process-independent effective charge: from qcd's green's ... · quark's gap equation...
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Process-independent effective Process-independent effective charge: charge: from QCD's Green's from QCD's Green's
functions to Hadron phenomenologyfunctions to Hadron phenomenology
INPC 2019; SEC, Glasgow; July 28th- August 2nd
Daniele Binosi, Cedric Mezrag, Joannis Papavassiliou, Craig D. Roberts, José Rodríguez-Quintero,K. Raya, J. Segovia, F. de Soto ...
[Phys.Rev. D99 (2017) no.5, 054026][Few Body Syst. 59 (2018) no.6, 121]Preliminary results...
QCD's running coupling. Motivation.
Q2
High
Low
The QCD Holy Grail: the understanding of hadrons in terms of its elementary excitations; namely, quarks and gluons!
1
What happens down here?
QCD's running coupling. Motivation.
Q2
High
Low
The QCD Holy Grail: the understanding of hadrons in terms of its elementary excitations; namely, quarks and gluons!
1
What happens down here?
ConfinementConfinement
DCSBDCSB
Colored bound states have never been seen
to exist as particles in nature
Chiral symmetryappears dynamically
violated in the Hadron spectrum
Emergent phenomena playing a dominant role in the real world dominated by the IR dynamics of QCD.
Quark's gap equation
Let's start by the beginning:
● Dynamical chiral symmetry breaking generates the “constituent-quark” masses
● This is the most important mechanism for the mass generation in our Universe (responsible for around 98 % of the proton mass)
● The effect is realized through the quark's gap equation and is clearly achieved through the pure theory's dynamics (nothing needs to be added to QCD!!)
2
4 π I (k2) 1
k2 (δμ ν−kμk ν
k2 )
Beyond Rainbow-Ladder truncation: One-gluon exchange effective kernel + Tree-level quark-gluon vertex
Γμ = ΓμBC +Γμ
ACM
Model parameters: Λ = 0.234 GeVζ = 0.55 GeVω ∈ [0.4,0 .6 ] GeV
Fixed by the pion decay constant
● Ball-Chiu vertex [PRD(22)1980]● Anomalous Chromomagnetic vertex
Consistent with both linear and transverse STI
Quark's gap equation
2
Beyond Rainbow-Ladder truncation: One-gluon exchange effective kernel + Tree-level quark-gluon vertex
Quark's gap equation
2
Quark's gap equation: RGI interaction
3
Quark's gap equation: RGI interaction
4
Quark's gap equation: RGI interaction
5
Quark's gap equation: RGI interaction
Both top-bottom and down-up approaches deliver quark-gluon effective interactions which compare remarkably well with each other! 5
Let us now carefully examine the RGI Interaction:
D. Binosi, J. R-Q, C.D. Roberts, PRD95(2017)114009
QCD effective charge
6
Let us now carefully examine the RGI Interaction:
D. Binosi, J. R-Q, C.D. Roberts, PRD95(2017)114009
D. Binosi, L. Chang, J. Papavassiliou, C.D. Roberts, PLB 742 (2015)
QCD effective charge
6
Let us first carefully examine the RGI Interaction:
D. Binosi, J. R-Q, C.D. Roberts, PRD95(2017)114009
D. Binosi, L. Chang, J. Papavassiliou, C.D. Roberts, PLB 742 (2015)
Remarkable QCD feature: saturation of the RG key ingredient ̂d (0)
Define then the RGI invariant function
D. Binosi, C. Mezrag, J. Papavassiliou, J.R-Q, C.D. Roberts, arXiv:1612.04835
Extract the (process-independent) couplingUsing the quark gap equation
QCD effective charge
7
D. Binosi, C. Mezrag, J. Papavassiliou, J.R-Q, C.D. Roberts, arXiv:1612.04835
QCD effective charge: comparison.
8
PI-effective charge from lattice data with Nf=3 flavors at the physical point
● The ghost dressing function
9
≃
The IR running of the PI effective charge with momenta only depends on:
Preliminary results:
PI-effective charge from lattice data with Nf=3 flavors at the physical point
● The ghost dressing function● The PT-BFM function L
The IR running of the PI effective charge with momenta only depends on:
9
≃
Preliminary results:
PI-effective charge from lattice data with Nf=3 flavors at the physical point
● The ghost dressing function● The PT-BFM function L
9
≃
The IR running of the PI effective charge with momenta only depends on:
Its strength depends also on the saturation point at zero-momentum of the gluon propagator
Preliminary results:
PI-effective charge from lattice data with Nf=3 flavors at the physical point
● The ghost dressing function● The PT-BFM function L
9
≃
The IR running of the PI effective charge with momenta only depends on:
Its strength depends also on the saturation point at zero-momentum of the gluon propagator and on the Taylor coupling.
Preliminary results:
PI-effective charge from lattice data with Nf=3 flavors at the physical point
● The ghost dressing function● The PT-BFM function L
All put together:
Less uncertainties (that of the gluon mass is only left here) and still a better agreement with the world data for the experimental determination of the Bjorken sum-rule effective charge.
9
≃
The IR running of the PI effective charge with momenta only depends on:
Its strength depends also on the saturation point at zero-momentum of the gluon propagator and on the Taylor coupling.
Preliminary results:
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator
10
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
10
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
LFWF leading to asymptotic PDAs
10
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
LFWF leading to asymptotic PDAs
A more realistic pion 2-body LFWF
10
DCSB-induced hardening
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
LFWF leading to asymptotic PDAs
A more realistic pion 2-body LFWF
Direct computation of Mellin moments:
10
DCSB-induced hardening
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
LFWF leading to asymptotic PDAs
A more realistic pion 2-body LFWF
Direct computation of Mellin moments:
10
DCSB-induced hardening
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
A more realistic pion 2-body LFWF
Direct computation of Mellin moments:
10
LFWF leading to asymptotic PDAs
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
LFWF leading to asymptotic PDAs
A more realistic pion 2-body LFWF
Direct computation of Mellin moments:
10
One application: pion PDF DGLAP evolution
The pion PDF can be computed as the lightfront projection of the hadronic matrix element of a bilocal operator and, in the overlap representation at low Fock states, can be expressed in terms of 2-body LFWFs at a given hadronic scale
LFWF leading to asymptotic PDAs
A more realistic pion 2-body LFWF
Direct computation of Mellin moments:
10
ζH →ζ2=5.2 GeV
M n(t )=∫0
1
dx xnq(x , t )
t=ln (ζ2
ζ02 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )+…
Moments' evolution (1-loop):
One application: pion PDF DGLAP evolution
ζH →ζ2=5.2 GeV
11
M n(t )=∫0
1
dx xnq(x , t )
t=ln (ζ2
ζ02 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )+…
ddtq (x , t )=−
α(t )4 π ∫
x
1dyyq ( y ,t )P( x
y)+…
∫0
1
dx P(x) = γ0n
A master equation for the (1-loop) moments' evolution:
Moments' evolution (1-loop):
One application: pion PDF DGLAP evolution
ζH →ζ2=5.2 GeV
11
M n(t )=∫0
1
dx xnq(x , t )
t=ln (ζ2
ζ02 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )+…
ddtq (x , t )=−
α(t )4 π ∫
x
1dyyq ( y ,t )P( x
y)+…
∫0
1
dx P(x) = γ0n
A master equation for the (1-loop) moments' evolution:
Moments' evolution (1-loop):
P(x) = 83 ( 1+z2
(1−x)++3
2δ(x−1))
γn =− 43 (3+ 2
(n+2)(n+3)−4∑
i=1
n+11i )
One application: pion PDF DGLAP evolution
ζH →ζ2=5.2 GeV
11
M n(t )=∫0
1
dx xnq(x , t )
t=ln (ζ2
ζ02 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )+…
ddtq (x , t )=−
α(t )4 π ∫
x
1dyyq ( y ,t )P( x
y)+…
∫0
1
dx P(x) = γ0n
A master equation for the (1-loop) moments' evolution:
Moments' evolution (1-loop):
P(x) = 83 ( 1+z2
(1−x)++3
2δ(x−1))
γ0n =−4
3 (3+ 2(n+2)(n+3)
−4∑i=1
n+11i )
ddt
α(t ) =−α2(t )4 π
β0+…
α(t ) = 4 πβ0(t−tΛ)
+…
tΛ=ln( Λ2
ζ02 )
One application: pion PDF DGLAP evolution
ζH →ζ2=5.2 GeV
11
M n(t )=∫0
1
dx xnq(x , t )
t=ln (ζ2
ζ02 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )+…
ddtq (x , t )=−
α(t )4 π ∫
x
1dyyq ( y ,t )P( x
y)+…
∫0
1
dx P(x) = γ0n
A master equation for the (1-loop) moments' evolution:
Moments' evolution (1-loop):
P(x) = 83 ( 1+z2
(1−x)++3
2δ(x−1))
γ0n =−4
3 (3+ 2(n+2)(n+3)
−4∑i=1
n+11i )
ddt
α(t ) =−α2(t )4 π
β0+…
α(t ) = 4 πβ0(t−tΛ)
+…
tΛ=ln( Λ2
ζ02 ) M n(t ) = M n(t 0)( α(t)
α(t0))γ0n /β0
One application: pion PDF DGLAP evolution
ζH →ζ2=5.2 GeV
11
α(t) = 4πβ0(t−tΛ)
+…= 4π
β0 ln ( ζ2
Λ2)+…
Which value of Lambda?
12
One application: pion PDF DGLAP evolution
α(t) = 4πβ0(t−tΛ)
+…= 4π
β0 ln ( ζ2
Λ2)+…
Which value of Lambda? It depends on the scheme... Indeed, at the one-loop level, its value defines by itself the scheme!!!
α(t)=α(t ) (1+c α(t )+…)
ln( Λ2
Λ2 ) =4 πβ0 ( 1
α(t )− 1
α(t) )+… = 4 π cβ0
12
One application: pion PDF DGLAP evolution
α(t) = 4πβ0(t−tΛ)
+…= 4π
β0 ln ( ζ2
Λ2)+…
Which value of Lambda? It depends on the scheme... Indeed, at the one-loop level, its value defines by itself the scheme!!!
α(t)=α(t ) (1+c α(t )+…)
ln( Λ2
Λ2 ) =4 πβ0 ( 1
α(t )− 1
α(t) )+… = 4 π cβ0
ddtM n( t) =−
α(t )4π
γ0nM n(t)+…
ddt
α(t) =−α2(t )4π
β0+…
The evolution will thus depend on the scheme via the perturbative truncation
12
One application: pion PDF DGLAP evolution
α(t) = 4πβ0(t−tΛ)
+…= 4π
β0 ln ( ζ2
Λ2)+…
Which value of Lambda? It depends on the scheme... Indeed, at the one-loop level, its value defines by itself the scheme!!!
α(t)=α(t ) (1+c α(t )+…)
ln( Λ2
Λ2 ) =4 πβ0 ( 1
α(t )− 1
α(t) )+… = 4 π cβ0
ddtM n( t) =−
α(t )4π
γ0nM n(t)+…
ddt
α(t) =−α2(t )4π
β0+…
The evolution will thus depend on the scheme via the perturbative truncation and the usual prejudice is that truncation errors are optimally small in MS scheme.
PDG2018:[PRD98(2018)030001]
12
One application: pion PDF DGLAP evolution
One application: pion PDF DGLAP evolution
Then, one can evolve the pion PDF, e.g. the one obtained by direct computation of Mellin moments, by using DGLAP evolution from one unknown hadronic scale up to the relevant one for the E615 experiment:
13
ζH →ζ2=5.2 GeV
ΛQCD=0.234 ;ζH=0.349 .
GeVGeV
Optimal best-fitting parameters:
One application: pion PDF DGLAP evolution
Then, one can evolve the pion PDF, e.g. the one obtained by direct computation of Mellin moments, by using DGLAP evolution from one unknown hadronic scale up to the relevant one for the E615 experiment:
13
ζH →ζ2=5.2 GeV
ΛQCD=0.234 ;ζH=0.349 .
GeVGeV
Comparison with the three first moments obtained from lQCD
Optimal best-fitting parameters:
One application: pion PDF DGLAP evolution
Then, one can evolve the pion PDF, e.g. the one obtained by direct computation of Mellin moments, by using DGLAP evolution from one unknown hadronic scale up to the relevant one for the E615 experiment:
13
ΛQCD=0.234 ;ζH=0.349 .
GeVGeV
ΛQCD=0.234 ;ζH=0.374 .
GeV GeV
ζH →ζL=2 →ζ2=5.2 GeVGeV
Matching the three first moments obtained from lQCD
Optimal best-fitting parameters:
α(t) = 4πβ0(t−tΛ)
+…= 4π
β0 ln ( ζ2
Λ2)+…
Which value of Lambda? It depends on the scheme... Indeed, at the one-loop level, its value defines by itself the scheme!!!
α(t)=α(t ) (1+c α(t )+…)
ln( Λ2
Λ2 ) =4 πβ0 ( 1
α(t )− 1
α(t) )+… = 4 π cβ0
ddtM n( t) =−
α(t )4π
γ0nM n(t)+ …
ddt
α(t) =−α2(t )4π
β0+…
The evolution will thus depend on the scheme via the perturbative truncation
14
One application: pion PDF DGLAP evolution
The use of GeV can be thus interpreted as the choice of particular scheme, differing from MS.
Λ=0.234
α(t) = 4πβ0(t−tΛ)
+…= 4π
β0 ln ( ζ2
Λ2)+…
Which value of Lambda? It depends on the scheme... Indeed, at the one-loop level, its value defines by itself the scheme!!!
