Virtual pion scattering

11 September 1997

PHYSICS

ELSEVIER

LElTERS

B

Physics Letters B 408 (1997) 12- 18

Virtual pion scattering M.A. Kagarlis a,‘, V.F. Dmitriev

b

a Niels

Bohr Institute, Blegdamsvej 17. DK 2100 Copenhagen. Denmark b Budker Institute of Nuclear Physics. Novosibirsk-90, 630090 Russia

Received 25 February 1997; revised manuscript received 27 May 1997 Editor: J.-F? Blaizot

Abstract We explore coherent pion production in (3He?Hn+) and (p,nr+) reactions as effective virtual pion scattering, in calculations of cross sections to the ground state and low-lying excitations of the target nucleus. Our coupled-channel model includes a source term emulating a virtual-pion beam and an optical potential for the interaction of the pion in the medium. We demonstrate that the landmark lowering of the peak in the energy-transfer spectra is entirely determined by the rescattering of the pion, and that its low-energy wing is sensitive to the pion-nucleon s-wave amplitude including its off-shell behavior. @ 1997 Published by Elsevier Science B.V. PAW 25.55.G; 25.40.K~; 24.1O.Eq Keywords: Coherent pion production; Off-shell s-wave amplitude; &J; Coupled channels

For many years pions served as a unique mesonic probe for the study of nuclei, with significant success. The effect of nuclear matter on their interaction with nucleons, however, is of such complexity that it continues to elude a definitive understanding. More recently, Ericson proposed virtual pions as a complementary tool [ 11. In contrast to real pions which are strongly absorbed and do not penetrate beyond the surface of a nucleus, virtual pions may be formed in the interior and traverse long distances. Consequently, with virtual pions, a larger region of the nucleus may be probed. Coherent pion production finds its role in this context as a process involving virtual pions at an intermediate step. It comprises of a charge-exchange reaction and the production of a virtual pion, which subsequently propagates in the nucleus off the mass shell. ’ Current address: Laboratoire National Satume, 91191 Gif-surYvette CEDEX, France. E-mail:kagarlisOipno.in2p3.fr.

By taking up the required recoil, the nucleus provides the momentum to bring the pion on shell. The final real pion emerges leaving the nucleus in its ground state, a process which has often been compared to elastic pion scattering [ 21. Presently, we extend this picture to include low-lying excitations of the target nucleus. Coherent pion production is a small channel relative to inclusive reactions with pion production. It has, nonetheless, attracted a lot of attention as a signature of the hadronic dynamics. Theoretical studies have so-far focused on the relative strength of the dominant longitudinal (pionic) channel of the nucleonnucleon interaction, and the rather model-dependent transverse corrections [ 3-61. The potential of the intermediate virtual pion as a probe of the in-medium pion-nucleon interaction, however, has been largely overlooked. The off-shell behavior of the low-energy pion-nucleon amplitude, in particular, is of fundamen-

0370-2693/97/$17.00 6 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00773-9

MA. Kagarlis. V.F. Dmitriev/Physics

tal significance, and it is crucial in formulations ranging from PCAC and current-algebra applications [ 71 to three-body nucleon forces [ 81. The interest in this subject has recently been renewed in connection with the s-wave pion and kaon condensation [ 9,101, and modifications of the pion mass and the quark condensate in nuclear matter [ 1l-141. Bearing these issues in mind, we have developed a theory in the spirit of Ref. [ 11, aiming at the exploration of coherent pion production as an effective laboratory of virtual pion scattering. The model consists of the coupled-channel inhomogeneous Klein-Gordon equation in coordinate space, with an explicit virtual pion source and a rescattering optical potential. In this article we are not so much concerned with the details of the model, which will be presented elsewhere. We concentrate foremost on the low-energy regime of energy transfer in coherent pion production. We show that the characteristic lowering of the peak in the energy-transfer spectra is entirely attributed to the in-medium rescattering of the pion, and furthermore, that its precise position and shape contain so-far unappreciated information on the off-shell component of the pionic channel. The shift is toward considerably lower energy, where the s-wave pion-nucleon amplitude acquires strength. The low-energy tail of the peak is sensitive to the off-shell pion-nucleon amplitude and the interference between the s and p waves. The core of our theory is the equation 1 d *d -_r-r2 dr dr

