Hadron Phenomenology from QuenchedOverlap Fermions

D. Galletly1, M. Gürtler2, R. Horsley1, K. Koller3, P. Rakow4,G. Schierholz2, T. Streuer2

2NIC/DESY Zeuthen4University of Liverpool

1University of Edinburgh3University of Munich

Nicosia, Semptember 2005

Outline

Introduction

Hadron Masses

Non-Perturbative Renormalisation

Nucleon Structure

Simulation DetailsGauge action: Tadpole-Improved Lüscher-Weisz

S = β∑

plaq

(1 − 13 re tr Uplaq) + βrect

∑

rect

(1 − 13 re tr Urect)

+βpar

∑

par

(1 − 13 re tr Upar)

Fermion action: Overlap Operator

D = ρ(1 + γ5sgn(HW (−ρ)))

Lattices:V β confs.

12324 7.60 15016332 8.00 30024348 8.45 200

Pion mass range: 300MeV..850MeV

Simulation DetailsTwo Point Function:

C2(t) = 12(1 + γ4)αβ〈Bα(t)Bβ(0)〉

Bα(t): Jacobi-smeared Proton OperatorThree Point Function:

CO3 (t , τ) = Γαβ〈Bα(t)O(τ)Bβ(0)〉

with Γ = 12(1 + γ4) (unpolarised) or 1

2(1 + γ4)γ5γ2 (polarised)For 0 � τ � t : C3(t ,τ)

C2(t)∝ 〈p|O|p〉

0 10 20 30 40

t

0

0.5

1

Lattice Spacing from fπβ = 7.6 β = 8.0 β = 8.45

0 0.2 0.4amq

0.1

0.2

af π

0 0.2amq

0.1

af π

0 0.2amq

0.1

af π

fπ = f + c · mq

β a(fm)

7.60 0.229(4)8.00 0.154(2)8.45 0.105(1)

Hadron Masses

Pion Mass

β = 7.6 β = 8.0 β = 8.45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55amq

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m2 π(G

eV

2)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24amq

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m2 π(G

eV

2)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16amq

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m2 π(G

eV

2)

Quenched Chiral Log in mπ

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14amq

1.4

1.5

1.6

1.7

1.8

am2 π/m

q

Quenched Chiral Perturbation Theory:

m2π = Amq

(

1 + δ logAmq

Λ2χ

)

+ bm2q + . . .

with

δ =m2

048π2f 2

Fit yields δ = 0.18(2), corresponding to

m0 = 850(45)MeV

Vector Meson Mass

Quenched Chiral Perturbation Theory:

mρ = cρ0 + cρ1mπ + cρ2m2π + · · ·

0 0.1 0.2 0.3 0.4 0.5amπ

0.4

0.5

0.6

amρ

Nucleon Mass

Quenched Chiral Perturbation Theory:

mN = cN0 + cN

1 mπ + cN2 m2

π + · · ·

0 0.1 0.2 0.3 0.4 0.5amπ

0.4

0.5

0.6

0.7

0.8

0.9

amN

Scaling ?

Vector meson Nucleon

0 200 400 600 800 1000

mπ(MeV)

600

800

1000

1200

mρ(M

eV)

β=7.6β=8.0β=8.45

0 200 400 600 800 1000

mπ(MeV)

600

800

1000

1200

mρ(M

eV)

β=7.6β=8.0β=8.45

→ Discretization errors small

Non-perturbativeRenormalisation

Renormalisation - Axial Current

Ward Identity:

ZA〈∂4A4(t)P(0)〉 = 2m〈P(t)P(0)〉

0 0.05 0.1 0.15 0.2m

1.4

1.41

1.42

1.43

1.44

β ZA7.6 2.23(5)8.0 1.59(1)

8.45 1.42(1)

Renormalisation - Vector CurrentWard Identity:

ZV 〈N|V u−d4 |N〉 = 1

obtain ZV from ratio

Γαβ〈Bα(0)V u−d4 (τ)Bβ(t)〉

Γαβ〈Bα(0)Bβ(t)〉

0 0.05 0.1amq

1.35

1.4Z V

β ZV ZA7.6 2.00(12) 2.23(5)8.0 1.55(2) 1.59(1)

8.45 1.40(2) 1.41(1)

RenormalisationDivergent Operators: RI − MOM-Scheme (Martinelli,Sachrajda):Consider (non-singlet) quark propagator and quark Three-PointFunctions in Landau Gauge:

S(p) = 〈ψ(p)ψ(p)〉

GO(p) = 〈ψ(p)Oψ(p)〉

Renormalisation Condition:

ZψZO(µ)ΠO(ΓO(p)) = 1 at p2 = µ2

withΓO(p) = S−1(p)GO(p)S−1(p)

andΠO(ΓO(p)) =

112

tr(ΓO,Born(p)−1ΓO(p))

Renormalisation

Zψ determined from

ZψZAΠA(ΓA) = 1

Conversion to MS,2GeV:

Z RGIO = 1/Z RI−MOM,RGI

O (µ)Z RI−MOMO

Z MS(µ′) = Z MS,RGIO (µ′)Z RGI

O

ZS,RGIO (µ) =

(

2b1gS(µ)2)−

dO,12b1 e

R gS (µ)0 dξ

γSO(ξ)

βS(ξ)+

dO,1b1ξ

!

