-
Callan-Symanzik equations for infrared QCD
Axel Weber
Instituto de Fı́sica y Matemáticas (IFM)Universidad Michoacana
de San Nicolás de Hidalgo (UMSNH)
Seminar Theoretical Hadron PhysicsInstitut für Theoretische
Physik
Justus-Liebig-Universität GiessenJuly 4, 2018
Collaborators: Pietro Dall’Olio, Francisco Astorga, Juan Pablo
Gutiérrez
1 / 57
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Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
2 / 57
-
Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
3 / 57
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Quantization of continuum Yang-Mills theory
◮ SU(N) Yang-Mills theory in the continuum, Landau gauge
◮ Faddeev-Popov determinant ⇒ ghosts (and BRST symmetry)
◮ Faddeev-Popov action in D-dimensional Euclidean space-time
S =
∫
dDx
(
1
4F aµνF
aµν + ∂µc̄
a(Dµc)a + iBa∂µA
aµ
)
◮ gauge copies ⇒ restriction of the gauge field configurations
to the (first) Gribov regionΩ (properly to the fundamental modular
region) [Gribov 1978]
◮ Zwanziger’s horizon function, breaks the BRST symmetry; local
formulation ⇒additional auxiliary fields [Zwanziger 1989]
◮ condensates of the additional auxiliary fields: “refined
Gribov-Zwanziger scenario” ⇒effective mass term for the gluons
[Dudal et al. 2008]
4 / 57
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Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
5 / 57
-
◮ Dyson-Schwinger equations are not affected by the restriction
of the A-integral to Ω(without introducing the horizon function):
the contributions from the boundary of Ωvanish because det(−∂µDabµ
) = 0 at the boundary [Zwanziger 2002]
◮ the perturbative expansion of the correlation functions,
obtained from the iterativesolution of the Dyson-Schwinger
equations, is also unchanged
◮ what can change are the (re)normalization conditions; and the
BRST symmetry isbroken
6 / 57
-
How nonperturbative is IR Yang-Mills theory?
◮ effective description by a local renormalizable quantum field
theory: include a gluonicmass term in the Faddeev-Popov action (in
4-dimensional Euclidean space-time)
S =
∫
d4x
(
1
4F aµνF
aµν +
1
2Aaµm
2Aaµ + ∂µc̄aDabµ c
b + iBa∂µAaµ
)
(Curci-Ferrari model, perturbatively renormalizable)
◮ apply straightforward one-loop perturbation theory to this
action;adjust two constants, g,m2, at some renormalization scale(in
principle, m2 is nonperturbatively fixed in terms of ΛQCD , or
g)
◮ result: excellent fit to the lattice data for the propagators
in the IR [Tissier, Wschebor2010]
7 / 57
-
◮ notations: propagators
〈Aaµ(p)Abν(−q)〉 = GA(p
2)
(
δµν −pµpν
p2
)
δab(2π)4δ(p − q)
〈ca(p)c̄b(−q)〉 = Gc(p2) δab(2π)4δ(p − q)
and dressing functions
GA(p2) =
FA(p2)
p2, Gc(p
2) =Fc(p2)
p2
◮ one-loop contributions to the gluon self energy
and the ghost self energy
calculated with a massive gluon propagator
GA(p2) =
1
m2 + p2
8 / 57
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gluon propagator GA(p2) in 4 dimensions
Tissier, Wschebor 2010
SU(2) lattice data: Cucchieri, Mendes 2008a
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
G(p
)
p (GeV)
at tree level, GA(p2) ∝
1
m2 + p2
9 / 57
-
ghost dressing function Fc(p2) = p2Gc(p2) in 4
dimensionsTissier, Wschebor 2010
SU(2) lattice data: Cucchieri, Mendes 2008b
