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PhD Exam Talk
1. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Systematic comparison between non-perturbative
functional methods in low-energy QCD models
Jordi ParĀ“ıs LĀ“opez
Advisors: R. Alkofer and H. Sanchis-Alepuz
Karl-Franzens-UniversitĀØat Graz, Austria
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 1 / 33
2. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Content
Motivation and thesis objectives.
Basics of the functional methods.
Results using the Functional Renormalisation Group (FRG).
Comparison between functional methods.
Summary.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 2 / 33
3. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Motivation and thesis objectives
Many features from QCD still not completely understood.
Bound states inherently non-perturbative.
Large couplings in QCD at hadronic energies.
Non-perturbative approaches required ā Functional Methods.
No sign problem.
Wide range of scales.
Successful predictions in QCD: Observables, DĻSB,...
Diļ¬erent truncations and approximations.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 3 / 33
4. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Functional methods treated:
Dyson-SchwingerāBethe-Salpeter equations (DSE-BSE).
Functional Renormalisation Group (FRG).
Objectives
Obtain observables using the FRG in diļ¬erent approximations.
Compare both approaches in diļ¬erent low-energy QCD models.
Analyse viability of the methods: truncations, numerics, etc.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 4 / 33
5. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Basics of functional methods
Euclidean generating functional as starting point:
Z[J] = eW[J]
= DĻ eāS[Ļ]+ x JĻ
Eļ¬ective Action Ī[Ļ] from W[J] Legendre transformation:
eāĪ[Ļ]
= DĻ exp āS[Ļ + Ļ] +
x
dĪ[Ļ]
dĻ
Ļ
with Ī“Ī
Ī“Ļ ā” J , Ļ ā” Ī“W[J]
Ī“J = Ļ J .
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 5 / 33
6. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
The eļ¬ective action Ī[Ļ]:
Expressed as sum of 1PI Greenās functions.
Main object of interest in functional methods.
Calculation of Ī[Ļ] using functional equations:
DSE: coupled integral equations.
FRG: diļ¬erential equations containing integrals.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 6 / 33
7. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
The Functional Renormalisation Group (FRG)1
Main functional: scale dependent 1-PI eļ¬ective action: Ī[Ļ] ā Īk[Ļ].
Scale introduced via regulator āSk[Ļ].
Initial and ļ¬nal conditions are ļ¬xed in theory space:
Īk=Ī ā Sbare
Īkā0 ā” Ī
The choice of the regulator is not unique.
1
See, e.g., Gies, arXiv:hep-ph/0611146 for an introduction.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 7 / 33
8. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Using quadratic regulators āSk[Ļ] = p ĻRkĻ:
ātĪk =
1
2
Tr ātRk Ī
(2)
k + Rk
ā1
Wetterichās Flow Equation
with t = ln k
Ī and āt = kāk.
Euclidean non-perturbative 1-loop integral-diļ¬erential equation.
Leads to non-perturbative ļ¬ow equation for vertex functions:
ā1
=āt + + +
Truncation/approximation required.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 8 / 33
9. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Dynamical hadronisation
Convenient to work with macroscopic QCD degrees of freedom.
Mesons introduced from a 4-Fermi interaction via the
Hubbard-Stratonovich (HS) transformation.
Problem: non-zero 4-Fermi interaction ļ¬ow ātĪ»k
=ā HS transformation cancelled in every RG-step:
āt = + . . .
Solved by dynamical hadronisation.
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10. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Introduction of scale dependent bosonic ļ¬eld:
ātĻk = ātAk( ĀÆĻĻĻ)
Wetterichās ļ¬ow equation modiļ¬ed =ā Additional term in ātĪ»k:
ātĪ»k = Flow Ī»k ā hkātAk
!
= 0
Generalisation of HS transformation for every RG-step.
Greenās functions computed with meson exchange diagrams:
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11. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Results using the FRG
Gluons decoupled at low energies2. Low-energy QCD eļ¬ectively described
by fermionic NJL-like models. Mesons introduced via HS transformation.
