A measurement of the axial form factor of the nucleon by the p(e, e′ π +)n reaction at W = 1125 MeV
ˇ A. Liesenfeld a,1 , A. W. Richter a,1, S. Sirca b,1,2 , K. I. Blomqvist a, W. U. Boe
glin a, K. Bohinc b, R. B¨hm a, o M. Distler a, D. Drechsel a, R. Edelho? a, I. Ewald a, J. Friedrich a, J. M. Friedrich a, R. Geiges a, M. Kahrau a, M. Korn a, K. W. Krygier a, V. Kunde a, H. Merkel a, K. Merle a, U. M¨ller a, R. Neuhausen a, T. Pospischil a, M. Potokar b, u A. Rokavec b, G. Rosner a, P. Sauer a, S. Schardt a, H. Schmieden a, L. Tiator a, B. Vodenik b, A. Wagner a, Th. Walcher a and S. Wolf a
arXiv:nucl-ex/9911003v1 11 Nov 1999
Institut f¨r Kernphysik, Universit¨t Mainz, D-55099 Mainz, Germany u a
Joˇef Stefan Institute, SI-1001 Ljubljana, Slovenia z
The reaction p(e, e′ π + )n was measured at the Mainz Microtron MAMI at an invariant mass of W = 1125 MeV and four-momentum transfers of Q2 = 0.117, 0.195 and 0.273 (GeV/c)2 . For each value of Q2 , a Rosenbluth separation of the transverse and longitudinal cross sections was performed. An e?ective Lagrangian model was used to extract the ‘axial mass’ from experimental data. We ?nd a value of MA = (1.077 ± 0.039) GeV which is (0.051 ± 0.044) GeV larger than the axial mass known from neutrino scattering experiments. This is consistent with recent calculations in chiral perturbation theory.
PACS: 13.60.Le, 25.30.Rw, 14.20.Dh Keywords: nucleon axial form factor, coincident pion electroproduction
This paper comprises parts of the doctoral theses of A. Liesenfeld, A. W. Richter ˇ and S. Sirca. 2 Corresponding author (tel: +386 61 1773-731, fax: +386 61 219-385, e-mail: email@example.com).
Preprint submitted to Elsevier Preprint
8 February 2008
There are basically two methods to determine the weak axial form factor of the nucleon. One set of experimental data comes from measurements of (quasi)elastic (anti)neutrino scattering on protons , deuterons  and other nuclei (Al, Fe) [3,4] or composite targets like freon [5–8] and propane [8,9]. In the (quasi)elastic picture of (anti)neutrino-nucleus scattering, the νN → ?N weak transition amplitude can be expressed in terms of the nucleon electromagnetic form factors F1 and F2 and the axial form factor GA . The axial form factor is then extracted by ?tting the Q2 -dependence of the (anti)neutrinonucleon cross section, dσ = A(Q2 ) ? B(Q2 ) (s ? u) + C(Q2 ) (s ? u)2 , 2 dQ (1)
in which GA (Q2 ) is contained in the bilinear forms A(Q2 ), B(Q2 ) and C(Q2 ) of the relevant form factors and is assumed to be the only unknown quantity. It can be parameterised in terms of an ‘axial mass’ MA as GA (Q2 ) = GA (0)/(1 + 2 Q2 /MA )2 .
Fig. 1. Axial mass MA as extracted from (quasi)elastic neutrino and antineutrino scattering experiments. The weighted average is MA = (1.026 ± 0.017) GeV, or (1.026 ± 0.021) GeV using the scaled-error averaging recommended by Ref. .
Fig. 1 shows the available values for MA obtained from these studies. References [3,5,6,9] reported severe uncertainties in either knowledge of the incident neutrino ?ux or reliability of the theoretical input needed to subtract the background from genuine elastic events (both of which gradually improved in subsequent experiments). The values derived fall well outside the most probable range of values known today and exhibit very large statistical and systematical errors. Following the data selection criteria of the Particle Data Group , they were excluded from this compilation. 2
Another body of data comes from charged pion electroproduction on protons [11–17] slightly above the pion production threshold. As opposed to neutrino scattering, which is described by the Cabibbo-mixed V ? A theory, the extraction of the axial form factor from electroproduction requires a more involved theoretical picture.