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
F(p)
p (GeV)
at tree level, Fc(p2) = 1
10 / 57
-
gluon dressing function FA(p2) = p2GA(p
2) in 4 dimensionsTissier, Wschebor 2010
SU(3) lattice data: Bogolubsky et al. 2009
and Dudal, Oliveira, Vandersickel 2010
2 p G
(p)
p(GeV)
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8
at tree level, FA(p2) ∝
p2
m2 + p2
renormalization group improvement necessary for the UV
11 / 57
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Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
12 / 57
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Extreme IR regime
◮ successful description of the IR fixed point for all
dimensions (D ≥ 2) withCallan-Symanzik equations in an epsilon
expansion[Weber 2012 and Weber, Dall’Olio, Astorga 2016]
◮ normalization condition for the gluon propagator
GA(p2 = µ2) =
1
m2
corresponding to a “high-temperature” fixed point
◮ for D > 2 dimensions (“decoupling solutions”), use an
epsilon expansion in D = 2 + ǫdimensions with the normalization
condition for the ghost propagator
Gc(p2 = µ2) =
1
p2
◮ for D = 2 dimensions (“scaling solution”), use an epsilon
expansion in D = 6 − ǫdimensions with the normalization condition
for the ghost propagator
Gc(p2 = µ2) =
b2
p4
corresponding to a “Lifshitz point” and Zwanziger’s original
horizon condition
◮ in the following, describe the crossover from the UV to the IR
fixed point, and hence thecomplete momentum dependence of the
propagators, in dimension D = 4
13 / 57
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◮ “IR safe” renormalization scheme proposed by Tissier and
Wschebor [Tissier,Wschebor 2011]: normalization conditions for the
proper two-point functions
Γ⊥A (p2)∣
∣
p2=µ2= m2 + p2
Γ‖A(p2)
∣
∣
p2=µ2= m2
Γc(p2)∣
∣
p2=µ2= p2
◮ Γ⊥A
and Γ‖A
are the transverse and longitudinal parts of the proper gluonic
2-pointfunction
Γ(2)A,µν
(p) = Γ⊥A (p2)
(
δµν −pµpν
p2
)
+ Γ‖A(p2)
pµpν
p2
◮ the first two normalization conditions can be rewritten as
Γ⊥A (p2)− Γ
‖A(p2)
∣
∣
p2=µ2= p2
Γ‖A(p2)
∣
∣
p2=µ2= m2
◮ these combinations correspond to the decomposition of the
2-point function
Γ(2)A,µν
(p) =(
Γ⊥A (p2)− Γ
‖A(p2)
)
(
δµν −pµpν
p2
)
+ Γ‖A(p2)δµν
which is analogous to the grouping of terms in the classical
action
p2(
δµν −pµpν
p2
)
+ m2δµν
14 / 57
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◮ renormalized coupling constant defined from the renormalized
proper ghost-gluonvertex in the Taylor limit (ghost momentum p → 0)
where there are no loop corrections
to the vertex ⇒ g(µ2) = Z1/2A
(µ2)Zc(µ2)gB(alternatively, use the symmetry point p2 = q2 = k2
= µ2)
◮ calculate the flow functions at one-loop order; then, solve
the Callan-Symanzikrenormalization group equations for the
propagators
◮ example: the µ2-independence of the bare longitudinal proper
gluonic 2-point functionimplies
µ2d
dµ2Γ‖A(p2, µ2) = µ2
d
dµ2
(
ZA(µ2) Γ
‖ BA
(p2, µ2))
=
(
µ2d
dµ2ZA(µ
2)
)
Γ‖ BA
(p2, µ2)
= γA(µ2) Γ
‖A(p2, µ2)
◮ integrating between two renormalization scales µ̄2 and µ2,
Γ‖A(p2, µ2) = Γ
‖A(p2, µ̄2) exp
(
∫ µ2
µ̄2
dµ′ 2
µ′ 2γA(µ
′ 2)
)
◮ setting µ̄2 = p2 and using the normalization condition,
Γ‖A(p2, µ2) = m2(p2) exp
(
−
∫ p2
µ2
dµ′ 2
µ′ 2γA(µ
′ 2)
)
15 / 57
