Exclusive meson production in hadron-nucleus collisions

NUCLEAR PHYSICS ELSEVIER

A

Nuclear Physics A 592 (1995) 539-560

Exclusive meson production in hadron-nucleus collisions * B . K r i p p a 1, W . C a s s i n g , U . M o s e l lnstitut fu'r Theoretische Physik, Universitgt Giessen, D-35392 Giessen, Germany

Received 21 April 1995; revised 12 June 1995

Abstract The exclusive production of ~- and 7/ mesons in hadron-nucleus reactions is studied in the framework of the impulse approximation in momentum space using relativistic nucleon spinors. A comparison with the data on pion production for 800 and 489 MeV protons on 12C, 160 and l°B shows the relativistic effects to be predominant for reactions leading to the ground state as well as excited states of the residual nuclei. The exclusive production of low energy r/mesons in pion-nucleus interactions is investigated with particular emphasis on the N(1535) self-energy in the medium. Furthermore, first calculations for the reaction 12C(p,'t/)13Ng.s a r e presented for 0.8 and 1.0 GeV laboratory. The sensitivity of the calculated cross section to different limits for the NN~7 coupling constant and the role of one-step and two-step processes is studied.

1. Introduction The past twenty years have shown continuous interest in exclusive meson production in hadron-nucleus interactions. Since the residual nucleus is left in a specific bound state these processes require a large momentum transfer to the target nucleus up to a few times the Fermi momentum. The mechanism for the absorption of such high momenta by the target nucleus is the crucial point in A (p, m e s o n ) B reactions, which thus are expected to be sensitive to short-range a n d / o r collective phenomena. Until now the progress in this field has been almost exclusively related to A (p, ~r)B reactions and there exist a sizeable amount o f experimental data [ 1-4] for proton energies in the A(3,3) resonance region and below. A number o f theoretical models [ 5 - 9 ] have been developed to describe the * Supported by KFA Jiilich. J On leave from INR, Moscow. 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 5 ) 0 0 3 1 3 - 4

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B. Krippa et al./Nuclear Physics A 592 (1995)539-560

experimental data for these reactions. However, a general understanding of the reaction mechanism in a broader kinematical regime and for a variety of targets is still lacking. All models describing (p, 7r) reactions should include several basic ingredients: pion and nucleon distortions, the proper nuclear structure for the target and residual nucleus as well as a suitable ~rNN vertex. The traditional approaches have been based on the non-relativistic distorted wave Born approximation (DWBA) where the Hamiltonian for the 7rNN vertex is calculated using a non-relativistic reduction of the corresponding relativistic covariant interaction density. However, it is well known [ 10] that such a reduction is not unique and an ambiguity arises in the so-called non-static term. In a number of studies [5,6] it has been demonstrated that the results of the calculations depend to a significant extent on the choice of the parameter fo~ the non-static corrections. Moreover, for processes involving momentum transfers of more than 0.5 GeV]c the lower component of the target nucleon Dirac spinor is approximately of the same order of magnitude as the upper one and consequently both components of the nucleon spinor should be treated on the same footing. The difficulties of the non-relativistic approaches mentioned above indicate the necessity to retain the fully relativistic form of the ¢rNN vertex and to use Dirac spinors for all nucleon states, i.e. for the incoming proton as well as for the target nucleons. Such calculations have been performed within the "one-nucleon" model (ONM) [7] and "two-nucleon" model (TNM) [8] where the importance of relativistic effects has been demonstrated especially for incoming proton energies from threshold up to the delta resonance region. The new generation of proton accelerators like COSY or CELSIUS now will allow to study proton-induced exclusive pion production at much higher energies as compared to the present meson factories and relativistic effects should be even more im?ortant in the 1 GeV energy regime [9]. Apart from the more traditional (p, 7r) reaction, the new proton machines also allow to study the production of heavier mesons like r/'s. Nuclear interactions of the r/mesons have been the subject of several recent theoretical and experimental investigations. First, the Saclay measurements [ 11 ] of d(p, T1)3He close to threshold revealed reaction cross sections much larger than expected. Furthermore, Liu et al. [ 12] suggested that bound nuclear states with the r/ meson might exist. Meanwhile, a couple of calculations for differential r/ cross sections in few body systems [ 13,14] have been performed and it has been pointed out in Ref. [ 15] that the fiNN coupling constant might be extracted from the coherent process (p, pr/). Near threshold the r/ meson is produced predominantly through the formation and subsequent decay of the N*(1535) Sll resonance. Thus the observation of r/-meson production in nuclei can provide information on the N*-nucleus interaction, too. We note that, in principle, the properties of the N* resonance in nuclear matter can also be studied in high-energy pion-nucleus scattering. However, in this case the terms involving the N* resonance formation in the nuclear medium are only a small correction (compared to the A-excitation) whereas for the case of ~7-nucleus processes the formation and decay of the N* resonance is the dominant mechanism. Since the reactions like (p, cr) or (p,r/) are characterized by a large momentum

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

541

transfer, the corresponding nuclear formfactors will lead to a significant suppression of the production cross section in single-step reactions. However, the inclusion of intermediate processes - which by itself have a larger cross section - should give rise to an increase of the cross section since the inclusion of such "few-step" processes implies an effective "sharing" of the total momentum transfer in subsequent reaction steps. It is worth mentioning that such "few-step" contributions were found to be significant e.g. in the exclusive process 3He(Tr, r/)t [ 13] and even in inclusive reactions of the type A(p,K+)X,A(p,~7)X,A(p, to)X [16]. In this paper we will thus investigate quantitatively (a) the importance of relativistic effects in exclusive (p, ~-) reactions above the A resonance region, (b) the effects of pion and proton distortions and subsequent reaction channels, and (c) the exclusive ~7 production in pion-nucleus and proton-nucleus collisions with emphasis on the N(1535) self-energy and the NN~i coupling constant. Accordingly, our paper is organized as follows: In Section 2 we outline the formal aspects of our approach for the general case of exclusive meson production in proton-nucleus reactions and apply this formalism to the process of pion production. In Section 3 we discuss the results of our calculation for the (p, rr) reaction in comparison to the available experimental data, which provides a good test for the approximations performed in our approach. In Section 4 we then present the respective results for the (p,7/) reaction and briefly discuss the closely related process of exclusive pion-induced r/ production on nuclei. The summary is finally presented in Section 5.

2. General T-matrix formalism

We start from the Lippman-Schwinger equation for the T-matrix which in operator form can briefly be written as T = V + TGoV.

(1)

For the T-matrix describing the production of a meson (a) in a proton-nucleus interaction we get

T pa = V pa + ~ T b a G ~ V v° + Tp'aGg'Vp'p,

(2)

b

where the first term in Eq. (2) describes the direct "one-step" production of the final meson (a) by the incoming proton without any intermediate processes. The second term in Eq. (2) takes into account the formation of an intermediate meson (b), its propagation in the medium, and the production of the final meson (a) in a subsequent reaction of the meson (b). The third term in Eq. (2) describes the production of the final meson (a) by a proton which is distorted in the initial channel. The sum over b in Eq. (2) implies a summation over different isospin states as well as different intermediate meson species (~r or r/). Eq. (2) can be rewritten as

542

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560 T pa -~ v P a ( 1 - GgVPP) -1 "~ ~ - - ~ T b a a b v P b ( 1 -- GPVPP) -1

(3)

b

by including the distortions in the propagator. Now considering the matrix element (a, flTP~lp, i) and using the relation for the distorted wavefunction [p+),

]p+) = ( 1 --

GgV pp) - 1 [p),

(4)

we arrive at the following equation in momentum space

(a, flTpalp, i) = (a, flVP~lp +, i) + ~b /

~ d3q {a, fiT]b ' n'q)GO°'n(q'E)(b'n'qlVPblp+'i)'

(5)

where [p) ([a)) are the incoming proton (outgoing meson) plane waves. The distortion of the incoming proton wavefunction is thus included in the respective matrix element. Furthermore, we denote by n the intermediate nuclear state which is the ground state or of lp-lh, 2p-2h structure. The second term in (5) takes into account the rescattering of the produced intermediate meson b with momentum q where the sum includes intermediate meson states b and nuclear states n, respectively. We note again that the sum over b in Eq. (5) implies the summation over both the different isospin states of the meson as well as different intermediate mesons (~r or r/). In the case of pion production, neglecting the contributions from intermediate excited nuclear states, Eq. (5) can be written as

