The Dark Side of the Propagators - Analytical approach to ......Do all the work analytically and forget about Lattice People! Change the expansion point! Massive Expansion also known

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

The Dark Side of the PropagatorsAnalytical approach to QCD

in the infrared of Minkowski space

Fabio Siringo

Department of Physics and AstronomyUniversity of Catania, Italy

LATTICE 2016 - Southampton, 25-30 July 2016

Fabio Siringo

UNIVERSITY of CATANIA, ITALY

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Overview: the whole talk in a slide

The Dream:Do all the work analytically and forget about Lattice People!

Change the expansion point → Massive Expansionalso known as Optimized Perturbation Theory (OPT) andevaluate everything from first principles

Reality:Not self-consistent yet → Optimization by Lattice

but

we can analytically continue to Minkowski space!

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


The Dream:

Do all the work analytically and forget about Lattice People!



but


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.





but


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.





but


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.





but


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.





but


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:

Perturbation Theory (PT) works well in the IR(Tissier+Wschebor,2010,2011 - Landau Gauge)

OPT is an old story: (L0 + δLm) + (Lint − δLm)

[Massive propagator] + [mass counterterm δΓ = m2]

Nice features:The mass kills itself at tree-level:

Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2

Mass divergences cancel in loops:

—⃝— + —×⃝— = IR finite

The original Lagrangian is not modifiedStandard UV behaviorFrom first principles

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:

Perturbation Theory (PT) works well in the IR(Tissier+Wschebor,2010,2011 - Landau Gauge)OPT is an old story: (L0 + δLm) + (Lint − δLm)



Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:




Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:



Nice features:

The mass kills itself at tree-level:

Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:




Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:




Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:




Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite

The original Lagrangian is not modified

Standard UV behaviorFrom first principles

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:




Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite

The original Lagrangian is not modifiedStandard UV behavior

From first principles

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Breaking the Dogma:




Σtree = —×— = δΓ = m2 =⇒ (−p2 + m2)− Σ = −p2


—⃝— + —×⃝— = IR finite


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Do mass divergences cancel at any order?

A simple argument:

Mass divergences arise from the massive propagatorNo mass divergences in the exact (scaleless) theoryThe Lagrangian is not modified

The counterterms δΓ = m2 must cancel the divergences.How many do we need?

—⃝— =⇒ —×⃝—

1−p2 + m2 =⇒ 1

−p2 + m2 m2 1−p2 + m2

The integral is less divergent at each insertion.A finite number of insertions makes any loop integralconvergent: divergences must cancel at a finite order

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


A simple argument:



—⃝— =⇒ —×⃝—

1−p2 + m2 =⇒ 1

−p2 + m2 m2 1−p2 + m2


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


A simple argument:



—⃝— =⇒ —×⃝—

1−p2 + m2 =⇒ 1

−p2 + m2 m2 1−p2 + m2

The integral is less divergent at each insertion.A finite number of insertions makes any loop integralconvergent:

divergences must cancel at a finite order

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


A simple argument:



—⃝— =⇒ —×⃝—

1−p2 + m2 =⇒ 1

−p2 + m2 m2 1−p2 + m2


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

One-Loop third-order double expansion(Landau Gauge)

Yang-Mills → F.S., Nucl. Phys. B 907 572 (2016)QCD → F.S., arXiv:1607.02040

=Σ +gh

= +qΣ + +

++= + + +Π

++ + +

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

UNIVERSAL SCALINGIgnoring RG effects, α ∼ Nαs

Σ(p) = αΣ(1)(p) + α2Σ(2)(p,N) + · · · (1)

Σ(p)αp2 = −F(p2/m2) +O(α); F(p2/m2) = −Σ(1)

p2 (2)

∆(p) =Z

p2 − Σ(p)=

J(p)p2 (3)

Setting Z = z (1 + αδZ) (one-loop):

z J(p)−1 = 1 + α[F(p2/m2)− δZ

]+O(α2) (4)

z J(p)−1 = 1 + α[F(p2/m2)− F(µ2/m2)

]+O(α2) (5)

Must exist x, y, z:

z J(p/x)−1 + y = F(p2/m2) + F0 +O(α) (6)

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

UNIVERSAL SCALINGGLUON INVERSE DRESSING FUNCTION

0.1

1

10

100

0.1 1

J(p)

-1

p (GeV)

Duarte et al., SU(3)

Cucchieri and Mendes, SU(2)

Bogolubsky et al., SU(3)

F(p2/m2)+F0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

UNIVERSAL SCALINGGLUON INVERSE DRESSING FUNCTION

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3

J(p)

-1

p (GeV)

Duarte et al., SU(3)

Cucchieri and Mendes, SU(2)

Bogolubsky et al., SU(3)

F(p2/m2)+F0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

UNIVERSAL SCALINGGHOST INVERSE DRESSING FUNCTION

Denoting by G(s) the ghost universal function (F(s) → G(s))

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.5 1 1.5 2

χ(p)

-1

p (GeV)

Bogolubsky et al.Duarte et al.

Cucchieri-Mendes SU(2)Ayala et al. Nf = 0Ayala et al. Nf = 2

Ayala et al. Nf = 2+1+1G(p2/m2)+G0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


The ghost universal function is just

G(s) = 112

[2 + 1

s − 2s log s + 1s2 (1 + s)2(2s − 1) log (1 + s)

]

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.5 1 1.5 2

χ(p)

-1

p (GeV)


Cucchieri-Mendes SU(2)Ayala et al. Nf = 0Ayala et al. Nf = 2


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


The ghost universal function is just

G(s) = 112

[2 + 1

s − 2s log s + 1s2 (1 + s)2(2s − 1) log (1 + s)

]

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.01 0.1 1 10

χ(p)

-1

p (GeV)


Cucchieri and Mendes SU(2)Ayala et al. Nf = 0Ayala et al. Nf = 2


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

TABLE of OPTIMIZED RENORMALIZATIONCONSTANTS: z J(p/x)−1 + y = F(p2/m2) + F0arXiv:1607.02040

Data set N Nf x y z y′ z′

Bogolubsky et al. 3 0 1 0 3.33 0 1.57Duarte et al. 3 0 1.1 -0.146 2.65 0.097 1.08

Cucchieri-Mendes 2 0 0.858 -0.254 1.69 0.196 1.09

Ayala et al. 3 0 0.933 - - 0.045 1.17Ayala et al. 3 2 1.04 - - 0.045 1.28Ayala et al. 3 4 1.04 - - 0.045 1.28

Table: Scaling constants x, y, z (gluon) and y′, z′ (ghost). Theconstant shifts F0 = −1.05, G0 = 0.24 and the mass m = 0.73 GeV areoptimized by requiring that x = 1 and y = y′ = 0 for the lattice data ofBogolubsky et al. (2009)

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Running CouplingPure Yang-Mills SU(3)

RG invariant product (Landau Gauge – MOM-Taylor scheme):

αs(µ) = αs(µ0)J(µ)χ(µ)2

J(µ0)χ(µ0)2 What if δF0 = δG0 = ±25% ?

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.01 0.1 1 10

α s

µ (GeV)

µ0 = 2 GeV, αs = 0.37, data of Bogolubsky et al.(2009).

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.




J(µ0)χ(µ0)2 What if δF0 = δG0 = ±25% ?

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.1 1 10

α s

µ (GeV)

µ0 = 0.15 GeV, αs = 0.2, data of Bogolubsky et al.(2009).

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.




J(µ0)χ(µ0)2 What if δF0 = δG0 = ±25% ?

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.01 0.1 1 10

α s

µ (GeV)

µ0 = 0.67 GeV, αs = 1.21, data of Bogolubsky et al.(2009).

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

FAQ ISILLY POINTS MADE BY THE REFEREES(more serious list at the end)

Does the photon acquire a mass by the same method?

How can you get anything new by adding "zero" to theLagrangian ? (without inserting any physical ansatz or anymodel for NP physics)

The method must be wrong otherwise it would also explainChiral Symmetry Breaking

The method must be wrong otherwise the propagatorcould be analytically continued to Minkowski space wherethe gluon would get a physical dynamical pole

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.






Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.






Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.






Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.






Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ANALYTIC CONTINUATIONarXiv:1605.07357

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 4

?

p2 (GeV2)

GLUON PROPAGATOR - SU(3)

Lattice

Real Part

Imaginary Part

Lattice data are from Bogolubsky et al. (2009)

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 4

m2= (0.16 ± i 0.60) GeV2

p2 (GeV2)

GLUON PROPAGATOR - SU(3)

Lattice

Real Part

Imaginary Part


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3 3.5 4

m2= (0.16 ± i 0.60) GeV2

p2 (GeV2)

GLUON PROPAGATAOR - SU(3)

Real PartImaginary Part

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

GENERALIZED SPECTRAL FUNCTIONHOW TO DEFINE A SPECTRAL FUNCTION WITH COMPLEX POLES ?

If G(p) has complex poles then

G(p2) = GR(p2) + δG(p2)

where the rational function GR just contains the poles

GR(z) =R

z − α− iβ+

R⋆

z − α+ iβ

and the finite part δG satisfies usual dispersion relations

Re δG(p2) = PV∫ +∞

0

ρ(ω)

p2 − ωdω

ρ(ω) = − 1π

Im δG(ω + iϵ) = − 1π

Im G(ω + iϵ)

GR(p2) cannot be reconstructed from Im G

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ANALYTIC CONTINUATIONDispersion relations with complex poles → arXiv:1606.03769

0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0 1 2 3 4

Re

G(p

)

p2 (GeV2)

Lattice

Re G(p)

GR(p)

Re δG=Re G - GR


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

BACK TO EUCLIDEAN SPACE

GR(z) =R

z − α− iβ+

R⋆

z − α+ iβ=⇒ p2

E + (α+ tβ)p4

E + 2αp2E + (α2 + β2)

where t = (Im R)/(Re R) = tan[arg(R)] RGZ model!

0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0 1 2 3 4

Re

G(p

)

p2 (GeV2)

Lattice

Re G(p)

GR(p)

Re δG=Re G - GR

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ANALYTIC CONTINUATIONGhost dressing function: G(p2) = χ(p2)

p2

ρ(p2) = − 1π

ImG(p2 + iε) = χ(0) δ(p2)− 1π

Imχ(p2)

p2

-1

-0.5

0

0.5

1 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-2-1.5

-1-0.5

0 0.5

1 1.5

2

-Ιmχ(p)

Im p2 (GeV2)

Re p2 (GeV2)

-Ιmχ(p)

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ANALYTIC CONTINUATIONGhost dressing function: G(p2) = χ(p2)

p2

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

ZG

Reχ

(p),

-Z

G Ιm

χ(p)

p2 (GeV2)


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDGluon sector

Optimized by the Lattice Nf = 2, m = 0.8 GeV M = ?

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

-3 -2 -1 0 1 2 3

p2 (GeV2)

REAL PART

Lattice, Nf = 2

M (GeV) = 0.65

0.56

0.52

0.48

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5

p2 (GeV2)

IMAGINARY PART

M (GeV) = 0.4

0.5

0.6

0.7

Lattice data are for two light quarks, from Ayala et al. (2012)

What about poles ?2 pairs of compex conjugated poles

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDGluon sector

Optimized by the Lattice:m = 0.8 GeV, M = 0.65 GeVm2

1 = (0.54 ± 0.52i) GeV2, m22 = (1.69 ± 0.1i) GeV2

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2 2.5

Ιm G(p)

Re G(p)

↑p = 2M

↑p = m

Ιm G

(p),

R

e G

(p)

p2 (GeV2)

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDQuark sector

=Σ +gh

= +qΣ + +

++= + + +Π

++ + +

The counterterm δΓ = −M cancels the mass at tree-levelA massive propagator from loops → S(p) = Z(p)

p−M(p)

A new parameter x = M/m

but

Agreement not as good as for pure YM theory(Z(p) is decreasing)M(p) depends on αs

Optimization is not easy without RG corrections!

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDQuark sector

=Σ +gh

= +qΣ + +

++= + + +Π

++ + +

The counterterm δΓ = −M cancels the mass at tree-levelA massive propagator from loops → S(p) = Z(p)

p−M(p)

A new parameter x = M/m

butAgreement not as good as for pure YM theory(Z(p) is decreasing)M(p) depends on αs

Optimization is not easy without RG corrections!