Using perturbative (3-Loop) γ-Function (Gracey, Chetyrkin)

Renormalisation: Scalar Density

Z RI−MOMS Z RGI

S

0 10 20

p2(GeV2)

0.8

1

1.2

1.4

1.6

0 10 20

p2(GeV2)

0.8

1

1.2

1.4

1.6

β Z MSS (2GeV)

7.6 1.82(4)8.0 1.48(2)

8.45 1.29(1)

Renormalisation: Tensor Current

Z RI−MOMT Z RGI

T

0 10 20

p2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 10 20

p21.5

1.6

1.7

1.8

1.9

2

2.1

2.2

β Z MST (2GeV)

7.6 2.36(4)8.0 1.73(1)

8.45 1.54(1)

Renormalisation: v2

Z RI−MOMv2

Z RGIv2

0 10 20

p21

2

3

4

5

0 10 20

p21

1.5

2

2.5

3

3.5

4

4.5

5

β Z MSv2

(2GeV)

7.6 2.65(5)8.0 2.11(3)

8.45 1.90(2)

Renormalisation: a1

Z RI−MOMa1

Z RGIa1

0 10 20

p21

2

3

4

5

0 10 20

p21

1.5

2

2.5

3

3.5

4

4.5

5

β Z MSa1

(2GeV)

7.6 2.95(4)8.0 2.21(2)

8.45 1.98(1)

Renormalisation

Comparison to Tadpole-Improved Perturbation Theory(Perlt et al.)β = 8.45, MS(2GeV):

Matrix Element Z MS TI PT Z MS NPgA 1.30 1.42h1 1.33 1.54v2 1.40 1.90a1 1.41 1.98

Large Difference for Operators with Derivative!

Nucleon Structure

Nucleon Structure

Nucleon described by structure functions:I F1, F2 UnpolarisedI g1, g2 PolarisedI h1 Transversity

Operator Product Expansion:

2∫

dxF1(x ,q2)xn−1 = cn(q2)vn + O(1/q2) (n even)

and similar for other structure functions.cn(q2): Wilson Coefficient (perturbative)vn defined by

〈p|γ{µ1↔

Dµ2 . . .↔

Dµn} −Traces|p〉 = 2vn(p{µ1 . . . pµn} − Traces)

Nucleon Structure

Matrix Element OperatorgA ψγ5γ2ψh1 ψγ5σ24ψv2 O44 −

13 (O11 + O22 + O33)

a1 O524

with

Oµν = ψγ{µ↔

Dν} ψ

O5µν = ψγ5γ{µ

↔

Dν} ψ

Axial Charge gA

gA defined by〈p|Au−d

µ |p〉 = gAupγµγ5un

0 0.2 0.4 0.6 0.8 1

m2π(GeV)2

1

1.1

1.2

1.3

1.4

1.5

1.6

β=7.6β=8.0β=8.45

β gA7.6 1.21(7)8.0 1.37(5)

8.45 1.14(5)

Tensor Charge

〈p|ψσµνγ5ψ|p〉 =2

mph1sµpµ

0 0.2 0.4 0.6 0.8 1

m2π(GeV)2

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

β=7.6β=8.0β=8.45

β h17.6 1.33(8)8.0 1.35(4)

8.45 1.18(5)

v2

〈p|ψγ{µ↔

Dν} ψ|p〉 = 2v2p{µpν}

0 0.2 0.4 0.6 0.8 1

m2π(GeV)2

0

0.1

0.2

0.3

0.4

0.5

β=7.6β=8.0β=8.45

β v27.6 0.23(3)8.0 0.28(2)

8.45 0.26(2)

a1

〈p|ψγ5γ{µ

↔

Dν} ψ|p〉 = 2a1s{µpν}

0 0.2 0.4 0.6 0.8 1

m2π(GeV)2

0

0.1

0.2

0.3

0.4

0.5

β=7.6β=8.0β=8.45

β a17.6 0.56(7)8.0 0.58(4)

8.45 0.62(4)

Summary

I We computed hadron masses and the nucleon matrixelements gA, h1, v2, a1.

I Discretization errors in hadron masses smallI Non-perturbative Renormalisation is necessary (Large

difference to perturbation theory)