The axial nucleon charge gA (and its renormalization

The axial nucleon charge gA

(and its renormalization constant ZA)

using the point-split axial vector current

operator on the lattice

Stanislav Kazmin

with Arwed Schiller and Holger Perlt

University Leipzig

25.11.2016

Stanislav Kazmin (University Leipzig)

Content

1. Introduction

2. Nucleon Axial Charge gA

3. Summary and Outlook

Stanislav Kazmin (University Leipzig) 1 / 19

Introduction – Basics

Why Lattice QCD?

� fundamental quantum field theory of quarks and gluons

� large energies: asymptotically free → perturbation theory

� small energies: coupling large → nonperturbative techniques

Basic steps towards lattice QCD

1 discretize Euclidean space-time by a hyper-cubic lattice Λ

� lattice volume: V = L3 × T

� lattice spacing: a

2 construct a discrete version of the QCD action

3 quantize QCD using Euclidean path integrals

4 calculate expectation values using Monte Carlo techniques

Stanislav Kazmin (University Leipzig) Sources: [1, 2] 2 / 19


Fermion fields ψf (x)cα and ψf (x)cα

� live on lattice sites as Nc ⊗ Nd objects

Link variables Uµ(x)

� link variables Uµ(x) ∈ SU(3) as parallel gauge transporters

from point x to a neighboring point x + µ (Figure 1)

� connected to gauge fields Aµ(x) in the continuum by

Uµ(x) = exp(iaAµ(x)) (1)

x x + µUµ(x) x x + µU†µ(x + µ)

Figure 1: link variables on the links between two sites



Wilson plaquette U�(x)

� closed loops of Uµ(x) are gauge invariant

� smallest closed loop is the Wilson plaquette

U�µν(x) = Uµ(x)Uν(x + µ)U†µ(x + ν)U†ν(x) (2)

x µ

ν

a plaquette U�(x)

x µ

ν

b planar rectangle R�µν(x)

Figure 2: schematic Wilson plaquette U�(x) and planar rectangle R�(x)

Stanislav Kazmin (University Leipzig) Sources: [1] 4 / 19


Integration on lattice

∫d4x →

∑x∈Λ

(3)

Parameters

� Inverse coupling β =6

g2(4)

� Hopping parameter κf =1

2(mf + 4)(5)

� work with 2 + 1 flavors → two light quarks with isospin

symmetry and the strange quark


Introduction – Improved QCD Action

� gauge fields are tree-level Symanzik improved

� quark fields are clover improved and contain the Wilson term

� Df (x , y ) is the Dirac operator

S(U, ψ, ψ) = SF (U, ψ, ψ) + SG (U) (6)

SG (U) =β

3

∫d4x

(C0

∑µ<ν

Re Tr(1− U�µν(x)

)

+C1

∑µ<ν

Re Tr(1− R�µν(x)

))(7)

SF (U, ψ, ψ) =∑f

∫d4x

∫d4y ψf (x)Df (x , y )ψf (y ) (8)


Introduction – Improved QCD Action

Dirac Operator Df (x , y ) with clover improvement

Df (x , y ) =1δxy − κf∑µ

(1− γµ)(Uµ(x)δx+µ,y − U−µ(x)δx−µ,y

)− cSWκf

∑µ,ν

1

2σµν Fµν(x)δxy (9)

� cSW is the clover (or Sheikholeslami-Wohlert) coefficient

� Fµν(x) is the field strength tensor on lattice

� inverse Dirac Operator is the quark propagator S(x , y )

S(x , y ) = D−1(x , y ) = 〈ψ(x) ψ(y )〉F (10)

� the quark propagates from point y to point x


Introduction – Expectation Values

Expectation value of operator O

� path integral formalism as in continuum

〈O〉 =1

Z

∫DU Dψ Dψ O(U, ψ, ψ) e−S(U,ψ,ψ) (11)