Approximate eļ¬ective action of the Quark Meson model:
Īk
ĀÆĻ, Ļ, Ļ, Ļ = Ī
(int)
k,4Ļ [ ĀÆĻ, Ļ] +
p
Zk,Ļ
ĀÆĻ i/p Ļ +
+
1
2
p2 Zk,Ļ Ļ2 + Zk,Ļ Ļ2 + Vk[Ļ, Ļ] ā cĻ +
+
q
hk
ĀÆĻ
Ļ
2
+ iĪ³5ĻzĻz
Ļ
2
Comparison to the full calculation, see A.Cyrol et al, arXiv:1605.01856.
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12. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Multi-meson interactions introduced via O(N) potential following:
Vk(Ļ) =
ā
n=0
V
(n)
k
n!
(Ļ ā Ļ0)n
with Ļ = 1
2 Ļ2 + Ļ2 and Ļ0 scale independent expansion point.
Flow equations to solve:
Potential terms, ĖV
(i)
k with i = 0, ... , 8.
Wave function renormalisation, ĖZk,i with i = Ļ, Ļ, Ļ.
4-Fermi coupling, ĖĪ»k = Flow Ī»k ā hk
ĖAk ā” 0.
Yukawa coupling, Ėhk = Flow hk ā V
(1)
k
ĖAk.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 12 / 33
13. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Approximations used:
LPA: Scale-dependent potential, constant Yukawa coupling
hk(p2) = h, unit Zk,i(p2) = 1 and zero 4-Fermi coupling Ī»k = 0.
LPA+Y: Yukawa coupling includes scale dependence.
LPA+Yā: Yukawa coupling includes scale and momentum
dependence.
Full: Scale and momentum-dependent wave function renormalisations
Zk,i(p2) are included.
Full+DH: Dynamical hadronisation taken into account.
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14. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
0.0 0.2 0.4 0.6 0.8 1.0
k (GeV)
0.5
1.0
1.5
2.0
2.5
ĀÆmk(GeV)
Pion
Sigma Meson
LPA
LPA+Y
LPA+Yā
Full
Full+DH
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15. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
0.0 0.2 0.4 0.6 0.8 1.0
k (GeV)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
ĀÆmk,Ļ(GeV)
LPA
LPA+Y
LPA+Yā
Full
Full+DH
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16. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
p (GeV)
0.25
0.26
0.27
0.28
0.29
0.30
ĀÆmIR,Ļ(GeV)
LPA+Yā
Full
Full+DH
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 16 / 33
18. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Applying analytic continuation to obtain pole masses and comparing with
curvature āmassesā (CM) mk,i we obtained:
Particle CM (Input) Pole Mass Decay Width
Pion 138.053 137.6 Ā± 0.4 0.5 Ā± 0.5
Sigma meson 551.843 330 Ā± 15 30 Ā± 6
Table: Pole masses vs. curvature masses and decay widths, all in MeV.
Pion pole mass agrees with CM, decay width compatible with zero.
Sigma meson pole mass close to two pion decay threshold, pole
belonging to second Riemann sheet.
Analytic continuation used requires large number of data points.
Results compatible with QCD calculations.3
3
Comparison with fQCD calculations, see Alkofer et al, arXiv:1810.07955.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 18 / 33
19. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Comparison between functional methods
Formal comparison.
Practical comparison in truncated low-energy QCD models.
Nambu-Jona-Lasinio (NJL) model.
Gross-Neveu (GN) model.
Quark-Meson (QM) model.
Numerical comparison.
Intrinsic properties of the methods.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 19 / 33
20. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Dyson-Schwinger equations (DSEs)
Consequence from cancellation of path integral under total derivative:
DĻ
Ī“
Ī“Ļ
eāS[Ļ]+ x JĻ
= 0
DSEs for 1PI correlators:
Ī“Ī[Ļ]
Ī“Ļi
ā
Ī“S
Ī“Ļi
Ļ +
Ī“2Ī[Ļ]
Ī“ĻĪ“Ļj
ā1
Ī“
Ī“Ļj
= 0
Self-coupled integral equations not exactly solvable in general.
