Lorentz contraction, geometry and range in antiproton-proton annihilation into two pions

We present a geometric interpretation of the so-called annihilation range in reactions of the type p̄p → two light mesons based upon Lorentz effects in the highly relativistic final states (γ = Ecm/2mc ≃ 6.8 − 8.0). Lorentz-boosted meson wave functions, within the framework of the constituent quark model, result in a richer angular dependence of the annihilation amplitudes and thus in higher partial wave contributions (J > 1) than usually obtained. This approach sheds some light on what could be a "short" annihilation range and how it is influenced by the angular distribution of the final states. In this talk we summarize the results obtained in Refs. [1–3] while focusing on the details of Lorentz effects in reactions of the type p̄p → π−π+ or p̄p → K−K+ when the final states are highly energetic. The initial motivation is twofold: for one, a longstanding debate about the antiproton-proton range exists in which proponents of a very short annihilation range, of the order of the Compton wavelength of the annihilating baryons (rann. = m −1 N ≃ 0.1 fm), emphasize the role of analytical properties of Feynman graphs [4]. This should be a priori the range of baryon exchange in the t-channel. Comparison of baryon exchange model results [5–8] with p̄p → π−π+ and p̄p → K−K+ data on differential cross sections dσ/dΩ and analyzing powers A0n, as measured in the pre-LEAR experiments at CERN and KEK [9–11] as well as at LEAR [12], nonetheless clearly indicates that these models are too short ranged. They give vanishing total p̄p angular momentum J ≥ 1 contributions to the annihilation amplitudes, whereas partial wave analyzes [13–16] of both the pre-LEAR [9–11] and LEAR [12] data consistently point to higher partial wave amplitudes with J up to 3 or 4. Of course, compared with elastic proton-proton scattering, where partial waves up to J = 9−10 are necessary to fit the data, antiproton-proton annihilation is a short-range interaction. Yet, the experiments mentioned above seem not to confirm the extreme shortness implied by the baryon exchange models. It is worthwhile to note that the annihilation range is not merely given by the Compton wavelength of the baryons alone. In general, the annihilation vertices are regularized with appropriate form factors, which smear out the baryon-meson interaction. For example, in the case of Ref. [8] the form factor of the pseudovector pion-proton coupling is found to improve the fits to p̄p → π−π+ cross sections and analyzing powers. It is therefore not advisable to ascribe the annihilation range solely to heavy baryon exchange in the t-channel. The use of alternative geometric arguments, such as the minimal overlap for annihilation of an p̄p pair within a quark picture motivated by bag models, was investigated by Alberg et al. [17]. In their fit to p̄p scattering data the effective nucleon bag radius R takes on values of 0.6−1.1 fm. The crucial assumption is that p̄p annihilation proceeds only when the p̄ and p bags overlap so the quarks and antiquarks can rearrange or annihilate to form the observed mesons. In Ref. [17] the annihilation is found to peak at 0.7 fm or 0.95 fm depending on the bag-model form used. Hence, in geometric terms the annihilation range is roughly twice the nucleon radius which is much larger than the Compton wavelength of the nucleon. If one uses a chiral bag model then the pressure of the mesonic cloud about the core quarks decreases the nucleon bag radius. Since the external mesons are not believed to participate in the p̄p annihilation, the upper bound for the annihilation range is somewhat lowered. Nonetheless, quark model calculations also fail to reproduce the LEAR data [18, 19] as they too are rather short-ranged. The range, in this case, is an intrinsic property of overlap integrals involving (anti)quark wavefunctions with size parameters that reproduce the particle radii known from fits as the one just mentioned [17]. Yet another approach to define an annihilation range is due to Povh and Walcher [20] and was subsequently applied by the Bonn group [21] to the Bonn and Paris N̄N potentials. In a nutshell, the method consists of taking the quantity pl(r) = 2/h̄ WN̄NR ∗ l (r)Rl(r)r 2 as the annihilation probability for a partial wave of angular momentum l, where Rl(r) is the radial component of the (Bonn or Paris) N̄N wavefunction and WN̄N the imaginary part of the potential. Plotting pl(r) for several partial waves, they found for the Paris N̄N potential a maximum localized at about 0.5 fm and for the Bonn N̄N potential this value is about 1.1 fm. This range compares well with the quark model prediction discussed in the previous paragraph. The second motivation stems from the observation that the Hasan et al. LEAR data [12] reveals a strong left-right asymmetry with respect to the beam direction and displays large variations of the cross sections as a function of the c.m. angle. In our work, we try to determine how relativistic effects, namely Lorentz contractions in quark-model wavefunctions, influence the annihilation amplitudes and whether these effects are relevant to the annihilation range. QUARK MODEL AND INTRINSIC MESON WAVEFUNCTIONS In a quark model description of the p̄p annihilation process, one usually seeks guidance from Feynman diagrams in order to deduce appropriate transition operators (i.e. the operators linking the initial p̄p quarks and antiquarks so they form the right q̄q pairs). These so-called quark-line diagrams (QLD) are classified according to their flavor-flux topology into rearrangement or annihilation diagrams. In the former case a q̄q pair is annihilated and a quark and antiquark are rearranged to produce the final mesons, while in the latter case two q̄q pairs are annihilated and an q̄q state is created in the final state. The q̄q pairs annihilate into states with distinct quantum numbers JP and I. In a way, one can conceive of the model as an expansion of the annihilation amplitude in terms of increasing total angular momentum JP. Parity requires the q̄q pairs be annihilated/created in an S = 1 state. Thus the spin-multiplicity is fixed and for JP = 0+,1−... one gets P0, S1... annihilation operators. For a concise review of the QLD see, for instance, Ref. [22]. The resulting transition operators may be written in a Hamiltonian form for both the P0 and S1 cases as H (P0) = γ ∑ i jmn am(k ′)an(k)ai(p)b j(p′)σ ·(p−p′)(2π)3δ (k′−k−p′−p)+h.c. (1) H (S1) = κ ∑ i jmn am(k )an(k)ai(p)b j(p) [ −2σ i j ·k+ i(σ mn ×σ i j)·(k−k) ]