lp(b + 1) - U:,“$(r) r*

+ ki

>

a

which we will now review. To begin with, we point out that Eq. ( 1) separates conveniently the two processes which are probed by coherent pion production: the pion-nucleon and the nucleon-nucleon interaction (left- and right-hand side respectively). We discuss each of these below as building blocks for the description of effective virtual pion scattering. Setting the left-hand side of Eq. ( 1) equal to zero, we retain the homogeneous coupled-channel KleinGordon equation in coordinate space - a formalism which has been used extensively in modelling real pion-nucleus scattering processes [ 151. The angular-

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Letters B 408 (1997) 12-18

momentum indices 1 and I refer to pion and nucleus states, and J to coupled channels (isospin indices in this schematic discussion have been suppressed). The center-of-mass kinematics and the Coulomb interaction are contained in kg, as is the excitation energy of the nucleus. The initial and final channels in rescattering states are labelled LYand p, and the primed indices indicate the entrance channel with the target in its ground state. In real pion scattering ,J is invariant, so that J’ = J. The functions Rf$p$ (r) are the coupled-channel radial wavefunctions. The pion-nucleus optical potential U:,“&(r) consists of diagonal (cu = p) and off-diagonal ((w # p) components: Uf&(r)

=

Uo&p+

<

~p~ppJI%~ll,LJ > .

(2)

The diagonal optical potential UO is phenomenologi-

cal and it has been adjusted for agreement with elastic pion-nucleus scattering data. The off-diagonal component describes inelastic and charge-exchange pion scattering (here the isoscalar component only is relevant) and to lowest order it is derived as the matrix element of a microscopic coordinate-space operator VIN between pion-nucleus coupled-channel states /lZJ >. The operator VTN is obtained by a Fourier transform of the elementary pion-nucleon f,,v-matrix (including s and p, spin-no-flip and spin-flip components) with medium corrections, weighed by initial and final nucleon wavefunctions and one-body transition densities from the shell model, and summed over all valence nucleons to account microscopically for the transition between the target and residual nuclei. The optical potential (2) has been constructed gradually, incorporating higher-order phenomenological medium corrections as data in elastic, inelastic, single and double-charge exchange became available [ 151. It is especially successful for pion kinetic energies below the A resonance - corresponding to the low-energy wing of the energy-transfer spectrum in coherent pion production. The medium corrections of this formalism have been shown to be well-described in terms of pion absorption, the Lorentz-Lorenz effect, and Pauli correlations [ 161. For our present purposes we will refer to these medium corrections, determined from real pion-nucleus scattering, as conventional medium efiects.

In the absence of a real pion beam the pion-nucleus interaction can only be sustained by a (virtual) pion

14

MA. Kagarlis, V.F. Dmitriev/Physics

source, pib’;d;(r) on the right-hand side of Rq. ( 1) . The source berm probes the nucleon-nucleon interaction in the exchange of a space-like particle between the projectile and the target, a spin-isospin excitation of the target nucleus via an intermediate A resonance. In terms of our notation, the entrance channel, which was denoted by primed indices, now contains a virtual pion. The coupled-channel angular momentum is no longer an invariant, but observes the selection rule J = J’fl. The production of a pion at the entrance channel through an intermediate A-resonance excitation in the nucleus is expressed by tNN--rNNr

= tNN_NA

f e

(St.

R) (Pt.

9)

(

(3)

where tNN-+NA excites a A, and the remaining of Eq. (3) refers to the vertex of the A decay into a nucleon and a pion. In particular, 3 is the A spin, & the pion momentum, and ?,? the isospin of the A and the decay nucleon. While (3) is straightforward, the precise composition of tNN_,NA has been the subject of extensive theoretical work [3-61. We adopt a phenomenological model of the nucleon-nucleon interaction [ 3,171 2 tNN--rNA

x

= h

fit &?

