisation errors seem to be small

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Comparison with lattice results: isoscalar case

0 0.25 0.5 0.75 1

mPS

2 [GeV

2]

0.5

0.6

0.7

0.8v 2bM

S (2 G

eV)

β = 5.29, u+d

0 0.02 0.04

(a / r0

c )2

u+d

I Discretisation errors again smallI Indications for “bending-down”I But: neglected disconnected contributions

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Discussion

I Large uncertainties in chiral extrapolationsI Discretisation errors seem to be smallI Lattice volume large enough?I Chiral properties of fermions relevant?

0 0.2 0.4 0.6 0.8mΠ

2 @GeV2D

0.1

0.15

0.2

0.25

0.3

<x>

u-

d

L = ¥

L = 6 fm

L = 4 fm

L = 3 fm

Finite size effects predictions Results using quenched[W. Detmold et al., 2003] overlap fermions

[T. Streuer (QCDSF), 2005]

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Content

Introduction

Masses



Conclusions

33 / 34


Conclusions

I We presented most recent QCDSF results for mN , m∆, gA and〈x〉

I Progress on control of systematic errors• Finite size corrections

• Small for mN

• Large but essentially under control for gA

• Still to be investigated for m∆, 〈x〉• Discretisation errors

• Seem to be small• Limited control so far

• Quark mass dependence• Fits to χPT results → reasonable parameters values• Lattice results essentially outside quark mass region where χPT is

expected to converge

I Exploring quark mass region mPS . 300 MeV with Cloverfermions starts to become feasible

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probing the chiral limit with clover fermions ii: the ... fileintroduction masses ga hxi conclusions...

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