� Grassmann numbers (anti-commuting numbers) allow to

integrate out the fermionic part

� fermion determinant det Df (x , y ) as an effective action

〈O〉 =1

Z

∫DU O(Sf ,U) e−S(U) (12)

S(U) = SG (U) + SeffF (U) (13)

SeffF (U) = −

∑f

Tr ln Df (14)


Introduction – Expectation Values

Configuration ensemble

configuration Ui is a set of link variables

Ui = {Uµ(x) | x ∈ Λ, µ = 1, . . . , 4} (15)

ensemble U is a set of configurations Ui

� generate ensemble U with a Hybrid Monte Carlo algorithm

� configurations Ui distributed according to the weight

W =1

Ze−S(U) (16)

� use importance sampling to obtain expectation values 〈O〉and errors σ(O)


Introduction – Analysis Steps

1 configuration generation

� used configurations produced by the QCDSF

collaboration

2 measurements

� performed at the HRLN supercomputing system

� used the Chroma C++ package

� used self-written Chroma module for the vertex function

calculation from the point-split axial current operator

3 data analysis

� performed in Leipzig with self-written Julia code

Stanislav Kazmin (University Leipzig) Sources: [8–10] 10 / 19

Content

1. Introduction




Nucleon Axial Charge gA – Introduction

� axial vector current operator Aµ = ψ(x)γµγ5ψ(x)

� nucleon matrix element 〈N(p′, s ′)|Aµ|N(p, s)〉� proportional to the axial form factor GA(q2)

� proportional to nucleon beta decay rate

� axial charge is defined by gA = GA(q2 = 0)

� gA is a benchmark quantity in lattice QCD

� current lattice measurements usually underestimate the

experimental value gexpA = 1.2723(23)

Stanislav Kazmin (University Leipzig) Sources: [11–13] 11 / 19


Axial vector current operator Aµ on the lattice

� local operator Alocµ (x) or point-split operator Aps

µ (x)

Alocµ (x) = Aµ = ψ(x)γµγ5ψ(x) (17)

Apsµ (x) =

1

2

(ψ(x)γµγ5Uµ(x)ψ(x + µ)

+ ψ(x + µ)γµγ5U†µ(x)ψ(x))

(18)

Motivation for Apsµ :

� at comparable pion masses estimates, that are closer to the

experimental value than for Alocµ

� renormalization constant is closer to one



Ratio of correlation functions can be used to determine gA

R(t; τ) =G3(t; τ)

G2(t)= gA (19)

Correlation functions

G2(t) =∑x

e−ipx Γ(2)βα〈Nα(x)Nβ(0)〉 (20)

G3(t; τ) =∑x ,y

e−ipxe−ip′(x−y) Γ

(3)βα〈Nα(x)O(y )Nβ(0)〉 (21)

with the nucleon interpolators N and projectors Γ

0 t

d

u

u0 t

O(τ)d d

u

u

Figure 3: Two-point and three-point function of a nucleon


Nucleon Axial Charge gA – Measurements

Performed measurements

L3 × T β a / fm t, mπ /MeV meas.

323 × 64 5.50 0.0740t = 13, 15, 17, 19, 21 ≈ 2600

mπ = 465

483 × 96 5.80 0.0558t = 15, 17, 19, 21, 23 ≈ 600

mπ = 427

� varied separation time t between source and sink at

symmetric point (κl = κs = κ)

� got curves gA(τ) for each t and gA as the mean value in the

range 0� τ � t


Nucleon Axial Charge gA – Measurements

0 2 4 6 8 10 12 14 16 18 20 22τ = τ + 1

2 (23− t)

