PhD Exam Talk

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 Par´ıs López
Advisors: R. Alkofer and H. Sanchis-Alepuz
Karl-Franzens-Universität Graz, Austria
Jordi Par´ıs López Systematic comparison between functional methods in low-energy QCD 1 / 33

Content
Motivation and thesis objectives.
Basics of the functional methods.
Results using the Functional Renormalisation Group (FRG).
Comparison between functional methods.
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 → Functional Methods.
No sign problem.
Wide range of scales.
Successful predictions in QCD: Observables, DχSB,...
Diﬀerent truncations and approximations.

Functional methods treated:
Dyson-Schwinger–Bethe-Salpeter equations (DSE-BSE).
Functional Renormalisation Group (FRG).
Objectives
Obtain observables using the FRG in diﬀerent approximations.
Compare both approaches in diﬀerent low-energy QCD models.
Analyse viability of the methods: truncations, numerics, etc.

Basics of functional methods
Euclidean generating functional as starting point:
Z[J] = eW[J]
= Dφ e−S[φ]+ x Jφ
Eﬀective Action Γ[φ] from W[J] Legendre transformation:
e−Γ[ϕ]
= Dφ exp −S[ϕ + φ] +
x
dΓ[ϕ]
dϕ
φ
with δΓ
δϕ ≡ J , ϕ ≡ δW[J]
δJ = φ J .

The eﬀective action Γ[φ]:
Expressed as sum of 1PI Green’s functions.
Main object of interest in functional methods.
Calculation of Γ[φ] using functional equations:
DSE: coupled integral equations.
FRG: diﬀerential equations containing integrals.

The Functional Renormalisation Group (FRG)1
Main functional: scale dependent 1-PI effective action: Γ[φ] → Γk[φ].
Scale introduced via regulator ∆Sk[φ].
Initial and final conditions are fixed in theory space:
Γk=Λ ≃ Sbare
Γk≃0 ≡ Γ
The choice of the regulator is not unique.
1
See, e.g., Gies, arXiv:hep-ph/0611146 for an introduction.

Using quadratic regulators ∆Sk[φ] = p φRkφ:
∂tΓk =
1
2
Tr ∂tRk Γ
(2)
k + Rk
−1
Wetterich’s Flow Equation
with t = ln k
Λ and ∂t = k∂k.
Euclidean non-perturbative 1-loop integral-diﬀerential equation.
Leads to non-perturbative ﬂow equation for vertex functions:
−1
=∂t + + +
Truncation/approximation required.

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 ﬂow ∂tλk
=⇒ HS transformation cancelled in every RG-step:
∂t = + . . .
Solved by dynamical hadronisation.

Introduction of scale dependent bosonic field:
∂tφk = ∂tAk( ¯ψτψ)
Wetterich’s flow equation modified =⇒ Additional term in ∂tλk:
∂tλk = Flow λk − hk∂tAk
!
= 0
Generalisation of HS transformation for every RG-step.
Green’s functions computed with meson exchange diagrams:

Results using the FRG
Gluons decoupled at low energies2. Low-energy QCD eﬀectively described
by fermionic NJL-like models. Mesons introduced via HS transformation.
Approximate eﬀective action of the Quark Meson model:
Γk
¯ψ, ψ, σ, π = Γ
(int)
k,4ψ [ ¯ψ, ψ] +
p
Zk,ψ
¯ψ i/p ψ +
+
1
2
p2 Zk,σ σ2 + Zk,π π2 + Vk[σ, π] − cσ +
+
q
hk
¯ψ
σ
2
+ iγ5τzπz
ψ
2
Comparison to the full calculation, see A.Cyrol et al, arXiv:1605.01856.

Multi-meson interactions introduced via O(N) potential following:
Vk(ρ) =
∞
n=0
V
(n)
k
n!
(ρ − ρ0)n
with ρ = 1
2 σ2 + π2 and ρ0 scale independent expansion point.
Flow equations to solve:
Potential terms, ˙V
(i)
k with i = 0, ... , 8.
Wave function renormalisation, ˙Zk,i with i = σ, π, ψ.
4-Fermi coupling, ˙λk = Flow λk − hk
˙Ak ≡ 0.
Yukawa coupling, ˙hk = Flow hk − V
(1)
k
˙Ak.

Approximations used:
LPA: Scale-dependent potential, constant Yukawa coupling
hk(p2) = h, unit Zk,i(p2) = 1 and zero 4-Fermi coupling λk = 0.
LPA+Y: Yukawa coupling includes scale dependence.
LPA+Y’: 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.

