Scalar QCD - ectstar.eu · Axel Maas with Tajdar Mufti Scalar QCD Bound States, Elementary Particles & Interaction Vertices 5th of September 2013 QCD TNT III Trento Italy

Axel Maas with Tajdar Mufti

Scalar QCD

5th of September 2013QCD TNT III

TrentoItaly

Axel Maas with Tajdar Mufti

Scalar QCDBound States, Elementary Particles

& Interaction Vertices

5th of September 2013QCD TNT III

TrentoItaly

● Only confinement – no chiral symmetry breaking● Confinement independent of Lorentz

structure

Why Scalar QCD?


structure● Simple(r) tensor structures

Why Scalar QCD?


structure● Simple(r) tensor structures● Rich bound state spectrum

Why Scalar QCD?


structure● Simple(r) tensor structures● Rich bound state spectrum● Cheap lattice simulations

● Test case for functional equations

Why Scalar QCD?


structure● Simple(r) tensor structures● Rich bound state spectrum● Cheap lattice simulations

● Test case for functional equations● Limited by (possible) triviality

● But triviality cutoff can be high enough

Why Scalar QCD?

Scalar QCD

● Gauge theory

Scalar QCD

● Gauge theory

● Gluons

L=−14

Aμ νa Aa

μ ν

Aμ νa =∂μ Aν

a−∂ν Aμa

Aμa

WA

Scalar QCD

● Gauge theory

● Gluons

● Coupling g and some numbers f abc

L=−14

Aμ νa Aa

μ ν

Aμ νa =∂μ Aν

a−∂ν Aμa+gf bc

a Aμb Aν

c

Aμa

WAWA A

Scalar QCD

● Gauge theory

● Gluons● Scalar quarks● Coupling g and some numbers f abc

L=−14

Aμ νa Aa

μ ν+(Dμij h j) + Dik

μ hk

Aμ νa =∂μ Aν


a Aμb Aν

c

Dμij=δij∂μ

Aμa

hi

W

h

AWA A

Scalar QCD

● Gauge theory

● Gluons● Scalar quarks

● Coupling g and some numbers f abc and ta

ij

● Gauge group SU(2)

L=−14

Aμ νa Aa


μ hk

Aμ νa =∂μ Aν


a Aμb Aν

c

Dμij=δij∂μ−igAμ

a t aij

Aμa

hi

W

h

AW

W

A

h A

A

Scalar QCD

● Gauge theory


● Coupling g and some numbers f abc and ta

ij

● Gauge group SU(2)● No 'baryon number'

L=−14

Aμ νa Aa


μ hk

Aμ νa =∂μ Aν


a Aμb Aν

c


a t aij

Aμa

hi

W

h

AW

W

A

h A

A

Scalar QCD

● Gauge theory


● Couplings g, v, λ and some numbers f abc and ta

ij

● Gauge group SU(2)● No 'baryon number'

L=−14

Aμ νa Aa


μ hk+λ(ha ha+ −v2)2

Aμ νa =∂μ Aν


a Aμb Aν

c


a taij

Aμa

hi

W

h

AW

W

A

h

h h

A

A

Scalar QCD

● Gauge theory


● Couplings g, v=0, λ=0 and some numbers f abc and ta

ij

● Scalar self-interaction set to zero● Gauge group SU(2)● No 'baryon number'

L=−14

Aμ νa Aa



Aμ νa =∂μ Aν


a Aμb Aν

c


a taij

Aμa

hi

W

h

AW

W

A

h

h h

A

A

Symmetries

L=−14

Aμ νa Aa



Aμ νa =∂μ Aν


a Aμb Aν

c


a taij

● Local SU(2) gauge symmetry● Invariant under arbitrary gauge transformations

Aμa→Aμ

a+(δb

a∂μ−g f bc

a Aμc)ϕ

b

ϕa(x)

hi→hi+g t aijϕ

a h j

Symmetries

L=−14

Aμ νa Aa



Aμ νa =∂μ Aν


a Aμb Aν

c


a taij

● Local SU(2) gauge symmetry● Invariant under arbitrary gauge transformations

● Global SU(2) quark flavor symmetry● Acts as right-transformation on the quark field only

Aμa→Aμ

a+(δb

a∂μ−g f bc

a Aμc)ϕ

b

ϕa(x)

hi→hi+g t aijϕ

a h j

Aμa→Aμ

a hi→hi+aij h j+bij h j∗

Symmetries

L=−14

Aμ νa Aa



Aμ νa =∂μ Aν


a Aμb Aν

c


a taij

QCD-like vs. Higgs-like [Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07]

g(Classical gauge coupling)f(

Cla

ssic

al H

igg

s m

ass

)



Cla

ssic

al H

igg

s m

ass

)

Confinement “phase”



Cla

ssic

al H

igg

s m

ass

)

Higgs “phase”