1.050

1.075

1.100

1.125

1.150

1.175

1.200

1.225

gA

gA = 1.125(11), t = 15

gA = 1.138(13), t = 17

gA = 1.149(15), t = 19

gA = 1.183(21), t = 21

gA = 1.194(26), t = 23

dotted lines to guide the eye

Figure 4: gA for different separation times t for β = 5.80


Nucleon Axial Charge gA – Results

Excited states contribution

� gA depends on separation time t

� for small and large t values the contribution is large

� systematically analyzed with two different methods

1 Summation method

2 Global fit method

Global fit method

1 from G2 obtain the ground state mass and two excited state

masses Mi with a symmetric fit ansatz

2 insert the obtained masses into a global fit function which

involves all gA(t, τ) values



0 10 20 30 40 50 60 70 80 90t

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

G2

+G2

−G2

three mass fit

Figure 5: G2 three mass fit for β = 5.80



0 2 4 6 8 10 12 14 16 18 20 22τ = τ + 1

2 (23− t)

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

gA

t = 15t = 17

t = 19t = 21

t = 23

gfitA = 1.234(16)

Figure 6: Global fit method result for β = 5.80


Content

1. Introduction




Summary and Outlook

� excited states contributions can be treated with global fit

method

� higher gA estimate than with the local operator for

comparable pion masses (β = 5.50, renormalized)

gpsA = 1.224(11) gloc

A = 1.1203(95) gexpA = 1.2723(23)

� Renormalization constant closer to one (β = 5.50)

Z MSA = 1.0212(12)(47) Z MS,loc

A = 0.8728(06)(27)

Outlook:

� Feynman-Hellmann approach could improve the measurement

as it does not involve three-point correlation functions

� extrapolate gA to the physical point

� measure ZA for β = 5.80 (expect to be even closer to one)


Thank You for Your Attention!

Stanislav Kazmin

with Arwed Schiller and Holger Perlt

University Leipzig

25.11.2016


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Backup – Renormalization Constant ZA

� observables measured with lattice QCD must be

renormalized to compare them to real physical values

� many observables diverge in the limit a→ 0

Necessary steps for a lattice renormalization scheme

1 remove the ultraviolet divergence in the observables

� perturbatively: in general bad convergence

� nonperturbatively: used RI’-MOM scheme

(regularization independent momentum subtraction)

2 match a convenient continuum scheme

� common choice: MS scheme (minimal subtraction)

� typical comparison scale µR = 2 GeV



Axial current renormalization constant ZA

� ZA = 1 for chirally symmetric actions and conserved current

� stays finite in the continuum limit a→ 0

Calculation steps

1 calculate Z RI’

A (p2;κi) in RI’-MOM scheme for quark masses

along the symmetric line κl = κs = κ

2 calculate Z RI’

A (p2;κsymcr ) by taking the chiral limit m → 0 of

Z RI’

A (p2;κi) at each p2

3 rescale the Z RI’

A (p2;κsymcr ) with a scale function RRI ′→MS(p2)

which lead to Z MSA (p2)

4 analyze the Z MSA (p2) curve and obtain the final Z MS

A



RI’-MOM scheme

� applicable in a scale window

Λ2QCD � µ2

R �1

a2(22)

� needs gauge fixed configurations (used Landau gauge)

� ideally, Z MSA is scale independent but due to lattice artifacts

we see a dependence

� lattice artifacts considered to be small: linear fit ansatz

� choice of the fit range p2min . . . p2

max leads to systematic error


Backup – Measurements

Performed measurements

κ 0.12090 0.12092 0.12095 0.12099 0.121021

mπ /MeV 465 439 402 343 290

cfg. 18 9 18 10 9

� β = 5.50, lattice size: 323 × 64, a = 0.0740 fm

� diagonal momenta in the range 0 ≤ p2 ≤ 10

� total number of momenta increased to 32 by using twisted

boundary conditions

pi =2π

L(ni + θi) , p4 =

2π

T(n4 + θ4 +

1

2) (23)