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◮ γA(µ2) and γc(µ2) depend on g(µ2) and m2(µ2) which are
obtained from the
integration of the system of differential equations
µ2d
dµ2m2(µ2) = µ2
d
dµ2Γ‖A(µ2, µ2)
∣
∣
∣
gB ,m2B
= βm2(
g(µ2),m2(µ2), µ2)
µ2d
dµ2g(µ2) = µ2
d
dµ2g(µ2)
∣
∣
∣
gB ,m2B
= βg(
g(µ2),m2(µ2), µ2)
◮ all these equations are exact if the flow functions are exact;
here, calculate the flowfunctions to one-loop order and integrate
the Callan-Symanzik equations with theseapproximate flow functions:
“renormalization group improvement” of perturbation theory
◮ adjust g(µ0),m2(µ0) at some renormalization scale to fit the
lattice data;
note that the lattice propagators are not normalized and thus
can be arbitrarily rescaled(field rescalings)
◮ fitting strategy: fix g(µ0),m2(µ0) by adjusting to the data
for the ghost propagator and
the ghost dressing function;comparison to the data for the gluon
propagator and the gluon dressing function thenshows how successful
the renormalization scheme is in reproducing the lattice data
◮ in all of the following, the SU(2) lattice data are from
Cucchieri and Mendes [Cucchieri,Mendes 2008a, 2008b]
16 / 57
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running coupling constant g(µ) in D = 4 dimensions
10-4
10-3
10-2
10-1
100
101
102
103
104
p [GeV]
0
1
2
3
4
5
6
7
8
9
10
11
12g(
p)
17 / 57
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running mass parameter m2(µ) in D = 4 dimensions
10-4
10-3
10-2
10-1
100
101
102
103
p [GeV]
0
0.1
0.2
0.3
0.4
0.5
m2
[GeV
2 ]
18 / 57
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ghost dressing function Fc(p2) = p2Gc(p2), Taylor scheme
0 1 2 3 4 5p [GeV]
1
2
3
4
p2 G
c(p)
19 / 57
-
gluon propagator function GA(p2), Taylor scheme
0 1 2 3 4 5p [GeV]
0
1
2
3
4
GA
(p)
20 / 57
-
gluon dressing function FA(p2) = p2GA(p
2), Taylor scheme
0 1 2 3 4 5p [GeV]
0
0.5
1
1.5p2
GA
(p)
21 / 57
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Derivative schemes◮ in the decomposition of the gluonic 2-point
function
Γ(2)A,µν
(p) =(
Γ⊥A (p2)− Γ
‖A(p2)
)
(
δµν −pµpν
p2
)
+ Γ‖A(p2)δµν
replace the normalization condition
Γ⊥A (p2)− Γ
‖A(p2)
∣
∣
p2=µ2= p2
withd
dp2
(
Γ⊥A (p2)− Γ
‖A(p2)
) ∣
∣
∣
p2=µ2= 1
◮ complement with the normalization conditions
Γ‖A(p2)
∣
∣
p2=µ2= m2
d
dp2Γc(p
2)∣
∣
∣
p2=µ2= 1
⇒ quantitatively, almost no change
◮ generalize to the decomposition
Γ(2)A,µν
(p) =(
Γ⊥A (p2)− ζΓ
‖A(p2)
)
(
δµν −pµpν
p2
)
+ Γ‖A(p2)
(
ζδµν + (1 − ζ)pµpν
p2
)
and impose the normalization condition
d
dp2
(
Γ⊥A (p2)− ζΓ
‖A(p2)
) ∣
∣
∣
p2=µ2= 1
22 / 57
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◮ integrating the Callan-Symanzik equation for the proper
gluonic 2-point function yields
∂
∂p2
(
Γ⊥A (p2, µ2)− ζΓ
‖A(p2, µ2)
)
= exp
(
−
∫ p2
µ2
dµ′ 2
µ′ 2γA(µ
′ 2)
)
then integrate over p2 with the initial condition inferred by
locality
Γ⊥A (p2 = 0, µ2) = Γ
‖A(p2 = 0, µ2)
◮ in the IR limit µ2 ≪ m2,
βg = µ2 d
dµ2g =
g
2
(
γA + 2γc)
≈g
2γA =
g
2µ2
d
dµ2ln ZA
and to 1-loop order
µ2d
dµ2ln ZA =
Ng2
(4π)2
(
−1
12+
ζ
4
)
◮ IR safety (βg > 0) for ζ > 1/3, the simple derivative
scheme corresponds to ζ = 1;the positivity of the beta function
arises from the momentum dependence of the
longitudinal part Γ‖A(p2)!
