The Model for QCD Running Coupling Constant with Dynamically Generated Mass and Enhancement

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IHEP 98-41June 1998THE MODEL FOR QCD RUNNING COUPLING CONSTANT WITH DYNAMICALLY GENERATED MASS AND ENHANCEMENT IN THE INFRARED REGION 1Aleksey I.Alekseev Institute for High Energy Physics,Protvino,Moscow Region,142284Russia E-mail:alekseev@mx.ihep.su Abstract Nonperturbative studies of the strong running coupling constant in the infrared region are discussed.Starting from the analyses of the Dyson –Schwinger equations in the gauge sector of QCD,the conclusion is made on an incomplete ?xing of the perturbation the-ory summation ambiguity within ”(forced)analytization procedure”(called also a dispersive approach).A minimal model for ˉαs (q 2)is

proposed so that the perturbative time-like discontinuity is preserved and nonperturbative terms not only remove the Landau singularity but also provide the ultraviolet convergence of the gluon condensate.Within this model,on the one hand,the gluon zero modes are en-hanced (the dual superconductor property of the QCD vacuum)and,on the other hand,dynamical gluon mass generation is realized,with m g estimated as 0.6GeV .The uncertainty connected with the division into perturbative and nonperturbative contributions is discussed with the gluon condensate taken as an example.

1Introduction

The infrared region corresponding to large distances is of exceptional interest

by virtue of its responsibility for the con?nement mechanism and its inac-cessibility to perturbative methods.A great number of papers are devoted

to the infrared behaviour of gluon Green functions but a commonly accepted opinion is absent in the literature.

Recently a possibility of power infrared behaviour of the gluon propagator

in the covariant Landau gauge has been discussed,[1,2]with a partial account for ghosts.It was found that in the approximation considered,the gluon

propagator at small values of Euclidean momenta vanished(in this case,

according to gauge identities,all the gluon vertexes turn out to be singular), the ghost propagator was singular,and a running coupling constant had a

large but?nite value at zero.

In the axial gauge in the framework of nonperturbative Baker–Ball–

Zachariasen(BBZ)approach,[3]which seems to be adequate to discuss the

possibility of the infrared behaviour not being too singular,the problem of consistency of the behaviour D(q)～(q2)?c,q2→0was considered.It has been shown[4,5]that for quite a wide interval of non-integer(non-half-

integer)values of c,?1

Versions of the infrared behaviour of the gluon propagator mentioned

above do not exhaust all the possibilities that can be realized beyond per-turbation theory.

1

2Model for the Running Coupling Constant with Nonperturbative Contributions Sup-pressed in the Ultraviolet Region

Let us consider the model forˉαs introduced in Refs.[6]and discussed fur-ther in Ref.[9]We begin with”analytized”expression for the QCD running coupling constant[10]

ˉα(1)s(q2)=4π

ln(q2/Λ2)

+

Λ2

π

:F aμν(x)F aμν(y):|vac>≈(2)

≈2δab lim x→y

πlim

x→y

(δμν?xλ?yλ??xν?yμ)N?1 dA A aμ(x)A bν(y)e S[A]+JA|J=0?pert=(4)

=2δab

αs

πlim

x→y

(δμν?xλ?yλ??xν?yμ) dk π dk

=αs

π5

dk [Z (k )?Z pert (k )]=(9)=3π3

dy y ˉαnonpert s (y ).(10)We shall use Eq.(10)to evaluate the gluon condensate (2).The transi-tion from Eq.(7)to (8)is made with the assumption of proportionality of the nonperturbative part of the propagator to the free one.The projector in

Eq.(8)removes the terms of the free axial gauge propagator D (0)μν(k )which

depend on the gauge vector.Assuming Z is independent of gauge parameter y =(kη)2/k 2η2,the employment of the axial gauge lets one obtain Eq.(10)from (9),thereby expressing the gluon condensate in terms of nonperturba-tive part of running coupling constant ˉαs (k 2).The one-loop ”analytized”behaviour of Eq.(1)leads to a quadratic divergence in Eq.(10)at in?nity and this is true for the two-and three-loop expressions [10]of the analytiza-tion approach.According to the results of Refs.[7,3,8]let us add in Eq.(1)the isolated infrared singular term of the form 1/q 2.This term is harmless at zero and it can improve the behaviour of the integrand at in?nity and make the integral logarithmical divergent.To make the integral (10)convergent at in?nity,it is su?cient to add one more isolated singular term of a pole type with parameters chosen appropriately.In this sense the model we come to is minimal.The expression,we obtain for the running coupling constant,is the following:

ˉαs (q 2)=4π

ln (q 2/Λ2)+Λ2

q 2+(1?c )Λ2

c ?1.It is worth noting that an account of

nonperturbative contributions in Eq.(11)preserves a perturbative time-like cut of Eq.(1).With the given value of the QCD scale parameter Λ,the parameter c can be ?xed by the string tension κ,assuming the linear con-?nement V (r )?κr =a 2r at r →∞.We de?ne the potential V (r )of static q ˉq interaction by means of three-dimensional Fourier transform of ˉαs ( q 2)/ q 2with the contributions of only one dressed gluon exchange taken into account.This gives the relation c Λ2=(3b 0/8π)a 2.Taking a ?0.42GeV ,one obtains

3

c=Λ21/Λ2whereΛ21=3b0κ/8π?(0.434GeV)2(b0=9in the case of3light ?avours).For m g one obtains

m g=Λ2/

b0 1Λ2?k2+cΛ2k2+m2g =(13)

=ˉαpert

s (k2)+ˉαnonpert

s

(k2)=ˉαpert

s reg

(k2)+ˉαnonpert

s reg

(k2),(14)

where

ˉαpert s reg (k2)=

4π

ln(k2/Λ2)

+

Λ2

b0 Λ2k2+(1?c)Λ2

2See also Ref.

