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Asymptotic Limits and Sum Rules for the Quark Propagator

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Asymptotic Limits and Sum Rules for the Quark Propagator EFI 96-09 hep-th/9604021 ASYMPTOTIC LIMITS AND SUM RULES FOR THE QUARK PROPAGATOR 1 Reinhard Oehme and Wentao Xu Enrico Fermi Institute and Department of Physics University of Chicago Chicago, Illinois 60637, USA Abstract For the structure functions of the quark propagator, the asymptotic behavior is obtained for general, linear, covariant gauges, and in all directions of the complex k2-plane. Asymptotic freedom is assumed. Corresponding previous results for the gauge field propagator are important in the derivation. Except for coefficients, the leading asymptotic terms are determined by one-loop or by two-loop information, and are gauge independent. Various sum rules are derived. arXiv:hep-th/9604021v1 3 Apr 1996 1Work supported in part by the National Science Foundation, Grant PHY 91-23780. E-mail: [email protected] In a previous paper [1], we have obtained the asymptotic behavior of the gauge field propagator in general, covariant, linear gauges, and for all directions in the com- plex k2-plane. Given asymptotic freedom of the theory, we found interesting sum rules [1, 2]. Except for coefficients, the functional form of the leading asymptotic terms is gauge independent, and determined by the one-loop anomalous dimension coefficient γ00 and the corresponding β-function coefficient β0. The asymptotic prop- erties, and the related superconvergence relations, are of interest for the problem of confinement [3, 4]. If applied to certain models of N = 1 supersymmetric gauge theories [5, 6], the results obtained concerning the confinement of gauge field quanta, on the basis of superconvergence, are in agreement with those derived from duality [7, 8]. In particular, there are interesting relations between the one-loop anomalous dimension coefficient γ00 of the SUSY gauge theory and the one-loop β-function coefficient of the dual map [6]. These relations underline the relevance of the sign of the coefficient γ00 for supercovergence, duality and confinement. It is the purpose of this note to present general results for the quark propaga- tor. We obtain the asymptotic terms of the structure functions in general, linear, covariant gauges. Except for the case of the Landau gauge, which has been con- sidered before [9, 2], the previous results for the gauge field propagator play an important rˆole in the derivation. Again only anomalous dimension and β-function coefficients in lowest non-vanishing order of perturbation theory are relevant, and gauge dependence is limited. Our results are based upon general principles only. We use Lorentz-covariance, and minimal spectral condition as formulated in the state space of indefinite metric of the gauge theories considered [10]. For some results, we assume that the exact amplitudes connect with the weak coupling perturbative expressions in the limit of vanishing coupling, at least as far as the leading term is concerned. Our main tool is the renormalization group. We do not consider topological aspects of the theory. They should not be of importance within our general framework. In this report, we concentrate on the derivation of the asymptotic terms. Possible applications, in particular generalizations to SUSY theories and their dual maps, will be explored elsewhere. One may hope to find further correlations between the phase structure of the SUSY systems, as inferred from duality, and specific properties of 1 the propagators. Since we are interested in asymptotic expressions, it is sufficient to consider theories with no external mass parameters in the action. Possible non- perturbative mass generation is not excluded. We write the “quark” propagator αβ 4 ik·x α ¯β iSF (k)= d x exp 0 T ψ (x)ψ (0) 0 (1) − Z h | | i in the form iS (k)= A(k2)√ k2 + B(k2)γ k , (2) − F − · where gauge and spinor indices have been suppressed. With κ2 < 0 as the normal- ization point, we have the renormalization group equation ′ ′ ′2 2 ψ(x, g ,α , κ )= Z2ψ(x,g,α,κ ) , (3) q and corresponding relations for other fields. Here ′2 ′2 κ ′ κ Z2 = Z2 ,g,α , g =g ¯ ,g , κ2 ! κ2 ! κ′2 κ′2 α′ =α ¯ ,g,α = αR−1 ,g,α , (4) κ2 ! κ2 ! where R = k2D(k2, κ2,g,α) being the structure function of the transverse gauge − field propagator with the normalization R(1,g,α) = 1. It is convenient to introduce dimensionless structure function by k2 k2 k2A(k2, κ2,g,α)= S ,g,α , k2B(k2, κ2, g, 2) = T ,g,α . (5) − κ2 ! − κ2 ! Here we chose the normalization T (1,g,α)=1. (6) With Eqs. (1), (3) and (6), we can write the renormalization group equations for the functions T and S in the form 2 ′2 2 k κ k ′ ′ T ,g,α = T ,g,α T ′ ,g ,α , (7) κ2 ! κ2 ! κ 2 ! 