Renormalization Scale-Fixing for Complex Scattering Amplitudes

We show how to fix the renormalization scale for hard-scattering exclusive processes such as deeply virtual meson electroproduction by applying the BLM prescription to the imaginary part of the scattering amplitude and employing a fixed-t dispersion relation to obtain the scale-fixed real part. In this way we resolve the ambiguity in BLM renormalization scale-setting for complex scattering amplitudes. We illustrate this by computing the H generalized parton distribution at leading twist in an analytic quark-diquark model for the parton-proton scattering amplitude which can incorporate Regge exchange contributions characteristic of the deep inelastic structure functions. PACS. 11.55.Fv Dispersion relations – 11.10.Gh Renormalization 1 BLM renormalization scale setting A typical QCD amplitude for an exclusive process can be calculated as a power series in the strong coupling constant A = A +Aαs(μ ) +Aαs(μ ) + · · · (1) The renormalization scale μ of the running coupling in such processes can be set systematically in QCD without ambiguity at each order in perturbation theory using the Brodsky-Lepage-Mackenzie (BLM) method [1,2]. The BLM scale is derived order-by-order by incorporating the non-conformal terms associated with the β function into the argument of the running coupling. This can be done systematically using the skeleton expansion [3, 4]. The scale determined by the BLM method is consistent with (a) the transitivity and other properties of the renormalization group [5] (b) the renormalization group principle that relations between observables must be independent of the choice of intermediate renormalization scheme [6,7], and (c) the location of the analytic cut structure of amplitudes at each flavor threshold. The nonconformal terms involving the QCD β function are all absorbed by the scale choice. The coefficients of the perturbative series remaining after BLM-scale-setting are thus the same as those of a conformally invariant theory with β = 0. In practice, one can often simply use the flavor dependence of the series to tag the nonconformal β dependence in perturbation theory; i.e., the BLM procedure Send offprint requests to: resums the terms involving nf associated with the running of the QCD coupling. Non-Abelian gauge theory based on SU(NC) symmetry becomes an Abelian QED-like theory in the limitNC → 0 while keeping α = CFαs and nl = neff/2CF fixed [8]. Here CF = (N 2 C − 1)/2NC. The BLM scale reduces properly to the standard QED scale in this analytic limit. For example, consider the vacuum polarization leptonloop correction to ee → ee in QED. The amplitude must be proportional to α(s) since this gives the correct cut of the forward amplitude at the lepton pair threshold s = 4ml . Thus the renormalization scale μ 2 R = s is exact and unambiguous in the conventional QED GoldbergerLow scheme [9]. If one chooses any other scale μR 6= s, the scale μR = s will be restored when one sums all bubble graphs. The BLM procedure is thus consistent with the Abelian limit and the proper cut structure of amplitudes. 2 Difficulties in using the mean value theorem to set the BLM Scale The BLM scale at leading order has a simple physical interpretation: it is identical to the photon virtuality in QED applications and the mean gluon virtuality in QCD when one uses physical schemes which generalize the QED scheme such as the pinch scheme [2,10] and the αV [11, 4] scheme defined from the QCD static potential. The number of flavors active in virtual corrections to a given process is evident from the BLM scale choice: the BLM 2 Stanley J. Brodsky, Felipe J. Llanes-Estrada: Renormalization Scale-Fixing for Complex Scattering Amplitudes method sets the renormalization scale so that flavor number is changed properly in any renormalization scheme, including the MS scheme [12,2]. In effect, the BLM prescription identifies the renormalization scale and the gluon virtuality by eliminating the dependence on the number of flavors from the O(αs) (expanding αs itself) and O(α 2 s) nonconformal terms in the perturbative amplitude. Typically, a QCD amplitude involves an integral over the momentum running through the gluon propagator. Therefore the argument of the coupling, if taken to be the momentum flowing through the gluon, varies through the integration phase space. A mean Q̄2 can be extracted from the integral if the mean value theorem (MVT) of integral calculus can be applied. The essential requirement for the applicability of the MVT is that the function being evaluated at its mean value has to be continuous through the interval and the weight function has to be Riemann-integrable; this includes weight functions bounded and continuous in the range of integration. However, these necessary conditions are not a property of a principal value integrand associated with a pole which appears, for example, in Compton scattering and deeply virtual meson electroproduction (DVME); as we show explicitly in Section 2.1 below, the MVT does not apply for these amplitudes. It has been recently pointed out in Ref. [13] that the MVT prescription for determining the BLM procedure fails for amplitudes which are genuinely complex, that is, display non-vanishing real and imaginary parts. The authors then argue that there is no guarantee that the BLM prescription will yield the same answer for both parts of the amplitude. Worse, in the particular example that they study, ρ and π̂1 electroproduction, the scale obtained from the real part becomes discontinuous (zero to infinity) at a particular kinematical point due to a divergence in some of the intermediate functions. In this paper we note that in a quantum field theory, the real and imaginary parts of a scattering amplitude are not independent but are constrained due to causality, locality, and Lorentz invariance. This manifests itself in the form of the dispersion relations traditionally used in meson photoproduction [14,15] to link both parts of the amplitude. By examining a simple example in the next subsection, we show that the correct prescription for finding the BLM renormalization scales is to first fix the scale in the imaginary part, and then subsequently, the real part can be obtained by means of a dispersion relation. The BLM scales for the real and imaginary parts are thus not generally equal. We perform an explicit calculation for longitudinally polarized vector meson electroproduction at nonzero skewness in Section 2.2. 2.1 Example of the failure of the Mean Value Theorem To illustrate how the MVT can fail for an amplitudes which contains a pole, consider the following simple integral: