the dark side of the propagators - analytical approach to ......do all the work analytically and...
TRANSCRIPT
![Page 1: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/1.jpg)
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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
![Page 2: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/2.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 3: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/3.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 4: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/4.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 5: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/5.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 6: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/6.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 7: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/7.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 8: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/8.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 9: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/9.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 10: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/10.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 11: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/11.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 12: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/12.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 13: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/13.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 14: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/14.jpg)
UNIVERSITY of CATANIA, ITALY
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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 modified
Standard UV behaviorFrom first principles
Fabio Siringo
![Page 15: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/15.jpg)
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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 behavior
From first principles
Fabio Siringo
![Page 16: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/16.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 17: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/17.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 18: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/18.jpg)
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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
![Page 19: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/19.jpg)
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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
![Page 20: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/20.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 21: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/21.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 22: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/22.jpg)
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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
![Page 23: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/23.jpg)
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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
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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
![Page 25: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/25.jpg)
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UNIVERSAL SCALINGGHOST INVERSE DRESSING FUNCTION
Denoting by G(s) the ghost universal function (F(s) → G(s))
0.4
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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
![Page 26: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/26.jpg)
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UNIVERSAL SCALINGGHOST INVERSE DRESSING FUNCTION
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
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0.7
0.8
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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
![Page 27: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/27.jpg)
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UNIVERSAL SCALINGGHOST INVERSE DRESSING FUNCTION
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)
Bogolubsky et al.Duarte et al.
Cucchieri and Mendes SU(2)Ayala et al. Nf = 0Ayala et al. Nf = 2
Ayala et al. Nf = 2+1+1G(p2/m2)+G0
Fabio Siringo
![Page 28: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/28.jpg)
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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
![Page 29: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/29.jpg)
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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
![Page 30: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/30.jpg)
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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
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
![Page 31: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/31.jpg)
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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
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
![Page 32: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/32.jpg)
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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
![Page 33: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/33.jpg)
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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
![Page 34: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/34.jpg)
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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
![Page 35: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/35.jpg)
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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
![Page 36: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/36.jpg)
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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
![Page 37: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/37.jpg)
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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
![Page 38: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/38.jpg)
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ANALYTIC CONTINUATIONarXiv:1605.07357
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
Lattice data are from Bogolubsky et al. (2009)
Fabio Siringo
![Page 39: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/39.jpg)
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ANALYTIC CONTINUATIONarXiv:1605.07357
-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
![Page 40: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/40.jpg)
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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
![Page 41: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/41.jpg)
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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
Lattice data are from Bogolubsky et al. (2009)
Fabio Siringo
![Page 42: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/42.jpg)
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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
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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
![Page 43: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/43.jpg)
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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
![Page 44: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/44.jpg)
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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)
Lattice data are from Bogolubsky et al. (2009)
Fabio Siringo
![Page 45: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/45.jpg)
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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
![Page 46: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/46.jpg)
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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
![Page 47: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/47.jpg)
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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
![Page 48: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/48.jpg)
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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
![Page 49: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/49.jpg)
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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
![Page 50: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/50.jpg)
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CHIRAL QCDQuark sector – Nf = 2
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
![Page 51: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/51.jpg)
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CHIRAL QCDQuark sector – Nf = 2
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
Lattice data are: unquenched, Nf = 2, in the CHIRAL limitBowman et al. (2005)
Fabio Siringo
![Page 52: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/52.jpg)
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CHIRAL QCDQuark sector – Nf = 2
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)
Lattice data are: unquenched, Nf = 2, in the CHIRAL limitBowman et al. (2005)
Fabio Siringo
![Page 53: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/53.jpg)
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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
![Page 54: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/54.jpg)
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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
![Page 55: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/55.jpg)
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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
![Page 56: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/56.jpg)
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CHIRAL QCDQuark sector – Nf = 2
-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
![Page 57: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/57.jpg)
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CHIRAL QCDQuark sector – Nf = 2
-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
![Page 58: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/58.jpg)
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CHIRAL QCDQuark sector – Nf = 2
-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
![Page 59: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/59.jpg)
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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]
Positivity Conditions:
ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0
Fabio Siringo
![Page 60: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/60.jpg)
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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)
Positivity Conditions:
ρp(p2) ≥ 0, p ρp(p2)− ρM(p2) ≥ 0
Fabio Siringo
![Page 61: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/61.jpg)
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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
![Page 62: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/62.jpg)
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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
![Page 63: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/63.jpg)
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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
![Page 64: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/64.jpg)
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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
![Page 65: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/65.jpg)
UNIVERSITY of CATANIA, ITALY
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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
![Page 66: 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](https://reader033.vdocuments.us/reader033/viewer/2022052020/6033d069656c5b12502e9885/html5/thumbnails/66.jpg)
UNIVERSITY of CATANIA, ITALY
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.
.
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.
.
.
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.
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.
.
.
.
.
.
.
.
.
.
.
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