Quark confinement in QCD in the t Hooft limit

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arXiv:2210.02701v1 [hep-ph] 6 Oct 2022

Nuclear and Particle Physics Proceedings 00 (2022) 1–7

Nuclear

and

Particle

Physics

Proceed-

ings

Quark conﬁnement in QCD in the ’t Hooft limit

Marco Frascaa,1, Anish Ghoshalb, Stefan Grootec

aVia Erasmo Gattamelata, 3, 00176 Rome (Italy)

bINFN, Rome, Italy and Warsaw University, Poland

cUniversity of Tartu, Estonia

Abstract

We treat quantum chromodynamics (QCD) using a set of Dyson-Schwinger equations derived, in diﬀerential form, with the Bender-

Milton-Savage technique. In this way, we are able to derive the low energy limit that assumes the form of a non-local Nambu-Jona-

Lasinio model. The corresponding gap equation is then studied to show that such a model has no free quarks in the low-energy

limit.

Keywords:

1. Introduction

Understanding quark conﬁnement is one of the most outstanding problem in QCD. Some criteria have been devised

(e.g. [1, 2]) but a ﬁrst principle proof is not known yet. Some theory are shown to conﬁne as in supersymmetric Yang-

Mills theory [3–5] or standard Yang-Mills theory [6] where the exact beta function was obtained. Indeed, the gluon

propagator is also known in closed form with more or less ﬁtting parameters [7–11]). It should be emphasized that it is

essential to obtain the low-energy limit of QCD from ﬁrst principles as this opens up a wealthy number of applications

in several ﬁelds ranging from nuclear physics to cosmology. With the given results in Yang-Mills theory, this can be

accomplished. The relevant approximations involved are strong coupling limit and ’t Hooft limit of number of colors

running to inﬁnity keeping the product of the number of colors and the square of coupling constant [12, 13]. We will

obtain such a limit and prove quark conﬁnement in the ’t Hooft limit [14].

2. Bender-Milton-Savage technique

Our approach is based on the Bender-Milton-Savage (BMS) technique that permits to derive the Dyson-Schwinger

equations in PDE form [15]. This technique can be better explained referring to a scalar ﬁeld. Therefore, we consider

the following partition function

Z[j]=Z[Dφ]eiS (φ)+iRd4x j(x)φ(x).(1)

∗Talk given at 25th International Conference in Quantum Chromodynamics (QCD 22), 4 - 7 July 2022, Montpellier - FR

Email address: marcofrasca@mclink.it (Marco Frasca)

1Speaker, Corresponding author.

Marco Frasca /Nuclear and Particle Physics Proceedings 00 (2022) 1–7 2

We start from the equation of motion for th 1P-function

*δS

δφ(x)+=j(x),(2)

assuming

h...i=R[Dφ]...eiS (φ)+iRd4x j(x)φ(x)

R[Dφ]eiS (φ)+iRd4x j(x)φ(x)(3)

Then, we set j=0 to obtain the equation for the 1P-function. We derive this equation with respect to jto obtain the

equation for the 2P-function.The deﬁnition for the nP-functions is the following

hφ(x1)φ(x2). . . φ(xn)i=δnln(Z[j])

δj(x1)δj(x2). . . δ j(xn).(4)

This implies

δGk(...)

δj(x)=Gk+1(. . . , x).(5)

This procedure can be iterated to any desired order. These equations have the great advantage that permit to use

possible exact solutions to them providing closed form formulas for the correlation functions of the theory.

3. 1P and 2P functions for QCD

In order to make our computations simpler, as done in Ref. [10], we evaluate our equations in the Landau gauge.

The Bender-Milton-Savage method yields for the 1P-functions

∂2Ga

1ν(x)+g f abc(∂µGbc

2µν(0) +

∂µGb

1µ(x)Gc

1ν(x)−∂νGνbc

2µ(0)

−∂νGb

1µ(x)Gµc

1(x))

+g f abc∂µGbc

2µν(0) +g f abc∂µ(Gb

1µ(x)Gc

1ν(x))

+g2fabc fcde(Gµbde

3µν (0,0) +Gbd

2µν(0)Gµe

1(x)

+Geb

2νρ(0)Gρd

1(x)+Gde

2µν(0)Gµb

1(x)+

Gµb

1(x)Gd

1µ(x)Ge

1ν(x))

=gX

q,i

γνTaSii

q(0) +gX

q,i

¯qi

1(x)γνTaqi

1(x),(6)

and for the quarks

(i/

∂−mq)qi

1(x)+gT·/

G1(x)qi

1(x)+gT·/

q(x,x)=0.(7)

Here and in the following Greek indexes (µ, ν, . . .) are for the space-time and Latin index (a,b,...) for the gauge

group. We recognize immediately a known property of the Dyson-Schwinger equations that equations for the lower

order correlation functions depend on values of higher order correlation functions.

At this stage, we assume the mapping theorem as done in Ref. [10]. So, we assume

1ν(x)→ηa

νφ(x) (8)

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摘要：

arXiv:2210.02701v1[hep-ph]6Oct2022NuclearandParticlePhysicsProceedings00(2022)1–7NuclearandParticlePhysicsProceed-ingsQuarkconﬁnementinQCDinthe’tHooftlimitMarcoFrascaa,1,AnishGhoshalb,StefanGrootecaViaErasmoGattamelata,3,00176Rome(Italy)bINFN,Rome,ItalyandWarsawUniversity,PolandcUniversityofTartu,Es...

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