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21. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
DSEs in QCD:
ā1ā1
=
=
Quark Propagator
+
+
++
Quark-Gluon Vertex
...
Inļ¬nite tower of coupled equations.
Truncation is required.
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23. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Bethe-Salpeter equation: bound state equation for mesons:
Ī = KG0Ī
Pion BSE under Rainbow-Ladder truncation:
q q~
kq P P=
Ī Ī
ā0.20 ā0.15 ā0.10 ā0.05 0.00
p2 (GeV)
0.96
0.98
1.00
1.02
1.04
Ī»
ā0.25 ā0.20 ā0.15 ā0.10 ā0.05 0.00
p2 (GeV)
ā2000
ā1500
ā1000
ā500
0
500
1000
1500
2000
f(0)
(p2
,0,0)
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24. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
The NJL model
Fermion system with 4-Fermi interaction:
S[ ĀÆĻ, Ļ] =
p
ĀÆĻ(i/p + mq)Ļ + Ī» ĀÆĻĻ
2
Diagrammatic equations:
ā1
ā1ā1
=
=
āt
Quark DSE
+ +
Quark ļ¬ow equation
+
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25. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Same analytical expression obtained:
Proper interpretation of scale-dependent parameters.
Using constant Ī» ā c
Ī2 approximation.
0 1 2 3 4 5
c
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
M(GeV)
mq = 0
mq = 0
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26. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
The GN model
Fermion system with 4-Fermi interaction in 2-dimensions:
S[ ĀÆĻ, Ļ] =
d2p
(2Ļ)2
ĀÆĻ(i/p + mq)Ļ + Ī» ĀÆĻĻ
2
System is renormalisable.
Quark propagator dressings get momentum dependence.
2-loop terms appear.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 26 / 33
27. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
ā1ā1
ā1
=
=
=
=
āt
āt
DSE
+ + +
+ + + + +
+
+ + +
+ + +
+
FRG
aaa
a
a
a aaaa
aaaa
b
bb
b
b
b
b
b
b
b
b
b
b
b
c
cc
c
c
c
c
ccc
c
ccc
ddd
d
d
d dd
d
d
dd
d
d
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 27 / 33
28. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
10ā3
10ā2
10ā1
100
101
102
103
104
105
p2
(GeV)
2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M(p2
)(GeV)
FRG
DSE
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29. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
The Quark-Meson model
Bare action from bosonised NJL model:
S[Ļ, ĀÆĻ, Ļ, Ļ] =
p
ĀÆĻ Z2 i/p Ļ +
m2
2
ZĻ Ļ2
+ ZĻ Ļ2
+
q
ĀÆĻh
ZhĻ
2
Ļ + i ZhĻ Ī³5 Ļ Ļ Ļ
No bosonic kinetic terms.
Momentum-dependent quantities generated dynamically.
Self-coupled system of equations with zero quark-multi-meson vertex.
Jordi ParĀ“ıs LĀ“opez Systematic comparison between functional methods in low-energy QCD 29 / 33
31. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Intrinsic properties of the FRG
Dynamically generated kinetic terms.
Propagating degrees of freedom are preserved.
Probability amplitude conservation during ļ¬ow:
Zā2
k,Ļ +
1
4
Z2
k,Ļ +
3
4
Z2
k,Ļ ā” Zk,s = 1 āk
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32. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
0.0 0.2 0.4 0.6 0.8
k (GeV)
0.0
0.2
0.4
0.6
0.8
1.0
WaveFunctionRenormalisation
Zā1
k,Ļ
Zk,Ļ
Zk,Ļ
Zk,s
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33. Motivation and objectives Basics of functional methods Results using the FRG DSE-FRG comparison Summary
Summary
The FRG provides an alternative procedure to the BSE/Faddeev
equation to obtain resonance masses and decay widths.
Observables obtained are compatible with physical processes.
Approximations compatible in both functional methods can be found,
relating FRG with DSEs and BSEs.
The FRG reduces complexity of equations by introducing an
additional parameter.
Sophisticated numerical tools required in both functional methods.
THANK YOU FOR YOUR ATTENTION
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