QCD-like vs. Higgs-like

● (Lattice-regularized) phase diagram continuous

[Fradkin & Shenker PRD'79 Caudy & Greensite PRD'07]


Cla

ssic

al H

igg

s m

ass

)

Higgs “phase”


1st order

Crossover


● (Lattice-regularized) phase diagram continuous● Separation only in

fixed gauges



Cla

ssic

al H

igg

s m

ass

)

Higgs “phase”


1st order

Crossover

Landau gauge



fixed gauges



Cla

ssic

al H

igg

s m

ass

)

Higgs “phase”


1st order

Crossover

Landau gauge

Coulomb gauge



fixed gauges● Same physical state

space in confinement and Higgs pseudo-phases, irrespective of couplings



Cla

ssic

al H

igg

s m

ass

)

Higgs “phase”


1st order

Crossover



fixed gauges● Same physical state

space in confinement and Higgs pseudo-phases, irrespective of couplings● Asymptotic states depend on whether

ground states for given JPC

F are stable



Cla

ssic

al H

igg

s m

ass

)

Higgs “phase”


1st order

Crossover

Non-aligned gauges

● Explicit charge direction inconvenient beyond perturbation theory

[Maas, MPLA'12]

Non-aligned gauges


● Define a gauge without preferred direction

[Maas, MPLA'12]

Non-aligned gauges


● Define a gauge without preferred direction● Local part fixed to Landau gauge by

● Gribov-Singer ambiguity fixed by minimal prescription

● Introduces usual Faddeev-Popov ghosts

∂μ Aμa=0

[Maas, MPLA'12]

Non-aligned gauges


● Define a gauge without preferred direction● Local part fixed to Landau gauge by

● Gribov-Singer ambiguity fixed by minimal prescription

● Introduces usual Faddeev-Popov ghosts● Global part fixed by

● Aligned Landau gauges also possible

∂μ Aμa=0

[Maas, MPLA'12]

⟨h⟩=0

Differentiating phases [Maas, MPLA'12, Caudy & Greensite'07]

Differentiating phases

● How to distinguish phases?

[Maas, MPLA'12, Caudy & Greensite'07]


● How to distinguish phases?● Relative orientation

● is the magnetization

⟨∫hdx∫hdy ⟩

∫hdx



● How to distinguish phases?● Relative orientation

● is the magnetization● But not so important anyway...

⟨∫hdx∫hdy ⟩

∫hdx


Typical spectra

“Higgs”

[Maas, Mufti PoS'12, unpublished]

Typical spectra

“Higgs”

[Maas, Mufti PoS'12, unpublished, Maas MPLA'13]

Higgs

W

Typical spectra

“Higgs” “QCD”


Typical spectra

● Rather different low-lying spectra● 0++ lighter in (Landau gauge) QCD-like region● 1-- lighter in (Landau gauge) Higgs-like region



Typical spectra

● Rather different low-lying spectra● 0++ lighter in (Landau gauge) QCD-like region● 1-- lighter in (Landau gauge) Higgs-like region

● Use as operational definition of phase



Phase diagram [Maas, Mufti, unpublished]

Phase diagram

“Higgs”

“QCD”

● Complicated real phase diagram

[Maas, Mufti, unpublished]

Phase diagram

“Higgs”

“QCD”

● Complicated real phase diagram● QCD-like behavior even for negative bare mass


Phase diagram

“Higgs”

“QCD”

● Complicated real phase diagram● QCD-like behavior even for negative bare mass● Similar bare couplings for both physic types


Phase diagram

“Higgs”

“QCD”

● Complicated real phase diagram● QCD-like behavior even for negative bare mass● Similar bare couplings for both physic types


Propagators

● 3 propagators

Propagators

● 3 propagators● Gluon D

ab x− y= A

a x A

b y

Propagators

● 3 propagators● Gluon

● 1 scalar dressing function

D

ab x− y= A

a x A

b y

Dμ ν( p)=(δμν−pμ pν

p2)D ( p)

Propagators


● 1 scalar dressing function● Ghost

D

ab x− y= A

a x A

b y

DGab x− y = c

ax cb

y


p2)D ( p)

Propagators



● Negative semi-definite

D

ab x− y= A

a x A

b y

DGab x− y = c

ax cb

y


p2)D ( p)

−DG( p)

Propagators



● Negative semi-definite● Both renormalize multiplicatively

D

ab x− y= A

a x A

b y

DGab x− y = c

ax cb

y


p2)D ( p)

−DG( p)

Propagators



● Negative semi-definite● Both renormalize multiplicatively● Scalar

D

ab x− y= A

a x A

b y

DGab x− y = c

ax cb

y


p2)D ( p)

−DG( p)

DHij(x− y )= <hi

(x)h j+( y)>

Propagators



● Negative semi-definite● Both renormalize multiplicatively● Scalar

● Requires more complicated renormalization

D

ab x− y= A

a x A

b y

DGab x− y = c

ax cb

y


p2)D ( p)

