Strong Coupling Q C D

Strong Coupling QCDStrong Coupling QCD

ECT*, TrentoSeptember 2014


Michael PenningtonJefferson Lab


Strong Coupling QCDStrong Coupling QCD





u

u

d

QCD

FritzschFritzsch

Gell-MannGell-Mann

LeutwylerLeutwyler

q ( i D - m ) q

- F F

q

14

=QCD q=u,d,s,

c,b,t

QCD

pQCD

confinementasymptotic freedom

strong QCDstrong QCD

0

1

0 10-15

r (m)

strong coupling

strong couplingstrong couplingpQCDpQCD

Strong physics problems

d

u

u

s

u_





Schwinger-Dyson Equations

-1-1-

-1 -1

-

mass function

wavefunctionrenormalisation

Fermion propagator

-1 -1

-

Gauge variant quantities: only physical quantities are gauge independent

mass function


Fermion propagator

QCD


M

V

Bound State Equations

QCD


M

V


dressed quark propagator

QCD


M

V


dressed quark propagator

qq scattering kernel

QCD


P

V

f , m


SDE/BSE – ANL/KSU

pion/vectormesons

qq

Pq q

q q= +5 - -

qq

qqV = +

--

MV (GeV)

MP (GeV2)2

v

p2 GeV2

effective interaction strength

Maris & Tandy

10-3 103

p2 GeV2

Qin, Chang, Liu, Roberts, Wilson

effective interaction strength

10-3 103

q

q q

qq

q q

electromagnetic formfactors

Can Maris-Tandy (or Qin et al. ) modelling

be deduced from the SDE/DSEs?

q

q

Maris-Tandy model

V

q

q

q ( i D - m ) q - F F q=QCD q=u,d,s,

c,b,t

14

q

qq


2 equations

2 equations

12 equations

QEDQEDSchwinger-Dyson Equations

q

k p

pk k p=

-1 -1q

Ball & Chiu

Ward – Green –Takahashi Gauge Invariance

pk k p=

-1 -1q

Ball & Chiu


q

k p

pk k p=

-1 -1q


q

k p

1,2,..,8

q 0

-1 -1

-

mass function


Fermion propagator

how to regularize: d4k dnk

QEDQEDSchwinger-Dyson Equations

k2, q2 >> p2

k2, p2 >> q2

Gauge Invariance & Multiplicative Renormalizibility

Kizilersu & P

Unquenched Massless renormalised at: =0.2, : varying Kizilersu et al

Unquenched Massless renormalised at: =0.2, : varying Kizilersu et al

. . . .

Consistent truncation

Gauge Invariance &Multiplicative Renormalizibility

QED

(i) remove divergences (eg. quadratic div.)(ii) ensure correct gauge dependence (eg. transversality of boson)


Consistent Solutions ofConsistent Solutions of QCD

q ( i D - m ) q q=QCD q=u,d,s,

c,b,t

- F F 14

axial gauges


QCD

(q) orthogonal to q

and n - the axial vector

Baker, Ball & Zachariasen

axial gauges BBZ


QCD

(q) orthogonal to q


Slavnov-Taylor Identity

Richardson Potential

b

b_

heavy quark potential spectrum

bb

0.1 nm

positronium

V(r)

r

V(r)

r

g

g

e+

e-

1 fm

bottomonium

b

b_

b

b

1 fm

bottomonium

b

b_

b

b

gluon propagatorinterquark potential

rp ~ 1

r >> 1, p << 1

Coulomb : OBE

r << 1, p >> 1

rp ~ 1

r >> 1, p << 1

Coulomb : OBE

r << 1, p >> 1


interquark potential

axial gauges


QCD

(q) orthogonal to q


G1(q2, n.q), G2(q2, n.q)

axial gauges

QCD

(q) orthogonal to q


G1(q2, n.q), G2(q2, n.q)


G2(q2, n.q) = 0G1(q2, n.q) ~ 1/q2

ie ~ 1/q4


axial gauges

QCD

(q) orthogonal to q



West showed axial gauge could NOT be more singular than 1/q2

G1(q2, n.q), G2(q2, n.q)

G2(q2, n.q) = 0G1(q2, n.q) ~ 1/q2

ie ~ 1/q4


(q)

covariant gauges

QCD


(q)

covariant gauges

(q) = T + qq

q2q2

Gl (q)