◮ in the following, consider only the critical case ζ = 1/3
23 / 57
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running coupling constant g(µ)
10-4
10-2
100
102
104
p [GeV]
0
5
10
15g(
p)
24 / 57
-
running mass parameter m2(µ)
10-4
10-2
100
102
104
p [GeV]
0
0.2
0.4
0.6
0.8
1
m2
[GeV
2 ]
25 / 57
-
gluon propagator function GA(p2), critical derivative scheme
0 1 2 3 4 5p [GeV]
0
1
2
3
4
GA
(p)
26 / 57
-
gluon dressing function FA(p2) = p2GA(p
2), critical derivative scheme
0 1 2 3 4 5p [GeV]
0
0.5
1
1.5p2
GA
(p)
27 / 57
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Independent running couplings
◮ breaking of BRST symmetry destroys the relation between the
different renormalizedcoupling constants
◮ define the renormalized ghost-gluon coupling constant defined
from the renormalizedproper ghost-gluon vertex as before, define
the renormalized three-gluon couplingconstant from the renormalized
proper three-point vertex at the symmetry pointp21 = p
22 = p
23 = µ
2
◮ renormalized four-gluon coupling constant set equal to the
renormalized three-gluoncoupling constant for the time being
◮ integration of the Callan-Symanzik equations with two
independently running couplingconstants and a running mass
parameter: BRST symmetry and usual non-massivebehavior recovered in
the UV, only two adjustable parameters (fine tuning condition)
28 / 57
-
gluon propagator GA(p2) in 4 dimensions,
two coupling constants
0 1 2 3 4p [GeV]
0
1
2
3
4
GA
(p)
29 / 57
-
gluon dressing function FA(p2) = p2GA(p
2) in 4 dimensions,two coupling constants
0 1 2 3 4 5p [GeV]
0
0.5
1
1.5
p2 G
A(p
)
30 / 57
-
ghost-gluon vertex function at the symmetry point p2 = q2 =
k2
in 4 dimensions, two coupling constants,compared to an
approximate solution of the Dyson-Schwinger equations
SU(2) lattice data: Cucchieri, Maas, Mendes 2008
0 1 2 3 4 5 6 7p [GeV]
1
1.1
1.2
1.3
1.4
DA
cc(p
2 ,p2
,2π
/3)
1 coupling2 couplingsDSE
at tree level, the vertex function is equal to one
31 / 57
-
three-gluon vertex function at the symmetry point p21 = p22 =
p
23
in 4 dimensions, two coupling constants,compared to an
approximate solution of the Dyson-Schwinger equations
SU(2) lattice data: Cucchieri, Maas, Mendes 2008
0 1 2 3 4 5p [GeV]
-1
-0.5
0
0.5
1
1.5
2
DA
3 (p2
,p2 ,
2π/3
)
2 couplingsDSE1 coupling
32 / 57
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Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
33 / 57
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Dynamical quarks
◮ success in Yang-Mills theory motivates the use of one-loop
Callan-Symanzikrenormalization group equations in full QCD;hope to
describe dynamical mass generation by introducing running quark
massparameters, without a representation of its dynamical origin
(chiral condensate)
◮ first study by Peláez, Tissier and Wschebor [Peláez,
Tissier, Wschebor 2014]:reasonable representation of the mass
function M(p2), but not of the dressing function(field
renormalization) Z (p2) of the full quark propagator
Z (p2)
ip/+ M(p2)
(two-loop contributions important?)