Backup – Results

0 1 2 3 4 5 6 7 8 9 10p2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

ZA

κ = 0.120900κ = 0.120920κ = 0.120950κ = 0.120990κ = 0.121021

2.1 2.3 2.5 2.7 2.9p2

0.930

0.935

0.940

0.945

0.950

0.955

0.960

Figure 7: ZA(p2) for different kappa values (zoomed region highlighted)


Backup – Results

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5p2

0.85

0.90

0.95

1.00

1.05

1.10

ZA

ZMSA (p2)

fit: Z0A + B · p2

Z0A 1.0212(12)B −0.03106(33)dof = 11, χ2

red = 1.4082p2

min = 1.166 , p2max = 5.099

Figure 8: Z MSA (p2) curve and linear fit

Renormalization constant ZA result for β = 5.50

Z MSA = 1.0212(12)(47)


Backup – Sequential Propagator Technique

0 x

a Calculate the ordinary quark

propagator Sψ(x , t; 0)

0

N(0)

x

N(p, t)

b Construct the sequential source

Sψseq(x , t; 0; p)

0 x

O(y ) , y

c Compute the sequential

propagator Σψ(0; y , τ ; p, t) with

a fermion matrix inversion and

insert the operator O

0 x

O(y ) , y

d Complete the correlation

function with the ordinary quark

propagator Sψ(y , τ ; 0)

Figure 9: Sequential propagator technique



Ratio of correlation functions can be used to determine gA

R(t; τ ; Aµ; Γ(3)µ ) =

13

∑3µ=1 Im Gu−d3 (t; τ ; Aµ; Γ

(3)µ )

Re G2(t; Γ(2)µ )

= gA (24)

� polarized projector for the three-point function

Γ(3)µ =

1

2(1 + γ4)iγ5γµ (25)

� unpolarized projector for the two-point function

Γ(2)µ =

1

2(1 + γ4) (26)



Two-point correlation function

G2(p, t) =

∫d3x e−ipxεabc εa′b′c ′⟨

Tr(

Γ(2)Su(x , 0)aa′)

Tr(

Sd(x , 0)bb′Su(x , 0)cc

′)

+ Tr(

Γ(2)Su(x , 0)aa′Sd(x , 0)bb

′Su(x , 0)cc

′)⟩(27)

� the tilde operation is defined by

S = (Cγ5 S Cγ5)T (28)

� only one-to-all propagators

� low effort calculation



Three-point correlation function

Gψ3 (p, t; p′, τ ;O; Γ(3))

=

∫d3y e iqy

⟨Tr[Σψ(0; y , τ ; p, t)Oψ(y , τ) Sψ(y , τ ; 0)

]⟩(29)

� sequential propagator Σψ

Σψ(0; y , τ ; p, t) =

∫d3x Sψseq(x , t; 0; p)S(x , t; y , τ) (30)

� in sequential sources Sψseq skip disconnected terms as they

cancel each other in

Gu−d3 = Gu3 − Gd3 (31)



Sequential sources Sψseq for the nucleon

Sdseq(x , t; 0; p)a′a = e−ipxεabc εa′b′c ′

×[Su(x , t; 0)bb

′Γ(3)Su(x , t; 0)cc

′

+ Tr(

Γ(3)Su(x , t; 0)bb′Su(x , t; 0)cc

′)](32)

Suseq(x , t; 0; p)a′a = e−ipxεabc εa′b′c ′

×[Sd(x , t; 0)bb

′Su(x , t; 0)cc

′Γ(3)

+ Tr(

Sd(x , t; 0)bb′Su(x , t; 0)cc

′)Γ(3)

+ Γ(3)Su(x , t; 0)bb′Sd(x , t; 0)cc

′

+ Tr(

Γ(3)Su(x , t; 0)bb′Su(x , t; 0)cc

′)](33)



gA(t = 13) quark mass dependence

� no mass dependence measured (in the given error range)

