On the lattice Landau gauge quark propagator

orlando@fis.uc.pt University of Coimbra

A. Kizilersu M. Mueller-Preussker P. J. Silva J. Skullerud A. Sternbeck A. G. Williams

Gluon Propagator in Linear Covariant Gauges

orlando@fis.uc.pt University of Coimbra

P. Bicudo D. Binosi N. Cardoso P. J. Silva

arXiv:1505.05897

On going effort to investigate the quark gluon vertex on the lattice

�↵� µ = h0| ↵ � Aµ|0i 12 form factors

quark propagator gluon propagator+

close to the chiral limit

S(p) = Z(p2)�iak/+ aM(p2)

a2k2 +M2(p2)

quark wave function

running quark mass

several quenched calculations

Dynamical lattice simulations

improved staggered actionP. O. Bowman et al, Nucl. Phys. Proc. Suppl. 119, 323 (2003) [hep-lat/0209129] P. O. Bowman et al, Phys. Rev. D71, 054507 (2005) [hep-lat/0501019]

2+1 flavors Bare quark masses:

sea 16, 32, 47, 63 Mev + 79 MeV

a = 0.125 fm 203x64

Z(p2) suppressed in the IRM(p2) enhanced in the IR

chiral extrapolation

S. Furui, H. Nakajima, Phys. Rev. D73, 074503 (2006) [hep-lat/0511045]

M(0) ⋍ 380 MeV (chiral extrapolation)

FLIC overlap quark action

W. Kamleh et al, Phys. Rev. D76, 094501 (2007) [arXiv:0705.4129]

Nsea = 2 flavors

β a(fm) mπ(MeV) 123x24 4.00 0.120 806 (1.44 fm)3x(2.88 fm) 163x32 4.20 0.096 820 (1.44 fm)3x(2.88 fm)

50 configurations

M(0) < 300 MeV (chiral extrapolation)

CI action

M. Schrok, Phys. Lett. B711, 217 (2012) [arXiv:1112.5107]

Nsea = 2 flavors 125 configurations163x32 a = 0.144 fm mbare = 15.3(3) MeV

M(p2min) ⋍ 300 - 380 MeV

Clover action(O(a) improved Sheikholeslami-Wohlert action)

J. I. Skullerud et al, Phys. Rev. D63, 054508 (2001) J. I. Skullerud et al, Phys. Rev. D64, 074508 (2001) Ph. Boucaud et al, Phys. Lett. B575, 256 (2003)

mbare ⋍ 50 MeV, 58 MeV, 102 MeV, 118 MeV β = 6.0, 6.2 and 6.0 — 6.8

(1.5 fm)3x(4.5 fm) (1.6 fm)3x(3.3 fm)

X

f

X

x

(4 +m

f

) f

(x) (x)

� 1

2

X

f

X

x,µ

"

f

(x)(1� �

µ

)Uµ

(x) f

(x+ µ)

+

f

(x+ µ)(1 + �

µ

)U †µ

(x) f

(x)

#

� 1

2c

SW

X

f

X

x,µ⌫

f

(x)�µ⌫

F

µ⌫

(x) f

(x)

(x)(i@/+m) (x) + (x)@2 (x) + (x)�µ⌫Fµ⌫(x) (x)

Nf = 2 β = 5.29 κ = 0.13632

a = 0.072 fm

mbare = 6.66 MeV

mπ = 296 MeV mN = 1078 MeV

525 Landau gauge configurations

O(a) nonperturbative improved action

S(x, y) = h0| (x) (y) |0i

�! (1 + bqam)(1� cqaD/)

�! (1 + bqam)(1 + cqa �D/)

O(a) improvement of quark propagator

S(p) =X

x

e

�i p·xS(x, 0)

= Z(p2)�iak/+ aM(p2)

a2k2 + a2M2(p2)

akµ = sin(pµa)

tree level values cq = bq = 1/4

akµ = sin(pµa)Z(p2)�iak/+ aM(p2)

a2k2 + a2M2(p2)

Z(p2) ⇡ 1� a2k2

16+O(a2)

aM(p2) ⇡ am

1� am

2+

a2m2

16

k2 + 4m2 � 3k4/m2

k2 +m2� a2k2

16

!

