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TIU-PP-94-90 I. INTRODUCTION Oct 1994 Nuclcon-nucleon brcntsstrahlung has been extensively investigated both experimentally and theoretically during the past 30 years. For pp —> ppj the experimental data available A-Excitation and Exchange Corrections for NN-Bremsstrahlung covers an energy domain between 42 MeV [I] and 730 MeV [2], the most rrccnl results being obtained at incident proton energies around the pion production threshold [3-5]. Up to the pion production threshold, the NPf-j potential model using realistic NN-potcntials M. Jetter and H. W. Fearing for the description of the nuclear force plus first order approximation for the electromagnetic interaction gives the most successful description. TRWMF, W0\ Wesbrook Mail, Vancouver, B. G, A comparison of different NN-interactions yields widely equivalent WJVir-results in the whole kinematic range [6], so that bremsstrahlung provides a sensitive test for the dynam- Canada V6T SAS ical model used to describe the photon emission, lu the framework of the potential model, however, there is little controversy about the basic features and the correction terms to be used. In particular, relativistic spin corrections as derived in [7,8] and rescattering contri- butions [9] are used in the most recent analyses [6,10], see also [11,12]. Moreover, the role of Coulomb corrections in ppy [13] and exchange currents in the np-j cross section have been discussed [9,14]. Abstract The main theoretical problems left are presumably related to shortcomings in principle of the potential model description. For example, Lorcntz invariance can only be accounted The role of the relativistic amplitudes for a number of O(k) processes usually neglected for approximately by a covariant treatment of the kinematic transformations, inclusion of in potential model calculations of NN-bremsstrahlung is investigated. In particular, we higher order terms in (p/m) in the electromagnetic operator (relativistic spin corrections) consider the A-excitation pole contributions related to the one-pion and one-rho exchange and an appropriate description of the NN-intcracti»n. Further common features of existing and in addition include the exchange contributions induced by the radiative w, p -* -x-j potential model calculations are the absence of dynamical baryou resonances and the neglect decays. The contributions are calculated from relativislic Born amplitudes fitted to A- of two-body currents beyond the O(k°) terms given by the soft photon approximation (SPA). production and absorption data in the energy range up to 1 GeV and then used to supplement The role of the A-excitation was studied in the 1970*s in two different approaches. The potential model and soft photon calculations for nucleon-nucleon brctnsstrahlung. The authors of [15] used a dispersion analysis in order to correct the one-pion exchange neutron- effects on JV/V7-observables, although moderate in general, are found to be important in proton electromagnetic current. The effect of the A-resonance on the npy cïîss section at 200 some kinematic domains. MeV turned out to be small. In [16] and [17], the A-excitalion part of the ppj amplitude was derived from phenomcnological Lagrangians and combined with soft photon or one-boson (submitted to Physical Review C) exchange Horn calculations for the radiative background. A more sophisticated potential approach including effects of the A based on a coupled PACS: 13.75.Cs, 25.20.-x, 25.40.-h channel calculation of the half off-shell NN- and N A T-matrices together with a phenonieno- logical JVA7-vcrtcx has been published recently [18,19]. In contrast to the njry-results of [15], the authors find an appreciable A contribution to the 280 MeV ppj observables. For higher energies where the potential model is inappropriate, the closely related prob- lem of dileplon production has recently been studied in the framework of an effective one- boson exchange plus A-excitation Born approximation [20]. As mentioned above, a rigorous derivation of induced two-body currents is not possible to all orders. The reason is that, unlike e. g. for a one-pion exchange