IHEP 2000-40 hep-ph/0011178

In the framework of analytic approach to QCD, which has been recently intensively developed, the dependence of nonperturbative contributions in a running coupling of strong interaction on initial perturbative approximation to 3-loop order is studied. The nonperturbative contributions are obtained in an explicit form. In the ultraviolet region they are shown to be represented in the form of the expansion in the inverse powers of Euclidean momentum squared. The expansion coefficients are calculated for different numbers of active quark flavors nf and for different numbers of loops taken into account. For all nf of interest it is shown that 2-loop order and 3-loop order corrections result in partial compensation of 1-loop order leading in the ultraviolet region nonperturbative contribution. PACS number(s): 12.38.Aw, 12.38.Lg Talk presented at the XV International Workshop on High Energy Physics and Quantum Field Theory (QFTHEP’00), September 14 – 20, 2000, Tver, Russia Electronic address: [email protected] It is widely believed that unphysical singularities of the perturbation theory in the infrared region of QCD should be canceled by the nonperturbative contributions. The nonperturbative contributions arise quite naturally in an analytic approach [1] to QCD. The so-called ”analyticization procedure” is used in this approach. The main purpose of this procedure is to remove nonphysical singularities from approximate (perturbative) expressions for the Green functions of QFT. The idea of the procedure goes back to Refs. [2, 3] devoted to the ghost pole problem in QED. The foundation of the procedure is the principle of summation of imaginary parts of the perturbation theory terms. Then, the Källen – Lehmann spectral representation results in the expressions without nonphysical singularities. In recent papers [4, 5] it is suggested to solve the ghost pole problem in QCD demanding the ᾱs(q ) be analytical in q (to compare with dispersive approach [6]). As a result, instead of the one-loop expression ᾱ (1) s (q) = (4π/b0)/ ln(q /Λ) taking into account the leading logarithms and having the ghost pole at q = Λ (q is the Euclidean momentum squared), one obtains the expression ᾱ an (q ) = 4π b0 [ 1 ln(q2/Λ2) + Λ Λ2 − q2 ] . (1) Eq. (1) is an analytic function in the complex q-plane with a cut along the negative real semiaxis. The pole of the perturbative running coupling at q = Λ is canceled by the nonperturbative contribution (Λ |g2→0 ≃ μ 2 exp{−(4π)/(b0g )}) and the value ᾱ (1) an (0) = 4π/b0 appeared finite and independent of Λ. The most important feature of the ”analyticization procedure” discovered [4, 5] is the stability property of the value of the ”analytically improved” running coupling constant at zero with respect to high corrections, ᾱ (1) an (0) = ᾱ (2) an (0) = ᾱ (3) an (0). This property provides the high corrections stability of ᾱan(q ) in the whole infrared region. The 1-loop order nonperturbative contribution in Eq. (1) can be presented as convergent at q > Λ constant signs series in the inverse powers of the momentum squared. For ”standard” as well as for the iterative 2-loop perturbative input the nonperturbative contributions in the analytic running coupling are calculated explicitly in Ref. [7]. In the ultraviolet region the nonperturbative contributions can be also represented as a series in inverse powers of momentum squared. In this paper we extract in an explicit form the nonperturbative contributions to ᾱan(q ) up to the 3-loop order in the analytic approach to QCD, and investigate their ultraviolet behavior. To handle the singularities originating from the perturbative input, we develop here the method which is more general than that of Ref. [7]. According to the definition the analytic running coupling is obtained by the integral representation aan(x) = 1 π ∞ ∫ 0 dσ x+ σ ρ(σ), (2) where the spectral density ρ(σ) = Im aan(−σ − i0). It is seen that dispersively-modified coupling of form (2) has an analytical structure which is consistent with causality. According to the analytic approach to QCD we adopt that Im aan(−σ − i0) = Im a(−σ − i0) where a(x) is