Process-independent effective charge: from QCD's Green's ... · Quark's gap equation Let's start by...

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


2

Beyond Rainbow-Ladder truncation: One-gluon exchange effective kernel + Tree-level quark-gluon vertex


2

Quark's gap equation: RGI interaction

3


4


5


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, L. Chang, J. Papavassiliou, C.D. Roberts, PLB 742 (2015)


6

Let us first carefully examine the RGI Interaction:


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


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:


● The ghost dressing function● The PT-BFM function L


9

≃




9

≃


Its strength depends also on the saturation point at zero-momentum of the gluon propagator




9

≃


Its strength depends also on the saturation point at zero-momentum of the gluon propagator and on the Taylor coupling.




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

≃


Its strength depends also on the saturation point at zero-momentum of the gluon propagator and on the Taylor coupling.


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


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



LFWF leading to asymptotic PDAs

10




A more realistic pion 2-body LFWF

10

DCSB-induced hardening





Direct computation of Mellin moments:

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DCSB-induced hardening





10







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):


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:



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



P(x) = 83 ( 1+z2

(1−x)++3

2δ(x−1))

γn =− 43 (3+ 2

(n+2)(n+3)−4∑

i=1

n+11i )


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



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 )


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



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


ζH →ζ2=5.2 GeV

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α(t) = 4πβ0(t−tΛ)

+…= 4π

β0 ln ( ζ2

Λ2)+…

Which value of Lambda?

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+…= 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

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+…= 4π

β0 ln ( ζ2

Λ2)+…


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

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+…= 4π

β0 ln ( ζ2

Λ2)+…


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]

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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:

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ζH →ζ2=5.2 GeV

ΛQCD=0.234 ;ζH=0.349 .

GeVGeV

Optimal best-fitting parameters:



13

ζH →ζ2=5.2 GeV

ΛQCD=0.234 ;ζH=0.349 .

GeVGeV

Comparison with the three first moments obtained from lQCD




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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



+…= 4π

β0 ln ( ζ2

Λ2)+…


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+…


14


The use of GeV can be thus interpreted as the choice of particular scheme, differing from MS.

Λ=0.234


+…= 4π

β0 ln ( ζ2

Λ2)+…


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)


14


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 )


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 )


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


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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


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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



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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



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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

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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

Process-independent effective charge: from QCD's Green's ... · Quark's gap equation Let's start by...

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