[(~.8)(~.g)+(axci).(,Tx8)] (^7. o-Eh+12

.cA

?)

,

(4) where the strength g’ and the coupling constant A of the form factor ( A2-mi) / ( A2- t) (occuring once for each vertex) are treated as parameters. The propagator ( w - EA + irp /2) -’ accounts for the excitation of the A, and 6,? are isospin operators for the nucleon and the A respectively. The spin component of tNN+NA in (4) has a longitudinal (5 .Q) ( 3 .Q) and a transverse (& x 4) . (s x 8) channel, with &, 3 the spins of the nucleon and the A, and 4 the momentum transfer. In coherent pion production the longitudinal (pionic) channel is dominant, and the transverse channel is a small correction. We remark that though the transverse channel is model dependent and has been the subject of considerable debate in earlier studies cited above, these considerations are presently of marginal importance as

Letters B 408 (1997) 12-18

we are mainly concerned with the pion-nucleon aspects of coherent pion production. In so far as any model of the nucleon-nucleon interaction provides realistic cross sections for free NN and 3He-N scattering via the A-resonance channel, it satisfies our requirement of providing an effective virtual pion source, and our results below are insensitive to its details. To enforce the latter condition, we calculate the cross section for pp* rfpn with Eq. (4) and fit to the data of Ref. [ 191 - the only NN measurements in the literature exclusively via the A-resonance channel. This procedure yields an excellent fit with g’ = 0.43 and A = 800 MeV. Calculated cross sections for the reaction p( 3He,t)A+f with these values, after the effect of rescattering in the initial and final channels is included, are also in good agreement with the measurements of Ref. [ 201. With tNN_,,%,Nr fixed, the source term p;;,;;(r) of Eq. ( 1) is derived following the same procedure as for the off-diagonal pion-nucleus optical potential from tsN. The difference is that now, instead of an initial pion wavefunction, one has an effective virtual-pion wavefunction (and new boundary conditions) emulated by a convolution of the projectile and ejectile wavefunctions and a distortion factor in the eikonal approximation [ 4,181. The distortion factor accounts for the center-of-mass motion of the projectile-ejectile system for particles of finite size (such as 3He,3H). The source term so derived includes Fermi-motion corrections. The picture of coherent pion production that we have synthesized up to this point is purely conventional, meaning in the present context in accord with pion-nucleus and nucleon-nucleon phenomenology. If this description turns out to be sufficient, it will follow that coherent pion production cannot decipher the pion-nucleon interaction beyond real-pion scattering. On the other hand, the space-like features of the intermediate exchange pion open up the distinct possibility that the off-shell component of the pion-nucleon interaction play a role and result in a measurable effect. This possibility we examine next. The off-shell pion-nucleon amplitude is an illdefined object. This allows for an artificial modeldependence, which prevents settling the question of the pion mass and condensate in the medium (see e.g. the second of Refs. [ 121) . To illustrate this problem we recall that the off-shell pion-nucleon amplitude