potential [21], the gauge invariant replacement VN(P) -» VN (p*— e/l) in the argument of a general NN-potential Vtf [9,14] leads to au unique expression for the induced current only in SPA. To this order, the induced current approximates contributions due to the exchange of charged mesons and vanishes for ppi- The radiative decay processes we are considering here are thus not accounted for in the conventional potential model. So far, only the radiative ut —* x°7 decay for ppy has been considered as an example of such an O(Jt) internal radiation process [22|. The purpose of the present work is to provide an estimate of the rote of baryon reso- nances and internal radiation processes reliable up to photon energies of about 300 MeV corres|>oruling to the highest photon energy possible in the 730 MeV [ijrf-cxperiulcnt of
VOL 2 7 H» 1 7 [2]. From a comparison with recent analyses of pion-photoprodiiclion [23], we expect the involve only on-shell nucléons, pseudovector coupling for the pion instead of equation (3) A(]232)-resonancc and the radiative decay of the u) and p to be the leading corrections. would not change any of our results. From eq. (3) - (6), the vertex functions read A coupled channel calculation for the NN — NA system omitting the contribution of the AA-states [24] is typically represented by a set of Lippmann-Schwinger equations = -igpHtl \-f -
ÏAJV = VAN "I" V&NGNTNN + *^Û«A-^A/V» (2)
Inclusion of the A thus modifies the NN-amplitude TNf/ and yields an additional ampli- and one derives the transition amplitudes for diagram (a) in Fig. 1 [17]: tude TAN. The pure NN-interaction below the pion production threshold is known to be well described by phenomenological and/or meson theoretic potentials so that we feel safe Pi - Jt)^r£tt(
I T3 I Xy)(Xy I r3 \ Xr) = | II. INCLUSION OF THE A The full A excitation amplitude including the isospin factors and a relative minus for graphs involving interchange of the final state particles is then For the derivation of the A-excitation terms depicted in Fig. 1 we start from the inter- action Lagrangians for the isovector mesons x and p (sec J27J) in the notation of [28]:
(3) 3
T (4)
(5)
(6) l;p3,p4))}. (10) For the electromagnetic NA current of diagram (a) we use the parametrization [29] where V'A is the spin-3/2 Rarila-Schwinger field and f stands for the isospin operator for the transition of an isospin-1/2 and isospin-1 to an isospin-3/2 particle [27]. We have used the pscudscalar jrWiV-coupliiig for convenience. As the irAW-vcrticcs in our calculations ^-i-ff- W) + 2{{jrv - k • pg" s f-,#,»(<:,;») > equal to ryM^{k,p) up to a relative minus sign in the term proportional to G\. In writing down the Lorcnlz invariant propagators ( 15) and ( 16) we have dropped the static 2 1 The couplings G'i and G2 arc experimentally determined by a fit to the MI+/EI+ multipole approximation q* —• — q of the potential models A and B. As we an far from the pole in all pion-pholoproduclion cross section. Values for Gt and Gj range from G\ = 2.0, Gj = 0, our geometries and rescale our final amplitudes this has very little rlfccl on the end results. the prediction of the vector dominance model, to G| = 2.68, Gt = ~1.84 [18,29], recently For the calculations using model A, each NNirfNNp-veitex is multiplied by a monopole quoted values being G\ = 2.208, G = -0.55G [23J. 7 cutoff AÎ/(,/(AÎ/p - propagator requires some caution. In [17], the llarita-Schwinger vertices. Here again, we use Lorenlz invariant instead of static expressions. For model B, form 2 2 both the JV7V- and /VA-vertices are regularized by cutolTs [(A — »>Î/P)/(A — 9*)]" where 2- n = 1, 2 for x, p exchange respectively. ?