� compared only one separation time t = 13

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40m2π /GeV2

1.05

1.10

1.15

1.20

gA gA(m2π)

fit: g0A + B ·m2

π

g0A 1.103(36)B 0.12(20)dof = 1, χ2

red = 0.5143

Figure 10: gA for different quark masses


Backup – Plateau results for β = 5.80

Plateau values gA for β = 5.50 and β = 5.80

� chosen symmetrical time range τmin . . . τmax

� to remove contributions from excited states one has to fulfill

0� τ � t

� chosen the range according to this condition and to the good

fit condition χ2red ≈ 1

� gA is the mean value in the τ range


Backup – Plateau results for β = 5.80

0 2 4 6 8 10 12 14 16 18 20 22τ = τ + 1

2 (23− t)

1.050

1.075

1.100

1.125

1.150

1.175

1.200

1.225

gA

gA = 1.125(11), t = 15

gA = 1.138(13), t = 17

gA = 1.149(15), t = 19

gA = 1.183(21), t = 21

gA = 1.194(26), t = 23




Backup – Results for β = 5.50

0 2 4 6 8 10 12 14 16 18 20τ = τ + 1

2 (21− t)

1.075

1.100

1.125

1.150

1.175

1.200

gA

gA = 1.1364(64), t = 13

gA = 1.1591(93), t = 15

gA = 1.152(13), t = 17

gA = 1.189(22), t = 19

gA = 1.181(29), t = 21





0 10 20 30 40 50 60t

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

1020

G2

+G2

three mass fit

Figure 13: G2 three mass fit for β = 5.50



0 2 4 6 8 10 12 14 16 18 20τ = τ + 1

2 (21− t)

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

gA

t = 13t = 15

t = 17t = 19

t = 21

gfitA = 1.200(12)

Figure 14: Global fit method result for β = 5.50


Backup – Summation method

Summation method

� ansatz for the ratio (∆ is the energy gap between ground

state and first excited state)

R(τ, t) = gA + C1e−∆·τ + C2e−∆·(t−τ) + C3e−∆·t + . . . (34)

� sum over all ratio values R(τ, t) for each t (with cut tc)

S(t) =t−tc∑τ=tc

R(τ, t) (35)

S(t)t→∞−→ S(t) = gsum

A (t + 1− 2tc) + C sum (36)

� linear fit to S(t) yields gsumA

� still a systematic dependence on tc



13 14 15 16 17 18 19 20 21t

0

5

10

15

20

25

30

35

S(t)

tc = 1, gA = 1.209(31)

tc = 2, gA = 1.205(27)

tc = 3, gA = 1.200(25)

tc = 4, gA = 1.195(22)

tc = 5, gA = 1.189(18)

tc = 6, gA = 1.183(14)

Figure 15: Summation method fit for β = 5.50



15 16 17 18 19 20 21 22 23t

0

5

10

15

20

25

30

35

40

S(t)

tc = 1, gA = 1.243(41)

tc = 2, gA = 1.237(37)

tc = 3, gA = 1.227(34)

tc = 4, gA = 1.219(31)

tc = 5, gA = 1.209(27)

tc = 6, gA = 1.199(21)

Figure 16: Summation method fit for β = 5.80


Backup – Feynman-Hellmann Approach

� additional effective term in the Lagrangian

L→ L + λO (37)

the theorem states for any hadron state H

∂E

∂λ|λ=0 =

1

2E〈H|O(0)|H〉 (38)

� measurement steps:

1 perform hadron spectroscopy for multiple values of λ

2 observe the linear behavior in the resulting energy shifts

about λ = 0

� advantage: no three-point correlation functions needed and

therefore less noticeable excited states contributions

� disadvantage: new configuration are needed for each

operator and each λ


Backup – gA summary

exp

var

t=

13

t=

15

t=

17

t=

19

t=

21

t c=

0t c

=1

t c=

2t c

=3

t c=

4t c

=5 fit

t=

15

t=

17

t=

19

t=

21

t=

23

t c=

0t c

=1

t c=

2t c

=3

t c=

4t c

=5 fit1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1.24

1.26

1.28

1.30

gA

β = 5.50 β = 5.80

gA gsumA gA gsum

A

Figure 17: All obtained gA values and reference values


Backup – gA summary

Figure 18: gA measurements from different collaborations, point-split

operator is used for “QCDSF Nf = 2 + 1”