α(t)=α(t ) (1+c α(t )+…)
ln( Λ2
Λ2 ) =4 πβ0 ( 1
α(t )− 1
α(t) )+… = 4 π cβ0
ddtM n( t) =−
α(t )4π
γ0nM n( t)
The evolution will thus depend on the scheme via the perturbative truncation
14
One application: pion PDF DGLAP evolution
The use of GeV can be thus interpreted as the choice of particular scheme, differing from MS. Beyond this, the scheme can be defined in such a way that one-loop DGLAP is exact at all orders (Grunberg's effective charge).
Λ=0.234
α(t ) = 4π
β0 ln (mα2+ζ0
2 exp(t )
Λ2 )= 4π
β0 ln (mα2+k 2
Λ2 )
t=ln( k2
ζ02)
M n( t )=∫0
1
dx xnq(x , t )
One application: pion PDF DGLAP evolution
15
α(t ) = 4π
β0 ln (mα2+ζ0
2 exp(t )
Λ2 )= 4π
β0 ln (mα2+k 2
Λ2 )
t=ln( k2
ζ02)
M n( t )=∫0
1
dx xnq(x , t )
One application: pion PDF DGLAP evolution
15
α(0) = αPI (0) → mα=0.300 GeV
α(t ) = 4π
β0 ln (mα2+ζ0
2 exp(t )
Λ2 )= 4π
β0 ln (mα2+k 2
Λ2 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )
t=ln( k2
ζ02)
M n( t )=∫0
1
dx xnq(x , t )
γ0n =− 4
3 (3+ 2(n+2)(n+3)
−∑i=1
n+11i )
M n(t ) = M n(t0)exp(− γ0n
4 π ∫t0
t
dzα(z))Numerical integration with the effective charge
One application: pion PDF DGLAP evolution
15
α(0) = αPI (0) → mα=0.300 GeV
α(t ) = 4π
β0 ln (mα2+ζ0
2 exp(t )
Λ2 )= 4π
β0 ln (mα2+k 2
Λ2 )
ddtM n(t) =−
α(t )4 π
γ0nM n(t )
t=ln( k2
ζ02)
M n( t )=∫0
1
dx xnq(x , t )
γ0n =− 4
3 (3+ 2(n+2)(n+3)
−∑i=1
n+11i )
M n(t ) = M n(t0)exp(− γ0n
4 π ∫t0
t
dzα(z))Numerical integration with the effective charge
One application: pion PDF DGLAP evolution
15
α(0) = αPI (0) → mα=0.300 GeV
If one identifies: , all the scales (and the evolution between them) appear thus fixed, apart from (fixed by the scheme).
mα≡ζHΛQCD
One application: pion PDF DGLAP evolution
Then, one can evolve the pion PDF, e.g. the one obtained by direct computation of Mellin moments, by using DGLAP evolution from one unknown hadronic scale up to the relevant one for the E615 experiment:
16
ζH≡mα →ζ2=5.2 GeVΛQCD=0.234 ;GeV
If one identifies: , all the scales (and the evolution between them) appear thus fixed, apart from (fixed by the scheme). And the agreement with E615 data is perfect!!!
mα≡ζHΛQCD
One application: pion PDF DGLAP evolution
Then, one can evolve the pion PDF, e.g. the one obtained by direct computation of Mellin moments, by using DGLAP evolution from one unknown hadronic scale up to the relevant one for the E615 experiment:
16
ζH≡mα →ζ2=5.2 GeVΛQCD=0.234 ;GeV
If one identifies: , all the scales (and the evolution between them) appear thus fixed, apart from (fixed by the scheme). And the agreement with E615 data is perfect!!!
mα≡ζHΛQCD
ddtq(x , t ) =−
α(t )4π ∫
x
1dyyq( y , t )P ( x
y)+…
The same is obtained from the overlap of realistic pion 2-body LFWFs
and after integration of the DGLAP master equation
17
Conclusions
● The PI effective charge is shown to be equivalent, within the IR domain, to the one which can be defined to get an “exact” (at all order in perturbations) DGLAP evolution for the pion PDF.
● One can define a parameter-free effective charge (completely determined from 2-points gauge sector), with no Landau pole (physical coupling showing an IR fixed point) and smoothly connecting IR and UV domains (no explicit matching procedure)
● It is shown to be in good agreement with the Bjorken-rule effective charge and with the coupling from the light-front holographic model.
ΛQCD=0.234 ;GeV ζH≡mα →ζ2=5.2 GeV
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