◮ no dynamical mass generation in the chiral limit [Peláez,
Tissier, Wschebor 2015]
34 / 57
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quark mass function Mu,d (p2) in 4 dimensions for Nf = 2 + 1
flavors,
one-loop renormalization group improved:optimized fit, leading
to a less satisfactory fit of the unquenched gluon propagator
Peláez, Tissier, Wschebor 2014
lattice data: Bowman et al. 2004, 2005
0 1 2 3 4 5
0.05
0.10
0.15
0.20
0.25
0.30
p HGeVL
MHpLHG
eVL
35 / 57
-
quark mass function Mu,d (p2) in 4 dimensions for Nf = 2 + 1
flavors,
one-loop renormalization group improved:parameters adjusted for
an optimized fit to the unquenched gluon propagator
Peláez, Tissier, Wschebor 2014
lattice data: Bowman et al. 2004, 2005
0 1 2 3 4 5
0.05
0.10
0.15
0.20
0.25
0.30
p HGeVL
Mu,
dHpLHG
eVL
36 / 57
-
quark dressing function Zu,d (p2) in 4 dimensions for Nf = 2 + 1
flavors,
one-loop renormalization group improved:parameters adjusted for
an optimized fit to the quark mass function
Peláez, Tissier, Wschebor 2014
lattice data: Bowman et al. 2005
0 1 2 3 4 5
0.7
0.8
0.9
1.0
1.1
1.2
p HGeVL
Zu,
dHpL
37 / 57
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Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
38 / 57
-
General strategy
◮ analyze the description of dynamical mass generation through
the Callan-Symanzikequations in a simpler gauge theory:QED in D = 3
Euclidean space-time dimensions in the Landau gauge (with Juan
PabloGutiérrez)
◮ for a direct comparison with Dyson-Schwinger equations, use
the quenched and therainbow ladder approximations (tree-level
photon propagator and electron-photonvertex; lead to a closed gap
equation for the electron mass function)
◮ note: these approximations are not entirely consistent for the
renormalization of thetheory
39 / 57
-
Self-energy
◮ begin with the one-loop renormalization of QED3 in the
quenched and rainbow ladderaproximations
◮ one-loop electron self-energy in 3 dimensions
convergent for p2 > 0
◮ for a massless photon (quenched approximation),
Σ(p) =e202π
M0
parctan
(
p
M0
)
with p =√
p2, M0 the bare electron mass and e0 the bare coupling
constant
40 / 57
-
Renormalization◮ the bare one-loop electron propagator is
[
ip/+ M0 +e202π
M0
parctan
(
p
M0
)
]−1
◮ implement as a normalization condition that the renormalized
electron propagator formomenta p with p2 = µ2 (µ the
renormalization scale) is
Z (µ)
ip/+ M(µ)
◮ this condition defines Z (µ) = 1 and the renormalized mass
M(µ) = M0 +e202π
M0
µarctan
(
µ
M0
)
+O(e40)
◮ as a consequence of the rainbow ladder approximation and Z (µ)
= 1, therenormalized coupling constant
e(µ) = e0 ≡ e
◮ as part of the renormalization procedure, all quantities are
expressed in terms of therenormalized mass and coupling constant
(as the — in principle — experimentallyaccessible quantities), in
particular
M0 = M(µ)−e2
2π
M(µ)
µarctan
(
µ
M(µ)
)
+O(e4)
41 / 57
-
The NJL argument
◮ usually, one is not interested in the value of the bare mass
M0 as a function of therenormalized mass M(µ),
M0 = M(µ)−e2
2π
M(µ)
µarctan
(
µ
M(µ)
)
(1)
(at one-loop level in the renormalized theory, using the
quenched and rainbow ladderapproximations), but particularly in the
chiral limit M0 → 0, equation (1) is relevant
◮ the same equation (1) follows from the original argument of
Nambu and Jona-Lasinio(NJL) [Nambu, Jona-Lasinio 1961] applied to
QED3:add a contribution δM to the bare mass M0 such that the