2 2 ′2 2 k κ k ′ ′ S 2 ,g,α = T 2 ,g,α S ′2 ,g ,α , (8) κ ! κ ! κ ! −1 −1 because our normalization implies Z2 = T and Z3 = R. The primed parameters g′ and α′ are given in Eq.(4). In a situation, where the theory has unbroken chiral symmetry, the structure function A vanishes identically. But if we allow for the possible non-perturbative mass generation, we expect a nonzero function, which vanishes in the perturbative limit. As a consequence of Lorentz covariance and simple spectral conditions formu- lated in the state space of indefinite metric, it follows that the structure functions (distributions) B(k2 + i0) and A(k2 + i0) are boundary values of corresponding analytic functions, which are regular in the cut k2-plane with branch lines along the positive real k2-axis. In the following, we first use the renormalization group in order to obtain the asymptotic behavior of these functions for k2 along → −∞ the negative real k2 axis. We then generalize the results to all directions in the − complex k2-plane [1, 2]. For this purpose, we consider Eqs.(7,8) with κ′2 = k2 , −| | k2 = k2 eiφ, φ π, and find −| | | | ≤ k2 k2 T ,g,α = T ,g,α T (eiφ, g,¯ α¯), (9) κ2 ! κ2 ! k2 k2 whereg ¯ =g ¯( 2 ,g), andα ¯ =α ¯( 2 ,g,α ). There is a corresponding expression for | κ | | κ | the function S, but in the following, we first discuss only T. Assuming that the exact structure functions approach their perturbative limits for g2 0, at least as far as the leading terms are concerned, we have for vanishing → coupling: 2 2 k 2 k T ,g,α 1+ g γF 01α ln + κ2 ! ≃ κ2 ··· for α =0 2 2 6 k 4 k T ,g,α 1+ g γF 01 ln + κ2 ! ≃ κ2 ··· for α =0 (10) In these equations, the anomalous dimension coefficients of the quark field are de- fined by γ (g2,α) = γ (α)g2 + γ (α)g4 + , with γ (α) = γ + αγ , F F 0 F 1 ··· F 0 F 00 F 01 3 2 γF 1(α)= γF 10 + αγF 11 + α γF 12. We consider here QCD, or similar theories, so that we always have γ 0 and (16π2)γ = C (R) > 0, with C (R)=4/3 in QCD. F 00 ≡ F 01 2 2 For SUSY theories, on the other hand, we generally find γ = 0 for the Fermi F 00 6 field in the Wess-Zumino representation. These theories will be discussed elsewhere. Sinceg ¯2(u,g) ( β ln u)−1 andα ¯(u,g,α) = αR−1(u,g,α) 0 or α = γ /γ ≃ − 0 ≃ 0 − 00 01 for u , α 0, and in view of the analytic properties of the structure functions, →∞ ≥ we see that Eq.(9) gives the asymptotic behavior in all directions φ in terms of the limit for real k2 . → −∞ In order to make use of the renormalization group, we differentiate Eq.(7) with respect to k2, setting κ′2 = k2. Then we obtain the differential equation ∂T (u,g,α) u = γ (¯g2, α¯)T (u,g,α), (11) ∂u F where u = k2/κ2, k2 0 andg ¯2 =g ¯2(u,g),α ¯ =α ¯(u,g,α) = αR−1(u,g,α) and 2 2 ∂T (u,g,α) ≤ 2 ∂g¯ (u,g,α) γF (g ,α) u ∂u u=1, β(g ) u ∂u u=1. It is convenient to introduce ≡ | ≡ 2 | g¯2(u,g) as a new variable. With β(¯g2) u ∂g¯ , we write ≡ ∂u k2 T (¯g2; g2,α)= T ,g,α , (12) κ2 ! and obtain the equation ∂ ln T (¯g2; g2,α) γ (¯g2, α¯) = F . (13) ∂g¯2 β(¯g2) Here we have usedα ¯(u,g,α)=α ¯(¯g2; g2,α) = αR−1(¯g2; g2,α). If we consider R(¯g2; g2,α) as given, we can write the solution in the form g¯2 γ (x;¯α(x; g2,α)) T (¯g2; g2,α) = exp dx F , (14) 2 Zg β(x) where use has been made of the normalization condition (6): T (g2; g2,α) = 1. We are interested in the asymptotic expansion of T (¯g2; g2,α) forg ¯2 0. Be- → cause of the appearance of the gauge field structure function R for α = 0, we have 6 to consider separately the two cases for different signs of the one loop gauge-field anomalous dimension coefficient γ00 = γ0(α = 0), and the possibility that γ00 = 0. 2 2 2 k 2 2 2 For the expansion of R(¯g ; g ,α) = R( 2 ,g,α) = k D(k , κ ,g,α) in the limit κ − g¯2 0, we have obtained in Ref.[1] the leading terms → 4 2 2 2 ξ −1 2 β1 2 ξ+1 R(¯g ; g ,α) CR(¯g ) + CRβ0 γ10 γ12α0 γ00 (¯g ) + ≃ − − β0 ! ··· α α γ (α ) 1 + + 1 0 g¯2 + , (15) α α β 1 ξ ··· 0 0 0 − where α 0 and ξ γ00/β0 = 0 has been assumed.
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