-, In general, the relativistic off-shell A-partide is allowed to propagate with both spin 3/2 and spin 1/2. This extra freedom is reflected by additional off-shell terms in the interaction is used where the replacement m& —» m& — iT/2 is made in order to account for inelasticities Lagrangians; the more general chiral xN A-vertex functions corresponding to the propagator due to pion production. As we wish to use PA in the far ofF-resonance region (p1 « ma), we (12) read [32,33]: have to take the energy dependence of the width F —» F(g) into account. For our purpose, we rely on the parametrization of [30] requiring F to vanish below the pion production threshold: AÏ... = g, ) = 0, q2<0, W G""(Z,,P) = g"" - (Z.u + 3)7 V- (17) > 0. (13) Here, q,N denotes the maximum momentum in the xyV-subsystem of the process NN —» A generalization similar to ApNa (with parameter Z-,) applies for the (leading) G|-term NJVx: of equation (II). The ^-parameters are not well determined either theoretically or ex- perimentally. Whereas the simple coupling scheme corresponding to Z = —1/2 is gener- ally preferred in potential models, Olssen and Osypowski [33] find experimental values of a,s = [y/s — *"iv) , (14) Zx = 0 ± 1/4, Z, = 1/4 ± 1/4. Note that the off-shell parameters do not affect a cou- pled channel calculation at the OBB level but directly enter the Horn amplitude for the y/a is. the invariant energy in the TVJV-system and R, 7 are adjustable parameters. Values brenisstrahlung process where the A is off-shell. We therefore rely on the parameters of of 7 = 0.7lAfeV, R = 0.81 lead to a resonance width F = 120 MeV at swN = 1236 MeV Table 1 as fixed by NN-data (to all orders of a coupled channel calculation) but check the [30]. This value might be slightly modified by a variation of 7. It has been shown [31] that sensitivity of our results to a variation of Z. this parametrization is in good agreement with the experimental £33 phase as well as with a This completes the necessary ingredients for the A-excitalion part of the bremsstrah- A self energy calculation [24]. lutig amplitude (10). With the definitions of this chapter we have tried to incorporate as NN — A/A potential models differ in the choice of the A-coupling constants as well as in much as possible of the experimental information on the /VA excitation channels in the the definition of the meson propagators and vertex form factors. We use these models as a NN-interaction which have become available since the earlier Born calculations [17], [16]. starting point for our calculation defining the A-excitation amplitude up to a normalization. On the other hand, the relativistic approach allows us - in contrast to a pure potential In a reliable calculation, the final results should not depend too much on the underlying pa- model calculation [18] - to extend the analysis to energies far beyond the pion production rameter sets provided they agree comparably well with the empirical NN-data. We therefore threshold. compare two different recent coupled channel models: The OBE-model of [24], denoted as model A, and a coupled channel version of the Bonn potential (model I of [27], p. 353), denoted as model B. For model A, we use the meson propagators III. INCLUSION OF THE RADIATIVE u>/p-DECAY (15) We write the Lagrangian for the coupling of the u-meson to the nucléons in analogy to equation (4): The second term in P'9' drops out in the calculations of this and the following section. The propagators of model U take the mass difference between the nucleou and the A into (18) account: and parametrize the radiative decay vertices as in [23]
(20)
(i With the meson propagators as in (15) and a vertex function A£NH as in eq. (7) this yields the Horn scries. From a comparison of our Horn results using model A with the total A- for the diagram (c) of Pig. 1 absorption cross sections of [24] calculated in the iterated coupled channel model, one finds a factor g = 1.65 at lab energies below 300 McV. A calculation of the total A-production cross section and comparison with the experimental data at 800 MeV [25] and 970 MeV [26] suggests a weak energy dependence of g so that g(730 MeV) = 1.4. The latter value is in (21) dose agreement with the result of [20]. As we have to compensate also for the the change and consequently (see equation (10)) from static to Lorentz invariant propagators in eqs. (15), (16), we use ^(280 MeV) = 1.95 and s(730 MeV) = 1.65 respectively. If we suppose the higher order corrections to be of equal importance for both model A and B and for the w/p-decay amplitudes for which no experimental comparison could be made, the model is completely determined with the empirical