electron self-energycalculated with the new mass M = M0 + δM and
including the counterterm (−δM)vanishes,
Σ(p) = −δM +e202π
M
parctan
( p
M
)
= 0
◮ unlike in the theory originally considered by NJL, this is
only possible at one (arbitrarilychosen) momentum scale µ (on the
other hand, no momentum cutoff needs to beintroduced here),
thus
M0 +e202π
M
µarctan
( µ
M
)
= M ,
which is equation (1) with e0 = e and M = M(µ)
42 / 57
-
Solving for M(µ)
◮ in the chiral limit, equation (1) becomes
0 = M0 = M(µ)
[
1 −e2
2πµarctan
(
µ
M(µ)
)]
◮ there is a trivial solution, M(µ) = 0, and a nontrivial
one,
M(µ) =µ
tan(2πµ/e2)
as long as
e2
2πµ
π
2≥ 1 ⇔ µ ≤
e2
4;
in particular,
M(µ = 0) =e2
2π
◮ at µ = e2/4, the nontrivial solution coincides with the
trivial solution;for µ > e2/4 only the trivial solution
exists
43 / 57
-
Varying M0
◮ for M0 > 0, equation (1) can only be solved numerically;
however, two limits can beestablished analytically:
M(µ = 0) = M0 +e2
2π, M(µ → ∞) = M0
◮ a plot of M(µ) for several values of M0, in units of e2/2π
(both µ and M):
1 2 3 4 5
0.2
0.4
0.6
0.8
1.0
1.2
44 / 57
-
Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
45 / 57
-
Linear approximation
◮ Dyson-Schwinger equation for the mass function M(p2) in QED3
in the Landau gauge,using the quenched and the rainbow ladder
approximations:
M(p2) = M0 + 2e2
∫
d3k
(2π)31
(p − k)2M(k2)
k2 + M2(k2)
◮ in the chiral limit M0 → 0 with the (additional) linear
approximation
M(k2)
k2 + M2(k2)≈
M(k2)
k2 + M2(0),
one finds the exact solution
M(p2) =e2
4π
1
1 + (4πp/e2)2,
◮ in the extreme limits,
M(p2 = 0) =e2
4π, M(p2 → ∞) ∝
1
p2
46 / 57
-
Angular approximation
◮ alternatively, one may use (for M0 ≥ 0) the angular
approximation to evaluate thek -integral
∫
d3k
(2π)3M(k2)
k2 + M2(k2)
1
(p − k)2
=1
(2π)2
∫ ∞
0
dkk2M(k2)
k2 + M2(k2)
∫ 1
−1
d cos θ
k2 + p2 − 2p k cos θ
=1
(2π)2p
∫ ∞
0
dkk M(k2)
k2 + M2(k2)ln
k + p
|k − p|
◮ now
for k ≫ p , lnk + p
|k − p|≈
2p
k;
for k ≪ p , lnk + p
|k − p|≈
2k
p,
and the angular approximation is
lnk + p
|k − p|≈
2p
kΘ(k − p) +
2k
pΘ(p − k)
under the k -integral
47 / 57
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Comparison in the chiral limit
◮ in the angular approximation, the gap equation can be
converted (exactly) into adifferential equation which is
subsequently numerically solved; for large momenta, onefinds
analytically
M(p2 → ∞) = M0 ,
and particularly in the chiral limit M0 → 0,
M(p2 → ∞) ∝1
p2
◮ Comparison of the three mass functions, M(µ) from the one-loop
renormalization orthe NJL argument, and M(p2) from the
Dyson-Schwinger equation in the linear and theangular
approximations, in the chiral limit (in units of e2/2π):
0.5 1.0 1.5 2.0 2.5 3.0
0.2
0.4
0.6
0.8
1.0
48 / 57
-
Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
49 / 57
-
Callan-Symanzik equations
◮ the bare parameters and the bare n-point functions cannot
depend on therenormalization scale µ, in particular, at the
one-loop level,
µdM(µ)
dµ= µ
∂
∂µ
[
e202π
M0
µarctan
(
µ
M0
)
]∣
∣
∣
∣
∣
M0,e0 fixed
=e2
2π
[
M2(µ)
µ2 + M2(µ)−
M(µ)
µarctan
(
µ
M(µ)
)]
, (2)
expressing the bare quantities in terms of the renormalized ones
in the last step(renormalization group improvement)
◮ using the normalization condition for the renormalized
electron propagator
Z (µ)
ip/+ M(µ)
at µ = p, the renormalization group improved propagator is
1
ip/+ M(p)
where M(p) is the function obtained by integrating the
differential equation (2)