+O(a3)

0 0.5 1 1.5 2 2.5 3ap

0

0.1

0.2

0.3

0.4

0.5

aM(p²)

Bare0 0.5 1 1.5 2 2.5 3

ap0.6

0.7

0.8

0.9

1

1.1

Z(p²)

Bare

Preliminary data

H4 methodfundamental momenta invariants

p[2] =X

µ

p2µ, p[4] =X

µ

p4µ, p[6] =X

µ

p6µ, p[8] =X

µ

p8µ

Q(p[2], p[4], p[6], p[8]) = Q(p[2], 0, 0, 0) +@Q(p[2], 0, 0, 0)

@p[4]a2

p[4]

p2+ · · ·

continuum value

a2 corrections

a2 corrections:

p[4]

p2,

p[6]

(p2)2,

p[8]

(p2)3

problem: what about p2 corrections?

tree level quark propagator

S�1(p) =iak/A(ap) +B(ap)

z(ap)

S�1(p) =iak/+ aML(ap)

ZL(ap)

Z(0)(ap) = z(ap)/A(ap)

aM(ap) =B(ap)

A(ap)= am+ a�M (0)(ap) = amZ(0)

m (ap)

Z(ap) =ZL(ap)

Z(0)(ap)aM(ap) = aML(ap)� a�M (0)(ap)

aM(ap) =aML(ap)

Z(0)m

0 0.5 1 1.5 2 2.5 3ap

0

0.1

0.2

0.3

0.4

0.5

aM(p²)

BareTree Level SubTree Level Mult

0 0.5 1 1.5 2 2.5 3ap

0.6

0.7

0.8

0.9

1

1.1

Z(p²

)

BareTree Level

Preliminary data

p = (1, 1, 1, 1)

Momentum Cutsp[4]

(p[2])2=

1

4

p[4]

(p[2])2< 0.3

0 0.5 1 1.5 2 2.5 3ap

0.6

0.7

0.8

0.9

1

1.1

Z(p²

)

Tree LevelCuts

Z(p2) = z0 +z1p2

+z2p4

z0 = 1.0555(2)

z1 = 0.189(1)

z2 = �0.133(2)

�2 = 1.6

p > 1(= 2.741 GeV)0 0.5 1 1.5 2 2.5 3ap

0.6

0.7

0.8

0.9

1

1.1

Z(p²)

Cuts

Preliminary data

0 0.5 1 1.5 2 2.5 3ap

0

0.05

0.1

aM(p²)

Tree Level MultCuts

amq = (3.595 ± 0.014)x10-3

mq = 9.854 ± 0.038 MeV

a-1 = 2.741 GeV

0 0.5 1 1.5 21

1.05

1.1

1.15

1.2

1 2 3 4 51

1.05

1.1

1.15

1.2

2 3 4 5 6 71

1.05

1.1

1.15

1.2

2 3 4 5 6 71.06

1.07

1.08

1.09

1.1

1.11

Z(p2) as function of p[4]

ap = 1.063 ap = 2.026

ap = 2.663 ap = 2.932

Preliminary data

0 0.5 1 1.5 2 2.5 3ap

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Z(p²)

p6 Ext.

Z = 1.1480(40)

Preliminary data

0 0.5 1 1.5 2 2.5 3ap

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Z(p²)

p8 Ext.

Z = 1.1491(43)

Preliminary data

0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

0.03

1 2 3 4 50.0020.0030.0040.0050.0060.0070.0080.009

2 3 4 5 6 70.003

0.004

0.005

0.006

0.007

0.008

2 3 4 5 6 70.003

0.004

0.005

0.006

0.007

aM(p2) as function of p[4]

ap = 1.063 ap = 2.026

ap = 2.663 ap = 2.932

Preliminary data

0 0.5 1 1.5 2 2.5 3ap

0

0.05

0.1

aM(p²)

Preliminary data

Summary and outlook• don’t forget that it is a preliminary analysis • closer to the chiral limit • DCSB with a M(0)⋍ 322.1 ± 5.2 MeV • finite size effects for Z(p2) under control • need to be understood better for M(p2) • check other m and lattice spacing

0 0.5 1 1.5 2 2.5 3pa

0

0.5

1

1.5

2

2.5

3

3.5

4

λ 1(p²,0,p²)