MA. Kagarlis, V.F. Dmitriev/Physics

depends explicitly in w, due to time derivatives of the pion field. While the axial current is directly linked to the pion interpolating field in the PCAC approach, in effective chiral-Lagrangian models it is not [26]. This leads to different off-shell extrapolations, depending on whether one relies on properties of the axial current in the former case, or follows the rules of chiral symmetry in the latter. As a consequence, starting from PCAC or effective Lagrangians in chiral perturbation theory, leads to w* terms of opposite sign in the off-shell amplitude, even though they become identical in the on-shell limit. In principle this choice ought not to affect physical observables such as the pion mass and quark condensate [ 21 I. The off-shell contribution would be model-independent in an expansion to all orders of (q* - mi). In practice, however, PCAC allows to obtain terms up to 0( m$>, above which it is not evident whether PCAC or an effective chiral Lagrangian scheme should be preferred [ 121. This ambiguity can only be alleviated with experimental evidence. The simplest off-shell correction to the conventional pion-nucleon amplitude T$ is a T?r~= (q* - ma)fiAu + T”,;;, (5) where the operator ri depends on the specific choice of the pion field, and ii, u are nucleon Dirac spinors. Aside of the considerations of the previous paragraph, a second source of dependence on the pion interpolating field is in the factor (q* - mi), through which the off-shell effects of the operator A enter in the pionnucleus interaction to the second and higher orders in nuclear density. The higher orders in nuclear density are associated with other contributions as well, including the conventional ones discussed above. The Feynman diagrams corresponding to various contributions contain the internal pion lines and the pion-nucleon amplitude. Since in the internal lines the pions are off shell, these contributions depend on the particular offshell behavior of the pion-nucleon amplitude as well. Only the sum of all the contributions of a given order in nuclear density will be independent of the choice of the pion field. A procedure for the complete determination of the pion-nucleus amplitude may be the following: A pion field is chosen, resulting in a particular behavior of the off-shell amplitude (e.g. second derivative of the generating functional over the pseudoscalar source).

Letters B 408 (1997) 12-18

15

The amplitude is expanded with conventional and offshell corrections consistently to the same order in the kinematical variables, and any free parameters in the model are fitted simultaneously to pion-nucleus scattering and coherent pion production data. The latter can be very useful in constraining this procedure due to their sensitivity to off-shell effects. To demonstrate this sensitivity, we choose a particular pion field satisfying the PCAC relation. With this choice, the major component of the off-shell low-energy pion-nucleon amplitude is supplied by the Ward-Takahashi identity [ 221

T$Xp’,q’;p,q)

= TiL:’ +

q’* + q2 -

f27Tm;

m2 _ ?Tu’ua(t)6”‘a

l&Q’b(q’ + q) /J +fz,

2

where Tic:’denotes the Born term evaluated with pseudovector coupling, f* is the pion decay constant, a(t) is the nucleon sigma term and F$: are the remainder higher-order components in powers of pion momenta or mass (see Ref. [ 231 for a detailed discussion). For isospin-zero nuclei the isospin-even part of the amplitude (6) is relevant. The off-shell effects for forward scattering are proportional to q* - rn& which for real pion scattering is small. In this case the conventional corrections are most important and they are included in the expansion of the pion-nucleus optical potential as discussed above and elsewhere (see e.g. Ref. [ 241) . In coherent pion production, however, q* is large and negative, and results in a considerable increase of the s-wave repulsion. To make this discussion more quantitative, we construct a simple model in line with the above ideas. Neglecting the gradients of the density, nucleon Fermi motion and recoil in Eq. (6), we are left with the isoscalar forward scattering amplitude with all the pion invariant masses fixed at q’. Driving towards a simple correction such as Eq. (5), we follow the treatment of Ref. [ 121 and obtain, in the language of EricsonEricson [ 251, the s-wave correction

(7)

MA. Kagarlis, V.F. Dmitriev/Physics

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Letters B 408 (1997) 12-18

d) 1PC(SHe.3Hn+)‘2C’

c) '2C(3He.PHn+)'2C' ?

400

w (MeV)

0,

(ded

w

(MeV)

0

a0

40

8,

60

80

(de)

Fig. L. The (p,mr+) calculations are for Tp = 800 MeV and & = 0 deg. The angular distributions are for w = 230 MeV.