TP~r(p, k) = VP~(p, k) -~-(2zr)----~

~

d3q T~J(k,q) G~(E(k),q) VP~'(p,q),

(6)

isospin (j)

where p and k are the momenta of the initial proton and final pion, respectively. Here, q is the momentum of the intermediate pion in an isospin state j, T '~j is the operator describing the pion-nucleus interaction in the final state and G~ is the pion propagator of isospin j. According to the commonly accepted terminology we attribute the first term in Eq. (6) to the one-nucleon mechanism (ONM) whereas the second term includes meson distortions as well as the two-nucleon mechanism (TNM) of pion production since this term can also be used to calculate the process of or- production (double charge exchange) where the ONM fails by definition. We note that the effect of pion-nucleus rescattering can also be taken into account via a pion self-energy £ ~ ( p , g~), which is related to the pion-nucleus T-matrix by

T~r'rGo = wr~r( E - H - ~r(p, gt) ) -1, where p denotes the nuclear density and g~ stands for the Migdal parameter. This expression has been used in Ref. [9] for a detailed study of the exclusive (p, zr) reaction to investigate the sensitivity of the differential pion cross section on gt, In particular it was assumed in Ref. [9] that 2?~ and thus pion rescattering can be calculated via the

B, Krippa et al./Nuclear Physics A 592 (1995) 539-560

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excitation and decay of a/1(3, 3) resonance in the nuclear medium. Here instead we use the T-matrix and consequently the bare pion propagator to describe the pion-nucleus rescattering including not only the p-wave (for the A excitation) but also higher partial waves. The explicit form of the first term in Eq. (6) is given by

V t'~r = (¢r, flHNN~.Ip +, i),

(7)

where HNu~ is the covariant interaction Hamiltonian with pseudovector coupling

HN1V~r = f ~r~U'YSy#~'O#Orr~.

(8)

mTr

In Eq. (8) f ~ is the coupling constant for the ~rNN vertex taken as f~ = 1.008. The wave functions for the nucleons are obtained by solving a Dirac equation with scalar and vector potentials as described in detail in Ref. [9]. We note that the Ip +, i) state includes the wave function of the target and the proton distorted wave whereas the (Tr, fl state is the wave function for the residual nucleus and the pion plane wave. Using the partial wave representations for the distorted proton wave 7~,,t'(r) = Z

( fLs(r)Y~j(S2,), gL, j(r)Y~,j(12r) )~c

JLMm

×(lml/21zlJM) YLM(J2p)(i) L,

(9)

the bound nucleon wave function

~tj,,M,(r ) = ( F B ( r ) Yt;,i/2JB MB (g2r), _iGB(r)3)M, nBI/2jB(~r) )* Yo,

( 1O)

as well as for the pion plane wave

Ok(r) = ~(--i)lYl*m( Or)Ylm( O~)jt( kr),

(1l)

Im and carrying out the lengthy, but straightforward angular algebra one can represent the first term in Eq. (6) as

Vt'~ = Z

(i)g-l(-1)n"+l/2(tOl/21~[Jtz)

JLlm mTr ×(Jn - MnJlzllm)(Jnl/2J-

LJ:~ 1/21/0) ~m(/2k)

x (11P(LB, l, L') + I2P(L~, l, L) ), where P(Ln,I,L') = ½(1 + (--l)t~+l+L'); L' = 2 J and

11

f dr r2gLj(pr)Fn(r)(J(r)jl(kr), J

~. (12)

L, L~ = 2J~ - Ln, ) = x/2j + 1,

(13)

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B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

12 = f drr2 fLj(pr)U(r)Gb(r)jt(kr).