Fig. 2. Axial mass MA as extracted from charged pion electroproduction experiments. The weighted average (excluding our result) is MA = (1.068 ± 0.015) GeV, or (1.068± 0.017) GeV using the scaled-error averaging . Including our extracted value, the weighted scaled-error average becomes MA = (1.069 ± 0.016) GeV. Note that our value contains both the statistical and systematical uncertainty; for other values the systematical errors were not explicitly given. SP: soft-pion limit, DR: analysis using approach of Ref. , FPV: Ref. , BNR: Ref. .
The basic result about low energy photoproduction of massless charged pions can be traced back to the Kroll-Ruderman theorem , extended to virtual photons by Nambu, Luri? and Shrauner , who obtained the O(Q2 ) result e for the isospin (?) (see Ref. , p. 29 for notation) electric dipole amplitude at threshold E0+ (mπ = 0, Q2 ) =
egA Q2 2 1 Q2 1? + O(Q3 ) , (2) κv + rA ? 2 8πfπ 6 4M 2
where κv is the nucleon isovector anomalous magnetic moment, gA ≡ GA (0) is the axial coupling constant, and fπ is the pion decay constant. In the following years, improved models were proposed [16,20–22], most of them including corrections due to the ?nite pion mass ? = mπ /M. The values 3
of the axial mass were determined, within the framework of the respective model, from the slopes of the angle-integrated di?erential electroproduction cross sections at threshold,
2 rA = ?
6 dGA (Q2 ) GA (0) dQ2
12 . 2 MA
The results of various measurements and theoretical approaches are shown in Fig. 2. Note again that references [16,17] were omitted from the ?t for lack of reasonable compatibility with the other results. Although the results of these investigations deviate from each other by more than their claimed accuracy, the weighted averages from neutrino scattering and electroproduction give quite precise values of the axial mass. Comparing the average values of the two methods, one observes a signi?cant di?erence of ?MA = (0.042 ± 0.023) GeV, or (0.042 ± 0.027) GeV using the scaled-error averaging. Chiral perturbation theory (χPT) has recently shown a remarkable and modelindependent result that already at O(Q2 ), the NLS result of eq. (2) is strongly modi?ed due to pion loop contributions . These contributions e?ectively reduce the mean-square axial radius,
2 2 rA → rA +
12 3 1? 2 . 2 64fπ π
The loop correction in eq. (4) has a value of ?0.046 fm2 , which is a ?10 % 2 correction to a typical rA = 0.45 fm2 . Correspondingly, the axial mass MA = √ 2 1/2 12/ rA would appear to be about 5 % larger in electroproduction than in neutrino scattering, in agreement with the observed ?MA . The aim of the present investigation was to determine MA from new, high precision pion electroproduction data and thereby help verify whether this discrepancy was genuine. Since both the energy and momentum transfers were too high to allow (?) for a safe extraction of E0+ , these data were analysed in the framework of an e?ective Lagrangian model with the electromagnetic nucleon form factors, the electric pion form factor and the axial nucleon form factor at the appropriate vertices [23,24].
Kinematics of the experiment
The di?erential cross sections for π + electroproduction on protons were measured at an invariant mass of W = 1125 MeV and at four-momentum transfers 4
of the virtual photon Q2 = 0.117, 0.195 and 0.273 (GeV/c)2 . For each value of Q2 , we measured the scattered electron and the outgoing pion in parallel kinematics at three di?erent polarisations of the virtual photon, enabling us to separate the transverse and the longitudinal part of the cross section by the Rosenbluth technique. Table 1 shows the experimental settings.
Table 1 Experimental settings for the p(e, e′ π + )n experiment. The angles are measured with respect to the electron beam axis.