values of g(Ttl,t,) found above. lîPiiPï)- (22) The rescaled amplitudes (10) and (21) can thus be coherently added to the corresponding transition matrices of the potential model [6] and the soft-photon approximation [35,36] so The p-decay amplitudes are obtained from 7L,-, by interchange of the masses and cou- that the effects upon both cross sections and spin observables can be studied. The relevant pling constants and multiplication with the isospin operator fi • f2. The matrix elements of formula for the observables considered can be found in the literature ( [8,17]). For the this operator yields factors of 1 and 0 for the neutral and charged pion decay amplitude in potential model calculations we shall focus on the domain around pion production threshold PP7. For npy the corresponding values are -1 and 2. where both correction effects should be maximum; the soft photon model will be used to The coupling constants g^,^ and gpri are determined from the experimental decay widths reanalyze the experimental data of [2]. of the vector mesons [34]: In principle, with all conventions carefully chosen, the formulas in the last two sections give the correct relative signs for the corrections with respect to the leading (potential model 716 ±75 keV forw, or SPA) amplitudes. For an independent check we combined our amplitudes (10) and (21) with the pure one-pion exchange brcmsstrahlung amplitude (Fig. 1, diag. (a) with the in- 121 ±31 keV forp0, (23) ternal A-line replaced by a nucléon line). If the exchanged pion is numerically put on-shell 1 68±7 keV forp*. (q° + m*), this is equal up to a common factor to the pion-photoproduction ampli- tude with and without intermediate A excitation. In a near-threshold geometry [q1 ~ 0), This leads to the numerical values the A-amplitude gives enhancement of the cross section by a few percent, in agreement with Peccei's chirally invariant Lagrangian for r°p —» prj [37] whereas the w/p-decay amplitudes 3 2 3 2 3 2 interfere destructively, as in [23]. = 0.715 • 10- m; ; = 0.125 • 10-,-3m;m-2.; îetu. = 0 070 . io- m7 . (24) 4JT 4ir The sign of the ^-coupling constant given in eq. (24) is consistent with the pion- photoproduction data [23]. The sign of the much smaller radiative p-decay amplitude, For the remaining coupling constants and cutoffs, we again follow the rationale of however, is still the object of controversies [38). We adopt here the sign convention of Gari the previous section and use a complete set that has been successfully tested in pion- et al. [39] where the p-decay enhances the contribution of the u>-decay in ppr, as the p- photoproduction [23]. Thus we put contribution is only about 2 % of the u> (compare the respective coupling constants), a switch in the sign of gfrj would not alter our'conclusions.
= 0.563; Kp = 3.71; jp = 5.07; (25)
These values arc in agreement with the quark model and the vector dominance assump- A. Potential Model Results tion. All form factors in this section arc set to 1; the pion couplings arc taken from table 1. Note that a measurement of the vector meson radiative decay contributions in the pion All the calculations in this section are based on the inversion potential to the Nijmegen-II photoproduction process determines the product of the couplings in eqs. (24) and (25). NN-phase shifts used in Rcf. [6]. Using another realistic NN-potcntial would not change With the well-known empirical value of gr, the amplitude (21) is therefore essentially fixed. any of our results. The basic features of the potential model can be essentially represented in terms of two variables: the total energy of the process Tut, as a crucial parameter for the relative size of the various contributions in the amplitude, and the opening angles of the two protons determining the maximum photon energy and thus the off-shell signature of IV. RESULTS the process [6]. The two examples of Fig. 2 show the cojilanar, exclusive ppy cross section 8 at small proton angles as well as for the analyzing power Av at large proton angles. The V. CONCLUDING REMARKS A-contribulion is the leading effect but tends to be partly cancelled by the radiative decay contribution:). The maximum