Backup – RI’-MOM Scheme

consider a quark bilinear operator: O(z) = u(z)ΓOd(z) (39)

� renormalization condition at a scale µR

ZO

1

12Tr(〈u(p)|O(z)|d(p)〉〈u(p)|O(z)|d(p)〉−1

0

)|p2=µ2

R

m→0

= 1

(40)

� scale window: Λ2QCD � µ2

R � 1a2 (41)

� after some transformations and insertions

ZO(p) =12 Zq(p)

Tr(

ΛO(p)Γ−1B

)∣∣∣∣∣p2=µ2

R

m→0

(42)

Zq(p) =Tr(−i∑µ γµ sin(apµ)S−1(p)

)12∑µ sin2(apµ)

∣∣∣∣∣p2=µ2

R

(43)



� Born term is equal to the tree level matrix element

〈u(p)|O(z)|d(p)〉0 = ΓB (44)

� amputated vertex function

〈u(p)|O(z)|d(p)〉 = Z−1q ΛO(p)ΛO(p) = S−1(p)GO(p)S−1(p)

(45)

� propagator S(p)

S(p) =

∫d4x

∫d4y e−ip(x−y)S(x , y ) (46)

� vertex function GO

GO(p) =1

V

∫d4x

∫d4y

∑z

e−ip(x−y)〈u(x)|O(z)|d(y )〉

(47)



0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040m2π

0.932

0.934

0.936

0.938

0.940

0.942

ZA

ZA(p2 = 2.785, m2π)

fit: Z0A + sZ · m2

π

Z0A 0.93936(100)sZ −0.152(44)dof = 3, χ2

red = 0.0423

Figure 19: Chiral limit fit example


Backup – RI’-SMOM Scheme

� In RI’-MOM scheme:

p21 = p2

2 = µ2R , q = p2 − p1 = 0 (48)

� exceptional channel where q2 � µ2R

� asymmetric subtraction point

� chiral symmetry breaking effects decrease as 1/p2

� In RI’-SMOM scheme:

p21 = p2

2 = q2 = µ2R , q = p2 − p1 6= 0 (49)

� no exceptional channels

� symmetric subtraction point

� chiral symmetry breaking effects decrease as 1/p6

� some changes in the amputated vertex function are needed


Backup – Applications

Some applications of lattice QCD

� hadronic matrix elements

� confinement mechanism

� gluon self-interaction

� spontaneous chiral symmetry breaking

� . . .


Backup – Continuum QCD

QCD action in continuum

S(A, ψ, ψ) =1

2g2

∫d4x Tr (Fµν(x)Fµν(x)) +

+∑f

∫d4x ψf (x) (γµDµ(x) + mf )ψf (x) (50)

� fermion fields ψf and ψf with mass mf and flavor index f

� gauge field A

� gauge coupling constant g

� field strength tensor Fµν

� covariant derivative Dµ

Dµ(x) = ∂µ + iAµ(x) (51)



Gauge fields Aµ(x)

� represent the gluons

� elements of Lie algebra su(3)

� directional index µ

� N2c − 1 (= 8 for Nc = 3) generators TB of su(3)

Aµ(x) =8∑B=1

ABµ (x)TB (52)

� color components ABµ (x)



Field strength tensor Fµν(x)

� commutator [ Aµ(x) , Aν(x) ] does not vanish

Fµν(x) = −i[ Dµ(x) , Dν(x) ] (53)

Fµν(x) = ∂µAν(x)− ∂νAµ(x) + i[ Aµ(x) , Aν(x) ] (54)