Q2 [GeV2 /c2 ]
′ Ee [MeV]
θe [? ]
θπ [? ]
0.834 0.500 0.219 0.742 0.437 0.229 0.648 0.457 0.259
855.11 587.35 ?27.93 510.11 242.35 ?58.22 405.11 137.36
92.96 ?18.41 38.27 209.62 66.67 ?28.03 93.45 ?20.12 46.83 ?35.82 65.27 ?29.37 89.60 ?21.90 228.00
855.11 545.79 ?37.72 585.11 275.89 495.11 185.79
855.11 504.55 690.11 339.55 585.11 234.55
The measurements were performed at the Institut f¨ r Kernphysik at the Uniu versity of Mainz, using the continuous-wave electron microtron MAMI . The energies of the incoming electron beam ranged from 405 to 855 MeV, and the beam energy spread did not exceed 0.16 MeV. The 15 to 35 ?A electron beam was scattered on a liquid hydrogen target cell (5 × 1 × 1 cm3 with 10 ?m Havar walls in settings with the photon polarisation parameter ε = 0.437, 0.648, 0.457 and 0.259, and on a 2 cm-diameter cylindrical target cell with 50 ?m Havar walls in all other settings) attached to a high power target cooling system. Forced circulation and a beam wobbling system were used to avoid density ?uctuations of the liquid hydrogen. With this system, luminosities of up to 3.2 · 1037 cm?2 s?1 (32 MHz/?b) were attained.
The scattered electrons and the produced pions were detected in coincidence by the high resolution (δp/p ≈ 10?4) magnetic spectrometers A (SpecA) and B (SpecB) of the A1 Collaboration . In settings with ε = 0.834, 0.500 and 0.742, electrons were detected with SpecB and pions with SpecA, and vice versa in all other settings. The momentum acceptance ?p/p was 20 % and 15 % in SpecA and SpecB, respectively. Heavy-metal collimators (21 msr in SpecA, 5.1 or 5.6 msr in SpecB) were used to minimise angular acceptance uncertainties due to the relatively large target cells. A trigger detector system consisting of two planes of segmented plastic scinˇ tillators and a threshold Cerenkov detector were used in each spectrometer. The coincidence time resolution, taking into account the di?erent times of ?ight of particles for di?erent trajectory lengths through the spectrometer, was between 1.0 and 2.6 ns FWHM. In each spectrometer, four vertical drift chambers were used for particle tracking, measurement of momenta and target vertex reconstruction. Back-tracing the particle trajectories from the drift chambers through the magnetic systems, an angular resolution (all FWHM) better than ±5 mrad (dispersive and non-dispersive angles) and spatial (vertex) resolution better than ±5 mm (nondispersive direction, SpecA) and ±1 mm (SpecB) were achieved on the target. A more detailed description of the apparatus can be found in Ref. .
In the o?ine analysis, cuts in the corrected coincidence time spectrum were applied to identify real coincidences and to eliminate the background of accidental coincidences. A cut in the energy deposited in the ?rst scintillator plane in the pion spectrometer was used to discriminate charged pions against ˇ protons. The Cerenkov signal was used to identify electrons in the electron spectrometer and to veto against positrons in the pion spectrometer. The true coincidences were observed in the peak of the accumulated missing 2 mass distribution (Emiss ? |pmiss |2 )1/2 ? Mn , using an event-by-event reconstruction of (Emiss , pmiss ) = (ω + Mp ? Eπ , q ? pπ ). The cross sections were subsequently corrected for detector and coincidence ine?ciencies (between +1.7 % and +3.2 %) and dead-time losses (between +1.4 % and +4.6 %). The detector e?ciencies and their uncertainties were measured by the three-detector method, while the coincidence e?ciency of the setup and the uncertainty of the dead-time measurement were determined in simultaneous single-arm and coincidence measurements of elastic p(e, e′ p), which was also used to check on the acceptance for the extended targets. 6
The accepted phase space was determined by a Monte Carlo simulation which provided the (event-wise) Lorentz transformation to the CM system and incorporated radiative corrections and ionisation losses of the incoming and scattered electrons and pions. Full track was kept of the particles’ trajectories and their lengths in target and detector materials. Since exact energy losses are not known event-wise, we used the most probable energy losses for the subsequent energy loss correction in both simulation and analysis programs, and compared the corresponding missing mass spectra. The uncertainty estimates were based on relative variations of their content in dependence of the cut-o? energy along the radiative tail. The uncertainty of the integrated luminosity originates only in the target density changes due to temperature ?uctuations within the target cell, while the electron beam current is virtually exactly known. Finally, a computer simulation was used to determine the correction factors due to the pion decaying in ?ight from the interaction point to the scintillation detectors, taking into account the muon contamination at the target (correction factors ranging from ×2.23 to ×2.89 in di?erent settings). The systematical errors of the pion decay correction factors were estimated from the statistical ?uctuations of the back-traced muon contamination at the target.