Fig. 2. The (3He,3Hr+) calculations, for T3, = 2 GeV and k& = 0 deg, with au&u distributions for w = 230 MeV.

where E = m,lK, fr = 93 MeV and C,+I= a(0) = 45 MeV. In light of the unresolved questions discussed above, our correction (7) must be considered as a rough estimate for the purpose of illustrating the sensitivity of coherent pion production to off-shell effects. In Fig. 1 we verify that our model for (p,nrr+) is consistent with earlier work for energy-transfer spectra and angular distributions [ 3,6,27]. The full curves include rescattering with conventional medium effects, as defined earlier. The dashed curves show the spectra without rescattering, corresponding to transparent nuclei with the form factor as the only trace left of the medium. The effect is a trivial lowering of the peak (e.g. -20 MeV for r*C) relative to the position of the A in the NN measurements of Ref. [ 191. The rest of the shift is unambiguously understood as the result of pion rescattering in the target nuclei. In Fig. 2 we show the spectra and angular distributions for ( 3He,3H7rf) to the ground state and lowestlying 2+ and l+ excitations of t*C. The transitions to excited states are weak and confirm that the widths in measured spectra and angular distributions come essentially from the ground state [ 28,291. All the calculations of Fig. 2 include conventional medium effects, and the dashed curves in the lower pannel contain in addition the a-contribution correction. Refinements in

the estimate of Eq. (7) can readily be made, but it is evident that the low-energy tail of the peak is sensitive to off-shell effects probed by the virtual pion. The measurements of Ref. [ 281 do not permit distinguishing differences of the order represented in Fig. 2a. The situation will soon change with high-resolution data presently analysed [29]. These new measurements will provide guidance for further improvements in theoretical models of the off-shell effects. The low-energy sensitivity depicted in Fig. 2a is hardly a surprise. The range of w-transfer from pion threshold to -240 MeV, mainly affected by the offshell correction, corresponds to pion kinetic energies up to -100 MeV. This is precisely the range over which the sensitive interference between the falling s and the rising p waves is known to play an important role, and often results in dramatic effects, in the context of pion-nucleus scattering [ 151. This is also the domain of highest confidence in models of the conventional medium effects, since a large volume of data is available, and one is still sufficiently below the A as to not be masked by its resonant behavior [ 161. It is consequently the region where off-shell effects, all the more pronounced at energies near pion threshold, are most likely to be identified. The effect of each important component of the pion-

MA. Kagarlis, V.F. Dmitrieu/Physics

Letters B 408 (1997) 12-18

17

pion scattering. We have shown that the lowering of the peak in the energy-transfer spectra is entirely explained by the rescattering of the pion in the nucleus. The low-energy tail of the peak is sensitive to the interference between the s and p waves and off-shell effects. In this sense, virtual pions may prove to be a trully new probe of the nuclear medium, over a kinematical region inaccessible to real pions. Upcoming data will provide guidance in further understanding the origin and importance of the off-shell effects.

w (MeV)

Fig. 3. The detailed behavior of the pion-nucleus tescattering amplitude, as explained in the text. The reaction and kinematics are as in Fig. 2.

nucleus optical potential is shown in Fig. 3. Without rescattering, as in (p,nr+) of Fig. 1, the peak is -15 MeV below the position of the A as measured in p(3He,t)A++ [ZO] - a minimal shift due to the cutoff of the form factor. Turning on the s-wave only, first with conventional corrections and subsequently adding the off-shell estimate, succesively lowers the peak. The nucleus becomes gradually more opaque as the optical potential becomes more repulsive, but no angular momentum is imparted and the position is not affected. On the contrary, the lowest-order p wave alone results in a dramatic lowering of the peak. The addition of conventional p-wave corrections affects mainly the high-energy wing of the peak, in the domain of A-resonance dominance (-300 MeV) . With s and p waves and conventional corrections present, the s wave cuts from the low-energy tail, effectively reducing the width and pushing the peak upwards in u. This is the result of destructive s and p-wave interference. Adding the off-shell correction pushes the peak further upwards, essentially by extending the range over which the s-wave strength is considerable and its interference with the p wave effective. For q* near the peak of the coherent-pion spectrum, the correction becomes comparable to the p-wave part of the amplitude with opposite sign. In summary, we have used a coupled-channel model with an explicit source term and a rescattering optical potential to facilitate the exploration of coherent pion production reactions as laboratories of virtual