� representation with generators and structure constants f ABC

Fµν(x) =8∑B=1

FBµν(x)TB (55)

FAµν(x) = ∂µAAν (x)− ∂νAAµ(x)− f ABCABµ (x)ACν (x) (56)



Fermion fields ψf (x) and ψf (x)

� live on lattice sites

� positional index x ∈ Λ

� physical spatial distance is ax

Note:

� all quantities are measured in lattice units a

� fermion fields, derivatives and masses are dimensionless

m → am (57)

ψ(x)→ a3/2ψ(x) (58)

ψ(x)→ a3/2ψ(x) (59)



Lattice derivative ∂µ

forward-backward difference as derivative

∂µψ(x) =1

2(ψ(x + µ)− ψ(x − µ)) (60)

Covariant derivative Dµ(x)

use link variables Uµ instead of gauge fields Aµ and discretize

Dµ(x)ψ(x) =1

2(Uµ(x)ψ(x + µ)− U−µ(x)ψ(x − µ)) (61)



Wilson loop U�(x)

� closed loops of Uµ(x) are gauge invariant

� smallest closed loop is the Wilson loop

U�µν(x) = Uµ(x)Uν(x + µ)U†µ(x + ν)U†ν(x) (62)

x Uµ(x)

Uν(x + µ)

U†µ(x + ν)

U†ν(x)

Figure 20: Wilson loop U�µν(x) starting at position x


Backup – Wilson QCD Action

Wilson gauge action SWG (U)

SWG (U) =β

3

∫d4x

∑µ<ν


)(63)

Wilson fermion action SWF (U, ψ, ψ)

SWF (U, ψ, ψ) =∑f

∫d4x

∫d4y ψf (x) DW

f (x , y ) ψf (y ) (64)

� Wilson Dirac operator DW

f (x , y )

� full lattice QCD Wilson action

SW (U, ψ, ψ) = SWF (U, ψ, ψ) + SWG (U) (65)


Backup – Wilson QCD Action

Wilson Dirac operator DW

f (x , y )

� contains naive fermion action discretization and Wilson term

� Wilson term removes doublers from the action

DW

f (x , y ) = 1δxy − κf∑µ


)(66)

� Wilson action ready for calculations

� Wilson gauge action: O(a2) artifact effects

� Wilson fermion action: O(a) artifact effects

� improvements useful


Backup – Action Improvements

Clover term

SSWF (U, ψ, ψ) = −cSW∑f

κf

∫d4x ψf (x)

1

2σµν Fµν(x)ψf (x)

(67)

� improve fermion action to order O(a2)

� improved action: SF = SWF + SSWF� clover coefficient cSW

� matrices: σµν = 12 [ γµ , γν ] (68)

� field strength tensor on lattice Fµν(x)

Fµν(x) =1

8(Qµν(x)−Qνµ(x)) (69)

� Qµν(x) is a sum of plaquettes



Qµν(x) = U�µ,ν(x) + U�−µ,ν(x) + U�µ,−ν(x) + U�−µ,−ν(x) (70)

xµ

ν

Figure 21: clover term Qµν(x) starting at position x



Tree level Symanzik improved gauge action

SG (U) =β

3

∫d4x

(C0

∑µ<ν


)+

+C1

∑µ<ν


))(71)

� include planar rectangles R�µν(x)

� constants depend on each other C0 = 1− 8C1

� in tree-level Symanzik improvement C1 = − 112

� possible to cancel order O(a2) artifacts and stay with O(a4)



x µ

ν

a plaquette U�(x)

x µ

ν

b planar rectangle R�µν(x)

x µν

η

c extended rectangle R�µνη(x)

x µν

η

d extended bent rectangle R�µνη(x)

Figure 22: schematic Wilson loop U�(x) and rectangles R�(x)


Backup – Final QCD Action

S(U, ψ, ψ) = SF (U, ψ, ψ) + SG (U) (72)

