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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
1 / 34
Introduction Masses gA 〈x〉 Conclusions
Outline
Introduction
Masses
Nucleon axial coupling
Moments of unpolarised structure functions
Conclusions
2 / 34
Introduction Masses gA 〈x〉 Conclusions
Content
Introduction
Masses
Nucleon axial coupling
Moments of unpolarised structure functions
Conclusions
3 / 34
Introduction Masses gA 〈x〉 Conclusions
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
4 / 34
Introduction Masses gA 〈x〉 Conclusions
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
5 / 34
Introduction Masses gA 〈x〉 Conclusions
Content
Introduction
Masses
Nucleon axial coupling
Moments of unpolarised structure functions
Conclusions
6 / 34
Introduction Masses gA 〈x〉 Conclusions
Introduction
I mN and m∆ are “golden plate” quantitiesI Test case for control on systematic errorsI BχPT ⇒ non-trivial quark mass dependence
7 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
8 / 34
Introduction Masses gA 〈x〉 Conclusions
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
9 / 34
Introduction Masses gA 〈x〉 Conclusions
Comparison with lattice results
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
10 / 34
Introduction Masses gA 〈x〉 Conclusions
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
]
11 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
12 / 34
Introduction Masses gA 〈x〉 Conclusions
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
13 / 34
Introduction Masses gA 〈x〉 Conclusions
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
14 / 34
Introduction Masses gA 〈x〉 Conclusions
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
15 / 34
Introduction Masses gA 〈x〉 Conclusions
Content
Introduction
Masses
Nucleon axial coupling
Moments of unpolarised structure functions
Conclusions
16 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
17 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
18 / 34
Introduction Masses gA 〈x〉 Conclusions
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 .
19 / 34
Introduction Masses gA 〈x〉 Conclusions
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
20 / 34
Introduction Masses gA 〈x〉 Conclusions
Comparison with lattice results
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)
21 / 34
Introduction Masses gA 〈x〉 Conclusions
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
22 / 34
Introduction Masses gA 〈x〉 Conclusions
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
23 / 34
Introduction Masses gA 〈x〉 Conclusions
Content
Introduction
Masses
Nucleon axial coupling
Moments of unpolarised structure functions
Conclusions
24 / 34
Introduction Masses gA 〈x〉 Conclusions
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 ]
25 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
26 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
27 / 34
Introduction Masses gA 〈x〉 Conclusions
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).
28 / 34
Introduction Masses gA 〈x〉 Conclusions
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)
29 / 34
Introduction Masses gA 〈x〉 Conclusions
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
30 / 34
Introduction Masses gA 〈x〉 Conclusions
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
31 / 34
Introduction Masses gA 〈x〉 Conclusions
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]
32 / 34
Introduction Masses gA 〈x〉 Conclusions
Content
Introduction
Masses
Nucleon axial coupling
Moments of unpolarised structure functions
Conclusions
33 / 34
Introduction Masses gA 〈x〉 Conclusions
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
34 / 34
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