This work has been supported in part by the International Science Foundation grant NQEtXKl (VFD), and by the European Union program “Human Capital and Mobility”contract ERBCHEJICT930258 (MAK) . One of us (MAK) thanks Eulogio Oset for illuminating discussions and criticism. We are both indebted to Carl Gaarde for his invaluable support at all the stages of this project. References [ 11T.E.O. Ericson, Nucl. Phys. A 560 ( 1993) 458. [21 E. Oset, P Femftndez de Cbrdoba, B. Lopez Alvaredo and M.J. Vicente-Vacas, Nucl. Phys. A 577 (1994) 255~. [ 31 P Oltmanns, E Osterfeld and T. Udagawa, Phys. Len. B 299 (1993) 194; T. Udagawa, F! Oltmanns, E Osterfeld and SW. Hong, Phys. Rev. C 49 (1994) 3162; B. Kiirfgen, E Osterfeld and T. Udagawa, Phys. Rev. C 50 (1994) 1637. [4] V.F. Dmitriev, Phys. Rev. C 48 (1993) 357. 151 I. Delorme and P.A.M. Guichon, Phys. Lett. B 263 (1991) 157. [61 E. Oset, E. Shiino and H. Toki, Phys. Lett. B 224 (1989) 249; F? Femandez de C&doba, J. Nieves, E. Oset and M.J. Vicente-Vacas, Phys. Lett. B 319 (1993) 416; P Fem&ndez de C6rdoba, E. Oset and M.J. Vicente-Vacas, Nucl. Phys. A 592 (1995) 472. [ 7 I For a review, see S.L. Adler and RF Dashen, Current algebra and applications to particle physics (Benjamin, New York, 1968). [81 S.A. Coon, M.D. Scadron,EC. McNamee, B.R. Barrett, D.W.E. Blattand B.H.J. McKellar, Nucl. Phys. A 317 (1979) 242, and references therein. [91 D.B. Kaplan and A.E. Nelson, Phys. Lett. B 175 (1986) 57; D.B. Kaplan and A.E. Nelson, Phys. Len. B 179 (1986) 409E. [ 101 G.E. Brown, V. Koch and M. Rho, Nucl. Phys. A 535 ( 1991) 701. [ 11I J. Delorme, M. Ericson and T.E.O. Ericson, Phys. Len B 291 (1992) 379.

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Letters B 408 (1997) 12-18

[20] C. Ellegaard et al., Phys. Lett. B 154 (1985) 110. [211 V. Thomson and A. Wirzba, Nucl. Phys. A 589 (1995) 633; J. Delorme, G. Chanfray and M. Ericson, Nucl. Phys. A 603 (1996) 239; M. Kirchbach and A. Wirzba, Nucl. Phys. A 604 (1996) 395. [22] L.S. Brown, W.J. Pardee and R.D. Peccei, Phys. Rev. D 4 (1971) 2801. [23] J. Gasser, M.E. Sainio and A. Svarc, Nucl. Phys. B 307 (1988) 779. [24] L.L. Salcedo, K. Holinde, E. Oset and C. Schtltz, Phys. Lett. B 353 1 (1995). [25 ] M. Ericson and T.E.O. Ericson, Ann. Phys. 36 323 ( 1966). [26] V. Bernard, N. Kaiser and Ulf-G. MeiBner, Int. Joum. of Mod. Phys. E 4 (1995) 193. 1271 In the limit of no off-diagonal pion-nucleus scattering our model reduces to the DWIA, and bears a conceptual similarity to the work of Ref. [ 61. [28] T. Hennino et al., Phys. Lett. B 283 (1992) 42. T. Hennino et al., Phys. Lett. B 303 (1993) 236. [ 291 SPES IV-r collaboration, coherent pion production dedicated experiments in progress. The expected energy resolution is ~4 MeV, as compared to ~25 MeV of Ref. [28].