(14)

In Eqs. (13) and (14) we have furthermore adopted the notation 0 = 2 M + V[ + V~n + V.n - V.p and U = 2M + ~P + Vs~ - V.~ + V.p where V~.) and ~ . ) are the scalar (vector) components of the corresponding relativistic nucleon-nucleus mean field potential and fLJ(gr'J), FB(GB), are the upper (lower) components of the Dirac spinors for the scattering and bound nucleon wave functions, respectively. By integrating over the angular variables within the partial wave expansion we get the expression for the partial components TLLp of the amplitude Tp~

T~,,Lp(k,P) = V~,,G(k,P)+ Z

dqqZ T'iL~(k,q)Go(E,r(k ' q) V,L~Lp(q,P) j (15) (2~-)3 J.

where we have omitted the isospin index of the intermediate pionic state. The partial amplitude TiLe can now be related to the squared matrix element of the (p, 7r) reaction as

spins

ILtpE,rr:t) (_l)g+L.YLo(f~lr) I(k, fITP~[p, i) l2 = ~-'~TJ/T~L~'I2,rLp--*JL(47r) 3/2 ×U(LL~pjl/2; Lpf)U(LL~fl/2; L~j) (L~OL,rOILO) (L~OLpOILO).

(16)

Here U(~ilk; mn) is the standard Racah coefficient while ~ represents an average over the initial spins and summation over the final spins. The bare pion propagator in (15) can be written as the sum of off- and on-shell terms:

1

Go(E(k), q) = faE(q) _ E(k)

iTrS(E(q) - E(k)

),

(17)

where ga denotes the principle value. The off-shell behavior of T ~" is quite a matter of debate and cannot be evaluated without explicit assumptions about form factors, cut-off parameters, etc. In view of the energy regime of about 1 GeV we will neglect the contribution of the off-shell part in the following because the pion-nucleus dynamics in (6) to a large extent is governed by on-shell pions of high energy. Consequently, we expect to obtain a lower bound to the exclusive pion yield which will be compared to the available data (cf. Section 3). Furthermore, we use a Glauber model [ 17] to describe the effects of pion rescattering with the pion-nucleus amplitude given by

(Tr', fIFG( q) [Tr,i) = ~iPacfl d2bexp(iqb)(Tr'' f[ ]-I J

[1 - F ( b -

sj) ] - l lTr,i) ,

j=l ,A

(18) with 1 fd2kexp(_ikb)F=u(k, F( b ) = 2~rip2-------~

E2c).

(19)

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

545

Here P2c and Pac are the pion momentum in the pion-nucleon and pion-nucleus centerof-mass systems, respectively, and F,~N is the amplitude for pion-nucleon scattering. The amplitude F~, furthermore, is related to the T ~ matrix in Eq. (6) via a trivial kinematical factor, T = -27"rF/(tz(Q)~(Q~)) 1/2, w h e r e / z ( Q ) a n d / z ( Q ) ' are the reduced masses in the initial and final states, respectively. Since the cross section of high-energy pion-nucleus scattering drops very fast with growing scattering angle the contribution of the second term in Eq. (6) is basically determined by the contribution from pions which are rescattered in forward direction; in this special geometry the Glauber approximation is expected to be valid. The differential cross section for the (p, 7r) reaction is finally given by

do"

1

d---~ = (47r)2

MEAEBk ~-~ Etotp~ Z..,,. ITP~'I2, spins

(20)

where Ea, En, Etot are the energy of the target, final nucleus and total energy, respectively, and M is the nucleon mass.

3. Results for exclusive pion production Now we turn to the results of our calculations for the 12C(p, 7r) 13C reaction employing the spinors from Ref. [9]. The latter are fixed by changing the depths of the scalar and vector part of the mean field potentials for the bound-state nucleons in order to reproduce the binding energy and momentum distribution of the nucleons in the ground state of 13C (cf. Ref. [9] ). In this way we reproduce the experimental momentum distribution up to about 600 MeV/c (cf. Ref. [9] ) which appears sufficient since the typical momentum transfer for the (p, 7r) reaction in the energy region above the A resonance is in the range of 500-700 MeV/c. Taking into account the effect of the momentum "sharing" between different subprocesses we can expect that the relativistic nucleon wave functions should provide a realistic description of the target momentum distribution. We note that we restrict to the mean-field level for the description of the final 13C nucleus which is supposed to be given by an inert core plus an additional valence neutron. As a consequence we underestimate the momentum distribution above 600 MeV/c where two-body correlation effects more likely are responsible for the high momentum tails. For a more detailed discussion of the role of the nucleon relativistic spinors in the (p, 7r) reaction we refer the reader to Ref. [9]. To obtain the relativistic spinors for the distorted proton wave Ip +) we fitted the pA elastic scattering data by searching the proper set of parameters for the proton-nucleus relativistic optical potential; the spinors are always computed by integrating the Dirac equation with the corresponding optical potential. In Fig. 1 the calculated cross section for the exclusive reaction 12C(p, 7r)13Cg~ is shown for 800 MeV initial kinetic energy in comparison to the experimental data from Ref. [ 18]. The solid curve is the result of the full calculations according to Eq. (6) taking into account all hadron distortions; the dotted curve results from the first term

546

B. Krippa et aL /Nuclear Physics A 592 (1995) 539-560

, . -

~

,

,

,

.