an appropriately normalized perturbative running coupling. The behavior of the QCD running coupling αs(μ ) is defined by the renormalization group equation μ ∂αs(μ ) ∂μ = β(αs) = β0α 2 s + β1α 3 s + β2α 4 s + ..., (3) where the coefficients β0 = − 1 2π b0, b0 = 11− 2 3 nf , β1 = − 1 4π2 b1, b1 = 51− 19 3 nf , β2 = − 1 64π3 b2, b2 = 2857− 5033 9 nf + 325 27 nf . (4) The first two coefficients β0, β1 do not depend on the renormalization scheme choice. Here nf is the number of active quark flavors. The standard three-loop solution of Eq. (3) is written in the form of 1 expansion in inverse powers of logarithms [8] αs(μ ) = 4π b0 ln(μ2/Λ2) [ 1− 2b1 b0 ln[ln(μ/Λ)] ln(μ2/Λ2) + + 4b1 b0 ln (μ2/Λ2) × ( ( ln[ln(μ/Λ)]− 1 2 2 + b2b0 8b1 − 5 4 )] . (5) Let us introduce the function a(x) = (b0/4π)ᾱs(q ), where x = q/Λ. Then, instead of (5) one can write a(x) = 1 lnx − b ln(lnx) ln x + b ( ln(lnx) ln x − ln(lnx) ln x + κ ln x ) , (6) where the coefficients b and κ are equal to b = 2b1 b0 = 102− 38 3 nf (11− 23nf ) 2 , κ = b0b2 8b1 − 1. (7) At nf = 3 b0 = 9, and b = 64/81 ≃ 0.7901, κ ≃ 0.4147. At x ≃ 1 the perturbative running coupling is singular. At large x the 1-loop term of Eq. (6) defines the ultraviolet behavior of a(x) but for small x the behavior of the running coupling depends on the approximation we adopt, and at x = 1 there are singularities of a different analytical structure. Namely, at x ≃ 1 a(x) ≃ 1 x− 1 , a(x) ≃ − b (x− 1)2 ln(x− 1), a(x) ≃ b (x− 1)3 ln(x− 1). (8) This is not an obstacle for the analytic approach which removes this nonphysical singularities. By making the analytic continuation of Eq. (6) into the Minkowski space x = −σ − i0, one obtains a(−σ − i0) = 1 lnσ − iπ − b (lnσ − iπ)2 ln (lnσ − iπ) + b { ln(ln σ − iπ) (ln σ − iπ)3 − − ln(ln σ − iπ) (lnσ − iπ)3 + κ (lnσ − iπ)3 } . (9) Function a(x) in Eq. (6) is regular and real for real x > 1. So, to find the spectral density ρ(σ) we can use the reflection principle (a(x)) = a(x) where x is considered as a complex variable. Then ρ(σ) = 1 2i (a(−σ − i0)− a(−σ + i0)) . (10) By the change of variable of the form σ = exp(t), the analytical expression is derived from (2), (9), (10) as follows: aan(x) = 1 2πi ∞ ∫ −∞ dt e x+ et × { 1 t− iπ − 1 t+ iπ − −b [ ln(t− iπ) (t− iπ)2 − ln(t+ iπ) (t+ iπ)2 ] + b [ ln(t− iπ) (t− iπ)3 − ln(t+ iπ) (t+ iπ)3 − − ln(t− iπ) (t− iπ)3 + ln(t+ iπ) (t+ iπ)3 + κ (t− iπ)3 − κ (t+ iπ)3 ]} . (11) Let us see what the singularities of the integrand of (11) in the complex t-plane are. First of all the integrand has simple poles at t = lnx ± iπ(1 + 2n), n = 0, 1, 2, .... All the residues of the function exp(t)/(x + exp(t)) at these points are equal to unity. Apart from these poles the integrand of (11) has simple poles at t = ±iπ, the third order poles and logarithmic type branch points which coincide with the second order and third order poles. Let us cut the complex t-plane in a standard way, t = ±iπ − λ, with λ being the real parameter varying from 0 to ∞. The integrand in (11) multiplied by t goes to 2 zero at | t |→ ∞. That allows one to append the integration by the arch of the ”infinite” radius without affecting the value of the integral. Close the integration contour C1 in the upper half-plane of the complex variable t excluding the singularity at t = iπ. In this case an additional contribution emerges due to the integration along the sides of the cut and around the singularities at t = iπ. The corresponding contour we denote as C2. Let us turn to the integration along contour C1. For the integrand of Eq. (11) which we denote as F (t) the residues at t = ln(x) + iπ(1 + 2n), n = 0, 1, 2, ... are as follows ResF (t) |t=ln(x)+iπ(1+2n)= 1 ln(x) + 2πin − 1 ln(x) + 2πi(n+ 1) − −b [ ln(ln(x) + 2πin) (ln(x) + 2πin)2 − ln(ln(x) + 2πi(n+ 1)) (ln(x) + 2πi(n+ 1))2 ] + b [ ln(ln(x) + 2πin) (ln(x) + 2πin)3 − ln(ln(x) + 2πi(n+ 1)) (ln(x) + 2πi(n+ 1))3 − − ln(ln(x) + 2πin) (ln(x) + 2πin)3 + ln(ln(x) + 2πi(n+ 1)) (ln(x) + 2πi(n+ 1))3 + κ (ln(x) + 2πin)3 − κ (ln(x) + 2πi(n+ 1))3 ] . (12) By using the residue theorem one readily obtains the contribution ∆(x) to integral (11) from the integration along contour C1. It reads ∆(x) = 1 2πi ∫