−DG( p)

DH (μ)=DHtl (μ)

DH (μ) '=DHtl (μ) '

DHtl(p)=1 /(p2+mr

2)

μ=mr

DHij(x− y )= <hi

(x)h j+( y)>

Gluon propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished]

Gluon propagator

● Significantly volume-dependent● Decoupling-type

[Maas, EPJC'11 Maas, Mufti PoS'12, unpublished]

Gluon propagator

● Significantly volume-dependent● Decoupling-type● Positivity violating


Gluon propagator

● Significantly volume-dependent● Decoupling-type● Positivity violating● Little impact when changing scalar sector


Ghost propagator [Maas, EPJC'11 Maas, Mufti PoS'12, unpublished]

Ghost propagator

● Infrared enhanced● But likely not divergent


Ghost propagator


● Derive a running coupling from p6DG

2D


Ghost propagator



2D


Ghost propagator



2D

● Not strongest at lowest bound state masses


Scalar quark propagatorm

r=1 GeV


Scalar quark propagator

● Requires mass renormalization● Tree-level mass zero: “Mass generation”

mr=1 GeV




● No sign (yet) of positivity violation

mr=1 GeV




● No sign (yet) of positivity violation

mr=0.25 GeV


Vertices

Vertices● Three 3-point vertices

● 4-point vertices too expensive


● 4-point vertices too expensive● Ghost-gluon vertex <Aμ

a cb c̄c> =Dμ νad DG

beDGcfΓνd e f

[Cucchieri, Maas, Mendes, PRD'06,'08 Maas, Mufti PoS'12, unpublished]


● 4-point vertices too expensive● Ghost-gluon vertex <Aμ

a cb c̄c> =Dμ νad DG

beDGcfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)



● 4-point vertices too expensive● Ghost-gluon vertex

● 3-gluon vertex

<Aμa cb c̄c> =Dμ ν

ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)




● 3-gluon vertex

● Scalar-gluon vertex


ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi h j + > =Dμν

ad DHik DG

jmΓνdkm

GA hh +

=Γtl<A hh +

> /(ΓtlDDHDHΓ

tl)




● 3-gluon vertex


● tF makes vertex flavor-conserving

● Flavor-violating vertex vanishes● Flavor conserved


ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)




● 3-gluon vertex


● Two momentum configurations


ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)




● 3-gluon vertex




ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)




● 3-gluon vertex




ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)

A Ap p




● 3-gluon vertex




ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)

A Ap=0 p




● 3-gluon vertex




ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)

A Ap=0

q k with q=-k

p




● 3-gluon vertex




ad DGbeDG

cfΓνd e f

GA c c̄=Γ

tl<A c c̄> /(Γ

tlDDGDGΓtl)

<Aμa Aν

b Aρc> = Dμα

ad Dνβbe Dρ γ

cfΓαβγd e f

GA3=Γ

tl<AAA> /(Γ

tlDDDΓtl)

<Aμa hi t Fh

j +> =Dμ ν

ad DHik DG

jmΓνdkm

GAhh +

=Γtl<A htF h

+> /(Γ

tlDDHDH Γtl)

A Ap=0

q k with q=-k

p

q k

p2=q2=k2


Ghost-gluon vertex

● Only small deviations from tree-level● Like in Yang-Mills theory● Strongest effect at bound state mass scale


3-gluon vertex

● Infrared suppressed● Sets in at bound state mass scale● Absence of (supposed) sign change of

Yang-Mills theory at small momenta?


Scalar-gluon vertex [Maas, Mufti PoS'12, unpublished]

● Essentially tree-level● No indications for infrared effects (yet?)

Scalar-gluon vertex [Maas, Mufti PoS'12, unpublished]


● Different than a (pseudo-)confining 1-gluon exchange

● As in the quenched case [Maas, PoS'11, unpublished]

Scalar-gluon vertex


● Different than a (pseudo-)confining 1-gluon exchange

● As in the quenched case [Maas, PoS'11, unpublished]


Summary● Scalar QCD a role model for QCD

● Scalar theory simpler...


● Scalar theory simpler...● ...but distinction to Higgs-like physics

complicated



complicated● Test case for functional methods




● Propagators in QCD-like region similar to Yang-Mills theory




● Propagators in QCD-like region similar to Yang-Mills theory● Positivity violating gluon● Scalar quark not obviously positivity violating





● Vertices Yang-Mills-like





● Vertices Yang-Mills-like● No infrared effects in the scalar-gluon vertex





● Vertices Yang-Mills-like● No infrared effects in the scalar-gluon vertex

● So far...no obvious confinement

Scalar QCD - ectstar.eu · Axel Maas with Tajdar Mufti Scalar QCD Bound States, Elementary Particles & Interaction Vertices 5th of September 2013 QCD TNT III Trento Italy

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