,

.

120 p ,Tt+) 130

-,r__~

.Q

TLab= 800 MeV ".... "", I

I

I

I

I

600

650

700

750

800

850

qc.m (MeV/c) Fig. 1. Differential cross section for ~'+ production in the reaction 12C(p, 7r+) 13Cg.s. versus the momentum transfer in the C.M. system at 800 MeV laboratory energy. The experimental data are taken from Ref. [ 18 ]. The full line represents the calculations including all hadron distortions, the dotted line represents the result of a calculation with only the first term of F__xl.(6) and the dashed line corresponds to the calculations including the effect of pion rescattering.

in Eq. (6) and the dashed curve corresponds to the calculations where the effect of pion distortions is not included. We observe that pion-nucleus rescattering significantly improves the description of the experimental data. On the contrary, taking into account the effect of proton distortions does not lead to any significant changes of the results. This is quite natural since the incoming proton wave function corresponds to a fast proton which is only moderately affected by the nuclear mean field. The deviation of the results of our calculations from the experimental data in the region of the larger momentum transfer is presumably due to the neglect of the off-shell term in the pion propagator (17) as well as to the combined contribution of the target emission diagrams [8,9], where the pion is ejected by a target nucleon, and those from the p-meson exchange diagrams which are not yet included in the present study. We note that the use of relativistic spinors is extremely important for the successful description of the data since the upper and lower components of the nucleon wave functions get mixed and give equally important contributions. Whereas the use of only the first term in Eq. (6) already gives a qualitatively reasonable description of the data, the use of non-relativistic wave functions in the same approximation fails dramatically even employing different kinds of non-relativistic states (cf. Ref. [5] ). This difference is quite remarkable in the context of the general problem concerning relativistic effects in modern nuclear physics. In reactions with relatively low momentum transfer, e.g. elastic proton-nucleus scattering, the relativistic effects were found to be important for subtle spin observables, only. Moreover, even this conclusion is still controversial since some spin observables can be quite well described in the framework of a non-relativistic theory provided that exact averaging over the target Fermi momenta is performed [ 19].

B. Krippa et aL /Nuclear Physics A 592 (1995) 539-560

I

I

I

+~ ~-

I

547

I

160(p,~+)170

10.

.Q

.c. "O

Tkab= 800 MeV I

I

I

600

650

700

750

800

850

qc.m (MeV/c) Fig. 2. Same as Fig. 1 for the reaction 160(p,~+)17Og.s .,

However, for reactions with large momentum transfer - as considered here - the use of relativistic spinors seems to be a necessary step to get a reasonable description of differential cross sections. The latter conclusion is not restricted to the 12C (p, ~-+)13C reaction since a quantitatively similar picture is also obtained for the reaction 16(p, Tr+)17Og.s" at 800 MeV as shown in Fig. 2. Also in case of the 17Qg.s. final state the picture of an "inert core plus valence neutron" should be justified. Now we turn to the results of calculations for the t°B target. In this case the residual nucleus can to a lesser extent be treated in the "core plus nucleon" framework since effects from configuration mixing should become more important. We note that including the configuration mixing effect calls for full relativistic RPA type calculations for the nuclear ,;tructure problem which is beyond our present approach. However, already within the simple "core plus nucleon" model we can get a reasonable description - at least in forward direction - for the reaction l°B (p, 7r+)llB as shown in Fig. 3. In order to explore the validity of our present approach at lower bombarding energy we show in Figs. 4 and 5 the calculated cross sections for 489 MeV laboratory energy on 12C and 160 targets in comparison to the available data. Since qualitatively the mechanism of the (p, ~-) reaction remains similar to that at 800 MeV we display in Figs. 4 and 5 only the results of the full calculations. However, the experimental data here are underestimated by roughly a factor of 2 which might be due to the neglect of the off-shell terms in the pion propagator (17), which become more important at lower bombarding energy. We now turn to the pion production processes where the residual nucleus is left in an exited state. In this case the "core plus nucleon" picture should also be less adequate.

