QCD RUNNING COUPLING: FREEZING VERSUS ENHANCEMENT IN THE INFRARED REGION

Alekseev A.I. QCD Running Coupling: Freezing Versus Enhancement in the Infrared Region: IHEP Preprint 97–90. – Protvino, 1997. – p. 9, figs. 2, refs.: 33. We discuss whether or not ”freezing” of the QCD running coupling constant in the infrared region is consistent with the Schwinger – Dyson (SD) equations. Since the consistency of the ”freezing” was not found, the conclusion is made that the ”analytization” method does not catch an essential part of nonperturbative contributions. Proceeding from the results on consistency of the infrared enhanced behaviour of the gluon propagator with SD equations, the running coupling constant is modified taking into account the minimality principle for the nonperturbative contributions in the ultraviolet region and convergence condition for the gluon condensate. It is shown that the requirements of asymptotic freedom, analyticity, confinement and the value of the gluon condensate are compatible in the framework of our approach. Possibilities to find an agreement of the enhanced behaviour of the running coupling constant with integral estimations in the infrared region are also discussed. aNNOTACIQ aLEKSEEW a.i. bEGU]AQ KONSTANTA SWQZI khd: ”ZAMORAVIWANIE” ILI USILENIE W INFRAKRASNOJ OBLASTI: pREPRINT ifw— 97–90. – pROTWINO, 1997. – 9 S., 2 RIS., BIBLIOGR.: 33. oBSUVDAETSQ WOPROS O SOGLASOWANNOSTI GIPOTEZY ”ZAMORAVIWANIQ” BEGU]EJ KONSTANTY SWQZI khd W INFRAKRASNOJ OBLASTI S URAWNENIEM –WINGERA-dAJSONA (–d) DLQ GL@ONNOGO PROPAGATORA. pOSKOLXKU SOGLASOWANNOSTX ”ZAMORAVIWANIQ” NE OBNARUVENA, SDELAN WYWOD O TOM, ˆTO METOD ”ANALITIZACII” NE UHWATYWAET SU]ESTWENNU@ ˆASTX NEPERTURBATIWNYH WKLADOW. wYRAVENIE DLQ BEGU]EJ KONSTANTY SWQZI khd MODIFICIROWANO S UˆETOM SOGLASOWANNOSTI USILENIQ INFRAKRASNOGO POWEDENIQ GL@ONNOGO PROPAGATORA S URAWNENIEM –d, PRINCIPA MINIMALXNOSTI NEPERTURBATIWNYH WKLADOW W ULXTRAFIOLETOWOJ OBLASTI I USLOWIQ KONEˆNOSTI GL@ONNOGO KONDENSATA. w RAMKAH RASSMATRIWAEMOGO PODHODA UDAETSQ SOGLASOWATX TREBOWANIQ ASIMPTOTIˆESKOJ SWOBODY, ANALITIˆNOSTI, KONFAJNMENTA I OCENKI WELIˆINY GL@ONNOGO KONDENSATA. oBSUVDA@TSQ TAKVE WOZMOVNOSTI SOGLASOWANIQ INFRAKRASNOGO USILENIQ BEGU]EJ KONSTANTY SWQZI S INTEGRALXNYMI OCENKAMI W INFRAKRASNOJ OBLASTI. c © State Research Center of Russia Institute for High Energy Physics, 1997 The phenomenon of asymptotic freedom [1] called forth an impressive success of perturbative QCD in the description of experimental data plethora. However, there is wide scope of phenomena which is intractable in the framework of perturbation theory. Nonperturbative effects modify the infrared behaviour of the quark and gluon Green‘s functions. With q decreas the renormalization group improved one-loop running coupling constant ᾱs(q ) = 4π b0 ln(q2/Λ2) . (1) (b0 = 11C2/3 − 2Nf/3) increases, which may indicate a tendency of unlimited growth of the interaction at large distances, leading to a confinement of coloured objects. However, at q = Λ in (1) the pole is present, which is nonphysical, at least, due to the fail of the perturbation theory, and the account of nonperturbative effects becomes obligatory. In recent papers [2] the solution of the problem of a ghost pole was proposed with the condition of analyticity in q being imposed. The idea of ”forced analyticity” goes back to [3,4] of the late 50s. They were dedicated to the problem of Landau-Pomeranchuk pole [5] in QED. Using for ᾱs(q ) a spectral representation without subtractions, the following expression for the running coupling constant was obtained in [2] ᾱ s (q ) = 4π b0 [ 1 ln(q2/Λ2) + Λ Λ2 − q2 ] . (2) This expression is analytic in the infrared region due to nonperturbative contributions and it has a finite limit at zero (although the derivative is infinite). Nowadays a possibility of ”freezing” the coupling constant at low energies is under discussion [6] in the framework of some scheme of approximate calculations ((16 2 −Nf ) expansion). In the approach [7] with confining background field, ”freezing” was also obtained, ᾱs(q ) = 4π b0 ln((mB + q 2)/Λ2) . (3) Here mB is a process-dependent constant of the order of 1 GeV. In the present paper we follow the approach of Refs. [8,9]. We discuss the problem of consistency of the constant behaviour of the running coupling constant in the infrared