Results and discussion
In the Born approximation, the coincidence cross section for pion electroproduction can be factorised as [23,28]
dσ dσv = Γv , ′ d?? d?e d?? π π
where Γv is the virtual photon ?ux and dσv / d?? is the virtual photon cross π section in the CM frame of the ?nal πN system. It can be further decomposed into transverse, longitudinal and two interference parts, dσv dσT dσL = + ε? + L ? ? d?π d?π d?? π 2 ε? (1 + ε) L dσLT dσTT cos φπ + ε cos 2φπ (6) ? d?π d?? π
with the transverse (ε) and longitudinal (ε? = Q2 ε/ω ?2) polarisations of the L virtual photon ?xed by the electron kinematics. The measured cross sections are listed in Table 2.
? In parallel kinematics (θπ = θπ = 0? ) the interference parts vanish due to ? ? their sin θπ and sin2 θπ dependence. At constant Q2 , the transverse and the
Table 2 Measured cross sections in the p(e, e′ π + )n reaction.
dσ/ d?? π [ ?b/sr ]
Stat. error [ ?b/sr ]
Syst. error [ ?b/sr ]
0.834 0.500 0.219 0.742 0.437 0.229 0.648 0.457 0.259
11.14 0.08 (0.7%) 8.40 0.11 (1.3%) 5.96 0.14 (2.3%) 7.48 0.11 (1.5%) 5.36 0.09 (1.7%) 4.55 0.10 (2.1%) 5.03 0.05 (1.1%) 4.19 0.06 (1.3%) 3.46 0.05 (1.4%)
0.41 (3.7%) 0.31 (3.7%) 0.19 (3.2%) 0.18 (2.4%) 0.10 (1.8%) 0.07 (1.6%) 0.08 (1.6%) 0.08 (1.8%) 0.07 (1.9%)
longitudinal cross sections can therefore be separated using the Rosenbluth method by varying ε, dσv dσT Q2 dσL = + ε ?2 . d?? d?? ω d?? π π π (7)
Not only the statistical, but also the ε-correlated systematical uncertainties of the data were considered in our ?t, whereas the ε-independent systematical errors were included in the ?nal uncertainty of dσT and dσL . The results are shown in Fig. 3 and Table 3. The Q2 -dependence of the separated transverse and longitudinal cross sections can be seen in Fig. 4, together with the theoretical ?ts used to extract the form factors. Since values of ν = ?Q2 /M 2 and W in this experiment were too high for a direct application of χPT, an e?ective Lagrangian model [23,24] was used to analyse the measured Q2 -dependence of the cross section, and to extract the nucleon axial and pion charge form factors. In the energy region of our experiment, the pseudovector πNN coupling evaluated at tree-level provided an adequate description of the reaction cross section. We included the sand u-channel nucleon pole terms containing electric and magnetic Sachs nucleon form factors of the well-known dipole form with a ‘cut-o?’ Λd = 0.843 GeV, the t-channel pion pole term with a monopole form factor Fπ (Q2 ) = 1/(1 + Q2 /Λ2 ), the contact (seagull) term with the axial dipole form factor π 8
Fig. 3. Least-squares straight-line ?ts to the measured cross sections for p(e, e′ π + )n at W = 1125 MeV, for three values of Q2 . The smaller error bars correspond to statistical, the larger ones to the sum of statistical and systematical errors. Table 3 The results of the L/T separation based on the least-squares ?t to the data.