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDQuark sector – Nf = 2

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4

m = 0.7 GeV

M = 0.65 GeV

Ζ (p

)

pE (GeV)

αs = 0.9

αs = 0.6

αs = 0.3

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 1 10

m = 0.7 GeV

Μ(p

) (

GeV

)

pE (GeV)

αs=0.6

αs=0.9

αs=1.2

αs=1.5

αs=1.8

Lattice

Lattice data are: unquenched, Nf = 2, in the CHIRAL limitBowman et al. (2005)

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 1 10

αs = 0.9

Μ(p

) (

GeV

)

pE (GeV)

m = 0.7 GeV

m = 0.5 GeV

m = 0.2 GeV

Lattice


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.1 1

αs = 0.9

αs = 0.75

αs = 0.6

αs = 0.45

αs = 0.3

m = 0.7 GeV, M = 0.65 GeV

Μ(p

) (

GeV

)

pE (GeV)


Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDQuark sector: ANALYTIC CONTINUATION TO MINKOWSKY SPACE

Quark propagator:

S(p) = Sp(p2)p + SM(p2)

NO COMPLEX POLES =⇒ Standard Dispersion Relations

ρM(p2) = − 1π

Im SM(p2)

ρp(p2) = − 1π

Im Sp(p2)

S(p) =∫ ∞

0dq2 ρp(q2)p + ρM(q2)

p2 − q2 + iε.

Positivity Conditions:

ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


Quark propagator:

S(p) = Sp(p2)p + SM(p2)


ρM(p2) = − 1π

Im SM(p2)

ρp(p2) = − 1π

Im Sp(p2)

S(p) =∫ ∞


p2 − q2 + iε.


ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


Quark propagator:

S(p) = Sp(p2)p + SM(p2)


ρM(p2) = − 1π

Im SM(p2)

ρp(p2) = − 1π

Im Sp(p2)

S(p) =∫ ∞


p2 − q2 + iε.


ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-4 -2 0 2 4

m = 0.7 GeVM = 0.49 GeVαs = 1.2

ρ p, R

e[S

p] (

GeV

-2);

ρM

, R

e[S

M]

(GeV

-1)

p2 (GeV2)

ρp

ρM

Re[Sp]

Re[SM]

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-4 -2 0 2 4

m = 0.7 GeVM = 0.65 GeVαs = 0.9

ρ p, R

e[S

p] (

GeV

-2);

ρM

, R

e[S

M]

(GeV

-1)

p2 (GeV2)

ρp

ρM

Re[Sp]

Re[SM]

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-4 -2 0 2 4

m = 0.7 GeVM = 0.94 GeVαs = 0.6

ρ p, R

e[S

p] (

GeV

-2);

ρM

, R

e[S

M]

(GeV

-1)

p2 (GeV2)

ρp

ρM

Re[Sp]

Re[SM]

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDQuark sector: Nf = 2, M = 0.65 GeV, m = 0.7 GeV

-0.4

-0.2

0

0.2

0.4

0.1 1 10

m = 0.7 GeVM = 0.65 GeVαs = 0.9

↑ p = m+M

↓p = m

(0.32 GeV)

ρ p, R

e[S

p] (

GeV

-2);

ρM

, R

e[S

M]

(GeV

-1)

p2 (GeV2)

ρp

ρM

Re[Sp]

Re[SM]


ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

CHIRAL QCDQuark sector: Nf = 2, M = 0.65 GeV, m = 0.7 GeV

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5

m = 0.7 GeV

M = 0.65 GeVαs = 0.9

p ρ p

(p)

- ρ M

(p)

(G

eV-1

)

p2 (GeV2)


ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

FAQ II.MORE SERIOUS LIST (that the Referees did not raise)

What about the Longitudinal Polarization ?The Landau gauge is very special!

Can we Optimize the expansion (without Lattice Data) ?May be by real observables, like glueball masses

Is there any proof of renormalizability ?

What about improving the expansion by RG ?

More questions and remarks are welcome!

THANK YOU

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.







THANK YOU

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.







THANK YOU

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.







THANK YOU

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.







THANK YOU

Fabio Siringo


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.







THANK YOU

Fabio Siringo

Documents

The Dark Side of the Propagators - Analytical approach to ......Do all the work analytically and forget about Lattice People! Change the expansion point! Massive Expansion also known