net effect of both contributions can increase the differential We have evaluated the A-excitation and radiative u/p-decay corrections to proton- cross section by 15 — 20% at medium photon angles but amounts to only a few percent in proton and neutron-proton bremsstrahlung for the energy range up to TM ~ 1 GeV. This the total cross section. was dune by calculating the relativistic Born amplitudes and adding them to potential model Among the examples of Fig. 2, only the 12.4°, 14° analyzing power shows a sizeable and SPA amplitudes. The Born amplitudes were normalized by calculating A-production dependence on the parametrization of the *• and p-exchange. The choice of different elec- and absorption cross sections in the same model and Fitting to experimental data and coupled tromagnetic couplings, e. g. G\ = 2.68, t7j = —1.84, increases the A-contribution in the channel predictions. cross sections by about 20 % which translates to an enhancement of the complete ppy cross Of the two processes considered, the A-excitation dominates but is generally partly section by about 5 % at maximum (Figure 3). Conversely, the choice of Z, = Z7 = —1/4 compensated by the radiative decay contributions. Both corrections together increase the [23] reduces the size of the A-contributions by roughly 20%. These two choices for the 280 McV pp7 integrated cross section dPa/diltdilj by an amount depending essentially on couplings and olf-shell parameters are likely to be rather extreme. Moreover, because of its the proton opening angles 0,- and reaching a maximum of roughly 7.5% at 0t ~ B2 ~ 20°. small relative size, a reseating error in the w/p-amplitude would not affect much the end The relative effect on iPo/dûidfydO-, is maximum for photon emission around ft, ~ 90° and results. One might therefore conclude that the corrections are reasonably well determined is typically < 20%. in our model. In the 730 MeV Rochester geometry, the corrections become relevant for photon energies In order to obtain a more general picture we calculate the double differential cross sec- above ~ 100 MeV but are net big enough to complement the soft photon approximation tion da/dilidili = f dO-,(d 10 agree only in relative size. REFERENCES Finally, a comparison with the coupled-channel A-results of [18] in the low energy region shows good agreement for the cross sections ami even for the analyzing powers Ay. This is [1] C. A. Smith, J. V. Jovanovich and L. G. Greeniaus, Pliys. Rev. C22 (1380), 2287 and encouraging for the feasibility of our method as well as for the various extensions made. earlier publications quoted therein. We wish to thank Dr. S. Schercr for valuable discussions. This work was supported in [2] B. M. K. Nefkens, O, R. Sander and D. I. Sober, Pliys. Rev. Lett. 38 (1977), 876 part by a. grant from the Natural Sciences and Engineering Research Council of Canada. [3] K. Michaelian et al., Phys. Rev. D41, 2689 (1990) [4] P. Hitching et al., Phys. Rev. Lett. 57, 23G3 (1986); Nucl. Phys. A463, 87 (1987) [5] D. v. Przewoski et al., Phys. Rev. C45, 2001 (1992) [6] M. Jetter and H. V. von Geramb, Phys. Rev. C49, 1832 (1994). See also: M. Jetter, II. Frcitag und H. V. von Geramb, Nucl. Phys. A553, 655c (1993) and Phys. Scr. 48, 228 (1993) [7] R.L. Workman and H. W. Fearing, Phys. Rev. C34, 780 (198G) (8] M. K. Liou and M. E. Sobel, Ann. Phys. 72, 323 (1972) [9] K. Nakayama, Phys. Rev. C39, 1475 (1989) [10] V.A. Herrir.ann and K. Nakayama, Phys. Rev. C45,1450 (1992), V. Herrmann, J. Spcth and K. Nakayama, Phys. Rev. C43, 394 (1091) [II] V. Brown, P. L. Anthony and J. Franklin, Phys. Rev. C44, 1296 (1991) [12] A. Katsogiannis and K. Amos, Phys. Rev. C47, 1376 (1993) [13] A. Katsogiannis, K. Amos, M. Jetter and H. V. von Geramb, Phys. Rev. C49, 2342 (1994) [14] V.R. Brown and J. Franklin, Phys. Rev. 8, 1706 (1973) [15] G. E. Bohannon, L. Heller and R. H. Thompson, Phys. Rev. C16, 284 (1977) [16] A. Szyjewicz and A. N. Kamal, in: Few Body Systems and Nuclear Forces I (Proc., Graz 1978), ed. by H. Ziegler, M. Haftel and II. Zankel; A. Szyjewicz and A. N. Kamal, in: Nucteon-Nucléon Interactions (Proc., Vancouver 1977), ed. by II. Fearing, D. Mcasday and A. Stralhdee, AIP conf. proc. no. 41 (1978) [17] L. Tiator, H. J. Weber and D. Drechsel, Nue. Phys. A306, 468 (1978), [18] F. de Jong, K. Nakayama, V. Herrmann and O. Scholten, Phys. Lett. B333, 1 (1994) [19] J. A. Eden et al. have recently performed a potential model calculation of A-effects in proton-proton bremsstrahlung (private communication). [20] M. Schâfer et al., Nucl. Phys. A575, 429 (1994) [21] D. O. Riska, Phys. Rep. 181, 207 (1989) [22] A. N. Kamal and A. Szyjewicz, Nucl. Phys. A285, 397 (1977) [23] II. Garcilazo and E. M. de Goya, Nucl. Phys. A562, 521 (1993) [24] H. ter Haar and It. Malllicl, PJiys. Ucj>. 