Setting (Q2 ) [ GeV2 /c2 ]
dσT / d?? π [ ?b/sr ]
(Q2 /ω ?2 ) dσL / d?? π [ ?b/sr ]
0.117 0.195 0.273
4.160 ± 0.165stat ± 0.202sys 3.080 ± 0.139stat ± 0.051sys 2.406 ± 0.088stat ± 0.056sys
8.394 ± 0.254stat ± 0.481sys 5.747 ± 0.284stat ± 0.246sys 4.012 ± 0.187stat ± 0.052sys
2 GA (Q2 ) = GA (0)/(1 + Q2 /MA )2 , and the s-channel ?-resonance term. Vector meson exchange contributions in the t-channel were found to play a negligible role in the charged pion channel.
Due to cancellations between higher partial waves and interference terms with the s-wave, dσT is predominantly sensitive to the E0+ (nπ + ) amplitude and therefore to MA . On the other hand, the pion charge form factor appears in the longitudinal amplitude L0+ (nπ + ) only at order O(?2 , ?ν), and the swave contribution to dσL amounts to 10 % only. Due to the contributions of the higher partial waves, however, the longitudinal cross section dσL is quite sensitive to Λπ , which is a bonus ‘by-product’ of the analysis. Since current conservation is violated if arbitrary form factors are included, gauge invariance was imposed by additional gauge terms in the hadronic cur9
8 dσT(0) fixed, MA fitted dσT(0) and MA fitted 6
Q [GeV /c ] (Q /ω ) dσL/d?π* [?b/sr]
Q [GeV /c ]
Fig. 4. Separated transverse and longitudinal cross sections. The solid line shows our ?t with dσT / d?? (Q2 = 0) ?xed to (7.22 ± 0.36) ?b/sr; the dotted line is the π unconstrained ?t. In the longitudinal part the ?ts are almost indistinguishable. The smaller error bars correspond to statistical, the larger ones to the sum of statistical and systematical errors.
rent. These terms modify the longitudinal part of the cross section and therefore in?uence the pion pole term and thus the extracted value of Λπ . However, this procedure does not a?ect the transverse part of the cross section. The axial mass can then be determined from the Q2 -dependence of dσT , and the result can be compared to the prediction of χPT. In the theoretical ?t, the transverse part is ?tted ?rst.
We have used two di?erent techniques to determine the axial mass: (I) Only the axial mass was varied, whereas the value of dσT at Q2 = 0 was ?xed by extrapolating the transverse cross section (i. e. the E0+ (nπ + ) amplitude) to the value of the photoproduction angular distribution at θπ = 0? . A precise cross section at the photon point (Q2 = 0) is very helpful for our analysis and can reduce the uncertainty in determining the axial mass considerably. Unfortunately, there are no experimental data available around W = 1125 MeV and forward angles. The only measurements in this energy region are performed at pion angles larger than 60? with large deviations among the di?erent data sets. Therefore we have used a value at the photon point obtained from di?erent partial-wave analyses of the VPI group  and of the Mainz dispersion analysis . This yields to a weighted-average cross section at the photon point of (7.22 ± 0.36) ?b/sr and this value was used as an additional data point. The corresponding value of E0+ (nπ + ) is also well supported by the studies of the GDH sum rule  and by the low energy theorem (Kroll-Ruderman limit). (II) The three data points alone were ?tted, while the value of the transverse cross section at Q2 = 0 was taken as an additional parameter, with the result dσT (0) = (7.06 ± 1.12) ?b/sr. The best-?t parameters for the transverse cross section were then used to ?t the longitudinal part. Using the ?rst and preferred procedure, we ?nd from the transverse cross sec2 tion MA = (1.077 ±0.039) GeV, corresponding to rA 1/2 = (0.635 ±0.023) fm. From the longitudinal part, we obtain Λπ = (0.654 ± 0.027) GeV, correspond2 ing to rπ 1/2 = (0.740 ± 0.031) fm. The second procedure leads to the follow2 ing results: MA = (1.089 ± 0.106) GeV or rA 1/2 = (0.628 ± 0.061) fm, and 2 1/2 Λπ = (0.658 ± 0.028) GeV or rπ = (0.734 ± 0.031) fm.