50 / 57
-
Results
◮ no mass is generated dynamically in the chiral limit through
the integration of theCallan-Symanzik equation (2)
◮ Comparison of the analytical solution of the Dyson-Schwinger
gap equation in thechiral limit with the solution of the
Callan-Symanzik equation with the same initial valueM(p2 = 0):
0.5 1.0 1.5 2.0 2.5 3.0
0.1
0.2
0.3
0.4
0.5
0.6
51 / 57
-
M(p2 = 0) as a function of M(p2 → ∞) = M0
in the one-loop renormalization of the theory or the NJL
argument,
the angular and the linear approximations of the Dyson-Schwinger
gap equation
and the Callan-Symanzik renormalization group equation
ææ
0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
2.0
52 / 57
-
Contents
Introduction
Perturbation theory for IR Yang-Mills theory
Callan-Symanzik equations
Dynamical generation of quark masses
Renormalization of QED3 and the NJL argument
The gap equation
Renormalization group equations
Conclusions
53 / 57
-
Summary
◮ quasi-analytic and systematic description of the correlation
functions of Landau gaugeYang-Mills theory: introduce a gluonic
mass term, solve the Callan-Symanzik equations
◮ this success motivates to try the same approach for the
description of dynamical quarkmass generation in QCD; a first
exploration by Peláez, Tissier and Wschebor isencouraging, but not
entirely successful
◮ look at a simpler theory first: QED in 3 space-time dimensions
in the quenched andrainbow ladder approximations (for comparison to
the solution of the Dyson-Schwingergap equation)
◮ the standard one-loop renormalization of the theory leads to
the same equation for theeffective mass as Nambu-Jona-Lasinio’s
original argument
◮ an effective (or renormalized) mass is generated even in the
chiral limit, but itsdependence on the momentum scale is not
satisfactory
◮ applying Callan-Symanzik renormalization group equations leads
to a much bettermomentum scale dependence, judging from a
comparison with the solution of theDyson-Schwinger gap equation
(here: in the linear and angular approximations)
◮ the value of the mass M(p2 = 0) generated by the
renormalization group is quitesimilar to the one generated by the
Dyson-Schwinger equation (in the angularapproximation) for not too
small bare masses (M0 & 0.1e
2/(2π)), but there is nodynamical mass generation in the chiral
limit by the Callan-Symanzik equations
54 / 57
-
Outlook
◮ Yang-Mills theory: current calculations in 3 space-time
dimensions
◮ continue work on the vertex functions, compare to new lattice
and Dyson-Schwingerresults
◮ proceed to two-loop level (with renormalization group
improvement)
◮ extend the formalism to 2 space-time dimensions (scaling
solutions)
◮ dynamical fermion mass generation: to complete the comparison
to theDyson-Schwinger gap equation, look at the numerical solutions
of the full equation
◮ before moving on to QCD, one may consider QED3 without the
quenched and rainbowladder approximations in the renormalization
group approach (those are not consistentapproximations in the
renormalization of the theory)
◮ consider alternative renormalization schemes, different
space-time dimensions andepsilon expansions
55 / 57
-
References
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57 / 57
IntroductionPerturbation theory for IR Yang-Mills
theoryCallan-Symanzik equationsDynamical generation of quark
massesRenormalization of QED3 and the NJL argumentThe gap
equationRenormalization group equationsConclusions