T (q) = gqq

q2-

D (q) = q2

Gh(q) QCD


first just gluons Pagels, Mandelstam, Bar-Gadda

Gl (q)

Studies in covariant gauges

first just gluons Pagels, Mandelstam, Bar-Gadda

Gl (q)

STIGl

~ 1/q4 possible

Studies in covariant gauges

(q)

covariant gauges

(q) = T + qq

q2q2

Gl (q)

T (q) = gqq

q2-

D (q) = q2

Gh(q)

=g

i



(q)

Landau gauge

(q) = T + qq

q2q2

Gl (q)

T (q) = gqq

q2-

D (q) = q2

Gh(q)


Brown & P (1988) Gh = 1


Brown & P1988

Gl (q)

Studies in the Landau gauge

q2 (GeV2)

Gl

R(q

)

Brown & P1988

Gl (q)

Nf = 2

s = 0.25

q2 (GeV2)

Gl

R(q

)

Nf = 2

s = 0.25

q2 (GeV2)

Gl

R(q

)

Studies in the Landau gauge


b

b_

heavy quark potential spectrum



von Smekal, Alkofer et al: ghosts are essentialghosts are essential

Landau gauge studiesLandau gauge studies

(k) = Gl (k) T(k) / k2

Gl (k) Gl (q) V(k,q,p)

q

k

(q) = Gl (q) T(q) / q2

Landau gauge studiesLandau gauge studies

(k) = Gl (k) T(k) / k2

Gl (k) Gl (q) V(k,q,p)

q

k

(q) = Gl (q) T(q) / q2

Model 1: V ~ 1

Tübingen, Graz, DarmstadtTübingen, Graz, Darmstadt

Gluon

Ghost

20 2 0.02 0.2distance (fm)

FischerDeep Infrared

scaling solution

Ghost

Gluon A(p)

Gluon B(p)

p2

Ghost

Gluon A(p)

Gluon B(p)

p2

Ghost

Gluon A(p)

Gluon B(p)

p2

Ghost

Gluon A(p)

Gluon B(p)

p2 von Smekal, Lerche



loss of symmetry

engineering to maintain scaling solution: V ~ Gh/Gl

Lattice QCD

p/a (GeV)

a2 D

(p2 )

V = 1284

Lattice Results: Cucchieri, Mendes

gluon

ghost

Bogolubsky et al. 2009Bogolubsky et al. 2009

p2

p2

Oliveira & Silva

p2 GeV2

Papavassiliou, BinosiBoucaud et al

Rodriguez Quintero

“massive”Solution of Gluon & Ghost SDEs

Tübingen, Graz, DarmstadtTübingen, Graz, Darmstadt

Gluon

Ghost

20 2 0.02 0.2distance (fm)

FischerDeep Infrared

scaling solution

p2 GeV2

Papavassiliou, BinosiBoucaud et al

Rodriguez Quintero

Wilson & P


V(k,q,p)

Model 1: V ~ 1

Model 2: V ~ Gh/ Gl

p

k

q

Gl (k)

Gl (q)

p

k

q

= bare vertex

Wilson & P

p

k

q

= bare vertex

p

k

q

= bare vertex

m2 + p2 [ 1 + ln( )]11 Nc g2

12 4p2 + m2

2

13

22

p2=Gl (p2)

Aguilar, Binosi, Papavassiliou

Wilson & P

m2 ~ 0.1 GeV2

q

k

p

To reproduce lattice results: ghost-gluon vertex has to have important non-Taylor terms

Coupled ghost equation

= ig fabc ( k – q FIR(k,p,q))k.q

q2

FIR 0, when k 0

p2 GeV2


Running coupling

Taylor coupling

Aguilar coupling

(GeV2)



Bloch

Meyers & Swanson

Adding quartic interactions


c,b,t

14

QCDAdding quartic interactions

m02 AA

+

QCD ?Meyers & Swanson

ghost

sunset

squint

Consistent Solutions ofConsistent Solutions of QCD

q ( i D - m ) q q=QCD q=u,d,s,

c,b,t

- F F 14

Truncation respects: Gauge invariance Multiplicative Renormalizability

Can Maris-Tandy (or Qin et al. ) modelling

be deduced from the SDE/DSEs?

q

q

Maris-Tandy model

V

q

q


c,b,t

14

q

qq

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Strong Coupling Q C D