[12]where the problem of perturbative and nonperturbative contributions

toˉαs is discussed and the de?nition of infrared?nite regularized perturbative part ofˉαs is suggested.

4

Figure1:The dependence of the gluon condensate fourth root K1/4onΛ. The parameter k0varies from0.5GeV(the lowest curve)up to3.0GeV(the highest one)at an interval of0.25GeV.The dashed line corresponds to the exceptional value k0=0.777GeV.The”standard”level K1/4=0.33GeV is indicated.

The arbitrariness of the procedure is parametrized by a scale parameter k0. With account for the stated above,let us calculate the gluon condensate(10). We obtain

K(Λ,k0)=4

Λ2?1

k20

Λgive the needed value of the gluon condensate.When k 0increases one of them increases and tends to Λ1and the other value of Λdecreases and,at k 0>2GeV,becomes less than 150MeV.

4Gluon Condensate and Vacuum Energy Den-

sity

Let us represent expression (11)

in an explicitly renormalization invariant form.It can be done without solving the di?erential renormalization group equations.In this order we write ˉαs (q 2)=ˉg 2(q 2/μ2,g )/4πand use the normalization condition ˉg 2(1,g )=g 2.Then we obtain the equation for the wanted dependence of the parameter Λ2on g 2and μ2:

g 2/4π=4π

ln(μ2/Λ2)+Λ2

μ2+(1?c )Λ2

?(x )+1

(c ?1)e ?(x )+1.

The solution of this equation at c >1is function ?(x ),which has the be-haviour ?(x )?1/x at x →0and ?(x )??ln(x/c )at x →+∞.The relation obtained ensures the renormalization invariance of ˉαs (q 2).At low g 2,we obtain Λ2=μ2exp {?4π/(b 0αs )},which indicates the essentially non-perturbative character of three last terms in Eq.(11)and these terms are absent in the usual perturbation theory.We will need further the β-function which can be found by the equation

u ?ˉg 2(u,g )

b 0 ?x +1?2(x )+1

((c ?1)e ?(x )+1)2

,

(19)

6

Figure2:Theβ–function for c=1,2,3,4,5.

which is simpli?ed using equation for?(x).Then,knowing the behaviour of ?(x)at x→0,∞,we can?nd

b0

β(g2)??

c(c?2)+O(1/ln g2),g2→∞.(21)

b0

Fig.2illustrates the dependenceβ(g2)for c=1,2,...,5.For all g2>0the β-function is negative de?nite.Let us consider the trace anomaly[14]for the energy-momentum tensor of the gluon?eld

β(g2)

Θμμ=

Figure3:K(μ)/K(μ→∞)=?(b0/16π2)g4/β(g2)and(K(μ)/K(μ→∞))1/4.

nonperturbative vacuum in some region(bag).The vacuum inside the bag is perturbative and its energy density is zero.The di?erence of the vacuum energy densities inside and outside the bag is the reason of external pressure on the bag.

It is important for us that because ofβ<0,the maximum of gluon condensate corresponds to the minimum of nonperturbative vacuum energy density.For our modelˉαs(q2)it means that the values

Λ=ˉΛ?375MeV,k0=ˉk0?777MeV,m g=ˉm g?600MeV(23) are advantageous from the viewpoint of the energy argument.Let us clarify what the dependence of the gluon condensate on renormalization parameter μis.As a physical quantity,the vacuum energy density is independent ofμ. The renormalized coupling constant g depends onμ,g2/4π=ˉg2(1,g)/4π=ˉαs(μ2),so asμ→∞g2→0and atμ→0g2→∞.Using(20),(21),we

can write

?vac=π2β(g2)

π

:F aμνF aμν:|vac>=(24)

8

=?b0

π

:F aμνF aμν:|vac>|μ→∞=(25)

=?

1

K(μ→∞)=?

9

β(g2)

,(27)

which describes the dependence of gluon condensate onμ.For theβ-function(19) in Fig.3by a solid line we show the ratio K(μ)/K(μ→∞)as function of μ.The parameter c=1,3476corresponds to the choiceΛ1=434MeV,

Λ=ˉΛ=375MeV.We can see that forμ>1GeV ratio(27)is practically unity.In the same?gure the dependence(K(μ)/K(μ→∞))1/4onμis shown by dots.

Acknowledgments

I am grateful to the Special Research Center for the Subatomic Structure

of Matter for support and hospitality during my stay at CSSM.I would like to thank the Organizing Committee of the Workshop on Methods in Nonperturbative Field Theory for all e?orts and care.

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