SG (U) =β

3

∫d4x

(C0

∑µ<ν


)+

+C1

∑µ<ν


))(73)

SF (U, ψ, ψ) =∑f

∫d4x

∫d4y ψf (x)Df (x , y )ψf (y ) (74)

Df (x , y ) =1δxy − κf∑µ


)−

− cSWκf∑µ,ν

1

2σµν Fµν(x)δxy (75)


Backup – Expectation Values

Expectation value of operator O

� path integral formalism

� definition as in continuum

〈O〉 =1

Z

∫DU Dψ Dψ O(U, ψ, ψ) e−S(U,ψ,ψ) (76)

� partition function Z

Z = 〈1〉 =

∫DU Dψ Dψ e−S(U,ψ,ψ) (77)

measures:

Dψ Dψ =∏x∈Λ

∏f ,α,c

dψf (x)cαdψf (x)cα , DU =∏x∈Λ

∏µ

dUµ(x)

(78)



� Grassmann numbers (anti-commuting numbers) allow to

integrate out the fermionic part

〈O〉 =1

Z

∫DU

∏f

det Df (x , y ) O(Sf ,U) e−SG (U) (79)

Z =

∫DU

∏f

det Df (x , y ) e−SG (U) . (80)

� in O fermionic part was integrated out

� O depends only on link variables and quark propagators

Quark propagator Sf (x , y )

Sf (x , y ) = D−1f (x , y ) = 〈ψf (x)ψf (y )〉F (81)



� integral∫DU very high dimensional

� use Monte Carlo techniques for calculation

� have to deal with the fermion determinant

Fermion determinant det Df (x , y )

1 simplest solution – quenched approximation: det Df = 1

2 treat as an effective action and add to the gluon action

SeffF (U) = −

∑f

Tr ln Df∏f

det Df = e−SeffF (82)

S(U) = SG (U) + SeffF (U) (83)

3 other possibilities



〈O〉 =1

Z

∫DU O e−S(U) (84)

Z =

∫DU e−S(U) (85)

this form calls for Monte Carlo technique

Monte Carlo basic simulation steps

1 configuration generation

2 operator measurement

3 data analysis


Backup – Configuration Generation

Configuration ensemble

configuration Ui is a set of link variables

Ui = {Uµ(x) | x ∈ Λ, µ = 1, . . . , 4} (86)

ensemble U is a set of Ncfg configurations Ui

� generate configuration ensemble U

� configurations Ui distributed according to the weight

W =1

Ze−S(U) (87)

� main difficulty:

� local link updates: many updates needed to uncorrelate

� global updates: low acceptance rate

� possible solution: Hybrid Monte Carlo algorithm


Backup – Configuration Generation

Hybrid Monte Carlo algorithm

� idea: do a step from U to U′ by performing a Hamiltonian

molecular dynamics in fictitious time

� use concept of pseudofermions

Basic steps:

1 start with configuration U

2 generate momenta p

3 evaluate the Hamiltonian H(U, p)

4 perform molecular dynamic in fictitious time t → U′, p′

5 evaluate the Hamiltonian H′(U′, p′)6 accept U′ according to Metropolis probability

P = min(1, eH(U,p)−H′(U′,p′))

7 if accepted: U := U′ an go to 1


Backup – Measurement

Importance sampling

� generated configurations Ui are distributed with 1Z e−S(U)

� expectation value estimator: 〈O〉 = 1Ncfg

Ncfg∑i=1

Oi (88)

� standard error: σO =

√〈O2〉 − 〈O〉2

Ncfg(Ncfg − 1)(89)

� often need bootstrap or jackknife method to obtain errors

use operators as two or three point correlators e.g. pion two point

correlation function with pion interpolator P(x) = d(x)γ5u(x)

O = 〈P(x)P(y )〉F = Tr (γ5Su(x , y )γ5Sd(y , x)) (90)


Documents

The axial nucleon charge gA (and its renormalization