149, 208 (1987) 11 12 125] J. Iludomalj-Gabitzsch cl a!., Phys. Rev. C6, 2CG6 (1978) FIGURES 126] V. Dmitriev, O. Suslikov and C. Gaardc, Nucl. Phys. A459, 503 (19S6) 1. A-cxcitation and internal radiation processes considered in this text. [27] R. Machleidt, in Advances in Nuclear Physics, Vol. 19, J. W. Negele and E. Vogt éd., 2. Coplanar pjrjr exclusive cross sections and analyzing powers at T| = 280 McV antl (19S9) ol for the smallest and largest proton angle pairs 0,, 02 of the TRIUMF experiment [3] [28] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New (the datapoints shown are rescalcd with a factor of 0.67 as in this reference). The York (1964) curves denote the pure potential model of Rcf. [6] (solid), potential model plus pure wfp-decay (long dashed), and the full model according to model A (doited) and B (29] II. F. Jones and M. D. Scadron, Ann. Phys. 81, 1 (1973) (dashed-dotted). [30] E. E. van Faassen and J. A. Tjon, Phys. Rev. C28, 2354 (1983) 3. Typical uncertainties of the A-contribution. The curves show the pure potential model (31] E. E. van Faassen and J. A. Tjon, Phys. Rev. C33, 2105 (1986) of Ref. [6] (solid) and the full calculation according to model A wilh standard param- eters (3, = 2.208, G2 = -0.278, Z, = 2, = -1/2 (long dashed). The upper bound of [32] R. Wittman, Phys. Rev. C37, 2075 (1988) the shaded area in the cross section and lower bound in A, corresponds to larger elec- [33] M. G. Olsson and E. T. Osypowskt, Nucl. Phys. B87, 399 (1975) tromagnetic couplings Gt = 2.68, G2 = -1.84. The lower bound in the cross section and upper bound in Ay is calculated with larger Z-parameters Z, = Z, = —1/4. [34] Particle Data Group, Phys. Rev. D45, part II (1992) 4. Relative change of the integrated, coplanar pjrjr cross section «Ptr/Jftt (/$!;,( with [35] II. W. Fearing, Phys. Rev. C 6, 1136 (1972); H. W. Fearing, in: Nucleon-Nucleon I corrections)/J a/dnl(/n2(wilhout corrections) at 280 MeV (dashed), 200 MeV Interactions (Proc., Vancouver 1977), ed. by 11. Fearing, D. Measday and A. Strathdee, (dashed-dotted) and 157 MeV (doited) for parameter set A as a function of the sym- AIP conf. proc. no. 41 (1978) metric proton angles 0\ = 0j. [36] B. M. K. Nefkens, O. R. Sander, D. 1. Sober and II. W. Fearing, Phys. Rev. C19, 877 5. Coplanar npy exclusive cross section and analyzing power at 7}at = 280 MeV. The (1979) curves denote the pure potential model of Rcf. [6] (solid), plus pure cu/p-decay (long- [37] R. D. Peccei, Phys. Rev. 181, 1902 (1969) dashed), and the full model according to model A and B (dashed-dotted and dotted). [38] P. Sarriguren, J. Martorell and D. W. I. Sprung, Phys. Lett. B228, 285 (1989) 6. 730 McV pprj measurements and soft photon approximation fn two typical geometries (see [36] (1979) and text). The curves denote the pure SPA results (solid), plus pure [39] M. Gari and II. Hyuga, Nucl. Phys. A264, 409 (1976) /à (long-dashed), and the full mode] according to model A (doited). Table I. Meson parameter of model A and B. For the definition of form factors, prop- agators etc. see text. Model A [24] Model B [27] ai A[MeV] ai A[McV] 4* 4» NNr 14.16 1140 14.4 1800 NNp 0.43 1140 0.7 2200 K=5.1 K=6.1 ATA* 0.35 910 0.35 920 NAp 4.0 13 14 lt [^b/jrfrad] * |l-^-w^_^-». H- s s i t s s, i i y y -0. 2 -0. 1 0. 0 0. 1 p p co - "ll ë- S3 s- a a °. U» =- i& Ti o -r! •=/ Relative A and u/p Effects O M * dV/dn^djO^ [yib/ir'rod] a S « 1.0 6 .0 2 3 r1 1 t O) (S S 0 . . . i S •A > G- ë ^ o [?• \\ \ ) \ m- \ 1 à t ^^ «s ^- H- 1 f u, 1 ; 2 _^ 8- V «• •//' B. o J ( 18 0 / \ Ul"