Summary and conclusions
We have measured the electroproduction of positive pions on protons at the invariant mass of W = 1125 MeV, and at four-momentum transfers of Q2 = 0.195 (GeV/c)2 and 0.273 (GeV/c)2 . In conjunction with our previous measurement at Q2 = 0.117 (GeV/c)2 , we were then able to study the Q2 dependence of the transverse and longitudinal cross sections, separated by the Rosenbluth technique for each Q2 . The statistical uncertainties were between 0.7 % and 2.3 %, an improvement of an order of magnitude over the result of Ref. . The systematical uncertainties were estimated to be between 1.6 % and 3.7 %, and are expected to decrease signi?cantly in the future experiments. 11
We have extracted the axial mass parameter MA of the nucleon axial form factor from our pion electroproduction data using an e?ective Lagrangian model with pseudovector πNN coupling. Our extracted value of MA = (1.077 ± 0.039) GeV is (0.051 ± 0.044) GeV larger than the axial mass MA = (1.026 ± 0.021) GeV known from neutrino scattering experiments. Our result essentially con?rms with the scaled-error weighted average MA = (1.068 ± 0.017) GeV of older pion electroproduction experiments. If we include our value into the database, the weighted average increases to MA = (1.069 ± 0.016) GeV, and the ‘axial mass discrepancy’ becomes ?MA = (0.043 ± 0.026) GeV. This value of ?MA is in agreement with the prediction derived from χPT, ?MA = 0.056 GeV. We conclude that the puzzle of seemingly di?erent axial radii as extracted from pion electroproduction and neutrino scattering can be resolved by pion loop corrections to the former process, and that the size of the predicted corrections is con?rmed by our experiment. Theoretical input needed to extract the axial mass and the pion radius has naturally led to some model dependence of the results. The dominant contribution to pion electroproduction at W = 1125 MeV is due to the Born terms. These are based on very fundamental grounds and the couplings are very well known. For our purpose the electric and magnetic form factors of the nucleons are also accurately known whereas the remaining two (the axial and the pion form factors) are the subjects of our analysis. In parallel kinematics we are in the ideal situation where the pion form factor contributes only to the longitudinal cross section, therefore reducing very strongly the model dependence of the transverse cross section and consequently of the determination of the axial mass. Furthermore, the ? resonance, which plays the second important role in our theoretical description, also contributes with a well-known M1 excitation to the transverse cross section, while the longitudinal C2 excitation gives rise to a larger uncertainty due to the less known C2 form factor, currently under investigation at di?erent laboratories. Altogether, the model dependence is smaller for extracting the axial mass than for the pion radius. This could partly explain the discrepancy in the di?erent values of the pion radius between our analysis and the analysis of pion scattering o? atomic electrons . However, as in the case of the axial mass, an additional correction of the pion radius obtained from electroproduction experiments is very likely. This should be investigated in future studies of chiral perturbation theory.
Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft, SFB 201.