0.0 0.2 0.4 0.6 0.8 1.0
k (GeV)
0.5
1.0
1.5
2.0
2.5
¯mk(GeV)
Pion
Sigma Meson
LPA
LPA+Y
LPA+Y’
Full
Full+DH

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
¯mk,ψ(GeV)
LPA
LPA+Y
LPA+Y’
Full
Full+DH

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
¯mIR,ψ(GeV)
LPA+Y’
Full
Full+DH

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
p2
(GeV)
2
0
1
2
3
4
5
¯Γ
(2)
IR,φ(GeV)
2
Pion
Sigma Meson
Full
Full+DH
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
p2
(GeV)
2
0
10
20
30
40
50
¯Γ
(−2)
IR,φ(GeV)
−2
Pion
Sigma Meson
Full
Full+DH
−0.142 −0.140 −0.138 −0.136 −0.134 −0.132
p2 (GeV)
−30000
−20000
−10000
0
10000
20000
30000
¯Γ
(−2)
IR,π(GeV)
−2
Padé
Modified Padé
0.0 0.1 0.2 0.3 0.4 0.5 0.6
p0 (GeV)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ρ(GeV)
−2
Pion
Sigma Meson

Applying analytic continuation to obtain pole masses and comparing with
curvature ”masses” (CM) mk,i we obtained:
Particle CM (Input) Pole Mass Decay Width
Pion 138.053 137.6 ± 0.4 0.5 ± 0.5
Sigma meson 551.843 330 ± 15 30 ± 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.

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.

Dyson-Schwinger equations (DSEs)
Consequence from cancellation of path integral under total derivative:
Dφ
δ
δφ
e−S[φ]+ x Jφ
= 0
DSEs for 1PI correlators:
δΓ[ϕ]
δϕi
−
δS
δϕi
ϕ +
δ2Γ[ϕ]
δϕδϕj
−1
δ
δϕj
= 0
Self-coupled integral equations not exactly solvable in general.

DSEs in QCD:
−1−1
=
=
Quark Propagator
+
+
++
Quark-Gluon Vertex
...
Inﬁnite tower of coupled equations.
Truncation is required.

Quark DSE under Rainbow-Ladder truncation:
-1 -1
= - α(k2
)
Solution for inverse quark propagator S−1(p) = A(p2)i/p + B(p2):
10−3
10−2
10−1
100
101
102
103
p2
(GeV)
2
1.0
1.1
1.2
1.3
1.4
1.5
A(p2
)
mq = 0
mq = 0
10−2
10−1
100
101
102
103
p2
(GeV)
2
10−3
10−2
10−1
M(p2
)(GeV)
mq = 0
mq = 0

Bethe-Salpeter equation: bound state equation for mesons:
Γ = KG0Γ
Pion BSE under Rainbow-Ladder truncation:
q q~
kq P P=
Γ Γ
−0.20 −0.15 −0.10 −0.05 0.00
p2 (GeV)
0.96
0.98
1.00
1.02
1.04
λ
−0.25 −0.20 −0.15 −0.10 −0.05 0.00
p2 (GeV)
−2000
−1500
−1000
−500
0
500
1000
1500
2000
f(0)
(p2
,0,0)

The NJL model
Fermion system with 4-Fermi interaction:
S[ ¯ψ, ψ] =
p
¯ψ(i/p + mq)ψ + λ ¯ψψ
2
Diagrammatic equations:
−1
−1−1
=
=
∂t
Quark DSE
+ +
Quark ﬂow equation
+

Same analytical expression obtained:
Proper interpretation of scale-dependent parameters.
Using constant λ ∝ c
Λ2 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

The GN model
Fermion system with 4-Fermi interaction in 2-dimensions:
S[ ¯ψ, ψ] =
d2p
(2π)2
¯ψ(i/p + mq)ψ + λ ¯ψψ
2
System is renormalisable.
Quark propagator dressings get momentum dependence.
2-loop terms appear.

−1−1
−1
=
=
=
=
∂t
∂t
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

10−3
10−2
10−1
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

The Quark-Meson model
Bare action from bosonised NJL model:
S[ψ, ¯ψ, σ, π] =
p
¯ψ Z2 i/p ψ +
m2
2
Zσ σ2
+ Zπ π2
+
q
¯ψh
Zhσ
2
σ + i Zhπ γ5 τ π ψ
No bosonic kinetic terms.
Momentum-dependent quantities generated dynamically.
Self-coupled system of equations with zero quark-multi-meson vertex.

10−5
10−4
10−3
10−2
10−1
100
p2
(GeV)
2
0
10
20
30
40
50
Gπ
(p2
)(GeV)
−2
FRG
DSE
10−5
10−4
10−3
10−2
10−1
100
p2
(GeV)
2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Gσ
(p2
)(GeV)
−2
FRG
DSE
10−5
10−4
10−3
10−2
10−1
100
p2
(GeV)
2
0.96
0.98
1.00
1.02
1.04
1.06
1.08
A(p2
)
FRG
DSE
10−5
10−4
10−3
10−2
10−1
100
p2
(GeV)
2
0.260
0.265
0.270
0.275
0.280
0.285
0.290
0.295
M(p2
)(GeV)
FRG
DSE

Intrinsic properties of the FRG
Dynamically generated kinetic terms.
Propagating degrees of freedom are preserved.
Probability amplitude conservation during ﬂow:
Z−2
k,ψ +
1
4
Z2
k,σ +
3
4
Z2
k,π ≡ Zk,s = 1 ∀k

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
Z−1
k,ψ
Zk,π
Zk,σ
Zk,s

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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