UncorrectedCorrected

zero gluon momentum

γμ

Gluon Propagator in Linear Covariant Gauges

orlando@fis.uc.pt University of Coimbra

P. Bicudo D. Binosi N. Cardoso P. J. Silva

arXiv:1505.05897

Landau gauge

0 0.5 1 1.5 2 2.5 3p [GeV]

0

1

2

3

4

5

6

7

8

9

10

11

D(p

) [G

eV-2

]

β = 6.3 - 1284 - L = 8.020 fmβ = 6.0 - 804 - L = 8.128 fmβ = 5.7 - 444 - L = 8.087 fm

Renormalized Gluon Propagator - µ = 4 GeV

effective gluon mass

M(p2) =m4

0

m20 + p2

O. Oliveira, P. Bicudo, J Phys G38, 045003 (2011)

0 0.5 1 1.5 2 2.5 3 3.5 4p [GeV]

0.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.62.72.82.93.0

p2 G(p²)

β = 6.3 - 1284 - 8.020 fmβ = 6.0 - 804 - 8.128 fm

Renormalized Ghost Dressing Function - µ = 4 GeV

Renormalizable Rξ gauges

F [g] =X

x,µ

Re Tr

g(x)U

µ

(x)g†(x+ µ)

�+X

x

Re Tr

ig(x)⇤(x)

�

๏Two integrations: links and gaussian distributed matrices Λ

P [⇤

a(x)] / exp

(�1

2⇠

X

a

(⇤

a(x))

2

)

A. Cucchieri, T. Mendes, and E. M. Santos, Phys. Rev. Lett. 103, 141602 (2009)

๏non trivial minimization problem

@A(x) = ⇤(x) Uµ(x) = exp iag0Aµ(x+ µ/2)

g = ⇧i�gi �gi = 1 + i

X

j

!

a(x)ta

choosen to minimize F[g]

�(x) =X

µ

g0rAµ(x)� ⇤(x)

✓ =1

NL

4

X

x

Tr⇥⇤(x)⇤†(x)

⇤< 10�15

Dµ⌫(p2) =

✓�µ⌫ � pµp⌫

p2

◆DT (p

2) +pµp⌫p2

DL(p2)

• Wilson action (50 configurations) • 324 β = 6.0 a = 0.1016(25) fm • 50 Λ configurations for each Uμ • FFT steepest descent + overrelaxation + stochastic overrelation + restart after random gauge transformation

• ξ = 0.1, 0.2, 0.3, 0.4, 0.5 • data renormalised at μ = 4.317 GeV

Lattice Setup + Gauge Fixing

ξ = 0.103(2), 0.203(2), 0.302(3), 0.402(3), 0.502(3)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.1 1 10 100

ξ=0.1ξ=0.2ξ=0.3ξ=0.4ξ=0.5

0

0.5

1

1.5

2

2.5

0.01 0.1 1 10 100

Landauξ=0.1ξ=0.2ξ=0.3ξ=0.4ξ=0.5

q2 [GeV2] q2 [GeV2]

q2∆

L(q

2)

q2∆

T(q

2)

Some differences with M. Q. Huber, Phys.Rev. D91, 085018 (2015)

Nice agreement A. Aguilar, D. Binosi, and J. Papavassiliou,

Phys. Rev. D91, 085014 (2015)

M2(p2) =

"a(⇠) + c(⇠)

✓p2

µ2

◆⇠log

p2

µ2

#M2

⇠=0(p2)

a(⇠) = 1 + a1⇠

c(⇠) = cNI⇠ cNI ⇡ 0.32

a1 ⇡ 0.26 fit mass at p2 = 0

0

0.02

0.04

0.06

0.08

0.1

0.001 0.01 0.1 1 10

ξ=0.1ξ=0.2ξ=0.3ξ=0.4ξ=0.5

q2 [GeV2]

m2(q

2)[G

eV2]

Summary and Outlook• Gauge fixing is the most intensive numerical part of the calculation

• The gluon propagators in Rξ gauges for a lattice volume large enough to access the IR

• Data shows an inflection point (few hundreds MeV) and a saturation in the IR

• Suggests that a dynamical gluon mass generation is a common feature of all Rξ gauges