 G. Fanourakis et al., Phys. Rev. D 21 (1980) 562; L. A. Ahrens et al., Phys. Rev. D 35 (1987) 785; L. A. Ahrens et al., Phys. Lett. B 202 (1988) 284.  S. J. Barish et al., Phys. Rev. D 16 (1977) 3103; K. L. Miller et al., Phys. Rev. D 26 (1982) 537; W. A. Mann et al., Phys. Rev. Lett. 31 (1973) 844; N. J. Baker et al., Phys. Rev. D 23 (1981) 2499; T. Kitagaki et al., Phys. Rev. D 28 (1983) 436; T. Kitagaki et al., Phys. Rev. D 42 (1990) 1331.  M. Holder et al., Nuovo Cim. A LVII (1968) 338.  R. L. Kustom et al., Phys. Rev. Lett. 22 (1969) 1014.  D. Perkins, in: Proceedings of the 16th International Conference on High Energy Physics, J. D. Jackson, A. Roberts (eds.), National Accelerator Laboratory, Batavia, Illinois, 1973, Vol. IV, 189.  A. Orkin-Lecourtois and C. A. Piketty, Nuovo Cim. A L (1967) 927.  S. Bonetti et al., Nuovo Cim. A 38 (1977) 260.  N. Armenise et al., Nucl. Phys. B 152 (1979) 365.  I. Budagov et al., Lett. Nuovo Cim. II (1969) 689.  C. Caso et al. (Particle Data Group), Review of Particle Properties, Eur. Phys. J. C 3 (1998) 9.  A. S. Esaulov, A. M. Pilipenko, Yu. I. Titov, Nucl. Phys. B 136 (1978) 511.  M. G. Olsson, E. T. Osypowski and E. H. Monsay, Phys. Rev. D 17 (1978) 2938.  E. Amaldi et al., Nuovo Cim. A LXV (1970) 377; E. Amaldi et al., Phys. Lett. B 41 (1972) 216; P. Brauel et al., Phys. Lett. B 45 (1973) 389; A. del Guerra et al., Nucl. Phys. B 99 (1975) 253; A. del Guerra et al., Nucl. Phys. B 107 (1976) 65.  P. Joos et al., Phys. Lett. B 62 (1976) 230.  S. Choi et al., Phys. Rev. Lett. 71 (1993) 3927.  Y. Nambu and M. Yoshimura, Phys. Rev. Lett. 24 (1970) 25.  E. D. Bloom et al., Phys. Rev. Lett. 30 (1973) 1186.  N. M. Kroll and M. A. Ruderman, Phys. Rev. 93 (1954) 233.
 Y. Nambu and D. Luri?, Phys. Rev. 125 (1962) 1429; e Y. Nambu and E. Shrauner, Phys. Rev. 128 (1962) 862.  G. Furlan, N. Paver and C. Verzegnassi, Nuovo Cim. A LXX (1970) 247; C. Verzegnassi, Springer Tracts in Modern Physics 59 (1971) 154; G. Furlan, N. Paver and C. Verzegnassi, Springer Tracts in Modern Physics 62 (1972) 118.  N. Dombey and B. J. Read, Nucl. Phys. B 60 (1973) 65; B. J. Read, Nucl. Phys. B 74 (1974) 482.  G. Benfatto, F. Nicol` and G. C. Rossi, Nucl. Phys. B 50 (1972) 205; o G. Benfatto, F. Nicol` and G. C. Rossi, Nuovo Cim. A 14 (1973) 425. o  K. I. Blomqvist et al. (A1 Collaboration), Z. Phys. A 353 (1996) 415.  D. Drechsel and L. Tiator, J. Phys. G: Nucl. Part. Phys. 18 (1992) 449.  V. Bernard, N. Kaiser and U.–G. Mei?ner, Phys. Rev. Lett. 69 (1992) 1877; V. Bernard, N. Kaiser and U.–G. Mei?ner, Phys. Rev. Lett. 72 (1994) 2810.  H. Herminghaus et al., Proc. LINAC Conf. 1990, Albuquerque, New Mexico; J. Ahrens et al., Nucl. Phys. News 2 (1994) 5.  K. I. Blomqvist et al., Nucl. Instr. Meth. A 403 (1998) 263.  E. Amaldi, S. Fubini and G. Furlan, Springer Tracts in Modern Physics 83 (1979) 6.  R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 53 (1996) 430; see also the web-page http://said.phys.vt.edu/analysis/go3pr.html.  O. Hanstein, D. Drechsel, L. Tiator, Nucl. Phys. A 632 (1998) 561.  D. Drechsel and G. Krein, Phys. Rev. D 58 (1998) 116009.  S. R. Amendolia et al., Phys. Lett. B 146 (1984) 116; S. R. Amendolia et al., Phys. Lett. B 178 (1986) 435.  G. Bardin et al., Nucl. Phys. B 120 (1977) 45.