548

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

100

I

I

10B(p,rt+)11B I

I

.O t'"O 10-

"6 "O v

~V~:

Tkab= 800 Me 1

550

I

I

I

I

600

650

700

750

800

qc.m (MeV/c) Fig. 3. Same as Fig. 1 for the reaction l°B(p, Tr+)llBg.s.

I

I

I

I

I

I

I

I

12C(p,lt÷)13C 100-

I..,.

..Q t"O

"6

TLab=489 MeV I

0,52

I

I

0,50 0,48

I

I

I

0,46 0,44 0,42

I

0,40

I

0,38

0,36

t (GeV2/c 2)

Fig. 4. Differential cross section for the 12C(p, zr+) 13Cg.s. reaction in comparison to data from Ref. [ 18] as a function of the C.M. angle of the outgoing pion at 489 MeV bombarding energy.

549

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

100

I

I

I

I

.Q c-

-o10v

TLab=489 MeV 1

0,7

|

0,6

I

I

0,5

0,4 t

I

I

0,3

0,2

0,1

(GeV2/c2)

Fig. 5. Same as Fig. 4 for the reaction

160(p, Tr+)lTOg.s..

However, since we incorporate the most important detail of the reactions with large momentum transfer, i.e. the relativistic structure of the nucleon spinors, one can hope to get at least a qualitative description of the available experimental data. In Fig. 6 we show the calculated cross section for 800 MeV initial kinetic energy of the proton with the residual 13C nucleus left in 1/2 + state of 3.09 MeV excitation energy. The corresponding experimental data [ 18] show an almost isotropic angular distribution which indicates the importance of configuration mixing effects since the single-particle wave functions lead to a typical (approximately exponential) fall off irrespective of the potential used. To compensate this exponential decay a configuration mixing has to be taken into account which, however, is beyond the scope of this paper. Our calculations in the framework of the "core plus nucleon" model for the residual nucleus thus only give the right order of magnitude of the differential cross section while the quantitative description of the data fails. In Fig. 7 we show additionally the results for the 160(p,'n'+)170~.87 MeV reaction where the ~70 is left with the excitation energy of 0.87 MeV. In this case the angular distribution is more likely of single-particle nature and consequently our calculations reproduce the magnitude and shape of the angular distribution to a greater extent. In this context we mention two additional problems that seem to be hard to resolve at present. First, for the case of transitions with the final nucleus left in its ground state we could check the reliability of our spinors by calculating the corresponding momentum distribution and comparing to experimental data. However, for transitions to some

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

550

I

I

I

1

I

I

2C(p ,rt ) 1 3 0 +

*

3.09

" ~ " t 0.

t'-

'e v

TLab = 800 MeV I

I

I

I

I

600

650

700

750

800

850

qc.m (MeV/c) Fig. 6. Differential cross section for the reaction ]2C(p, ~r+) 1303.09 MeV as the function of the momentum transfer at the laboratory energy of 800 MeV. The experimental data are taken from Ref. [ 18].

excited state of the residual nucleus this is no longer possible since the corresponding experimental data do not exist. Second, there are no systematic experimental data on inelastic high-energy pion scattering on nuclei which makes it rather difficult to check the validity of the Glauber approximation adopted for the intermediate pion propagation in Eq. (6). 10":

(D

I

'

I

'

16

I

=

I

'

17

I

'

) 0 0.89:

:

1

c-

"~

0,1 ¸

Tt_ab= 8 0 0 M e V 0,01

I

I

I

1

I

600

650

700

750

800

850

qc.m (MeV/c) Fig. 7. Same as Fig. 6 for the reaction 160(p,'¢/'+)1700.89 MeV.

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

551

4. P r o d u c t i o n o f r / m e s o n s in e x c l u s i v e pA a n d ~rA reactions

We now turn to the production of r/mesons in exclusive hadron-nucleus interactions. For the case of r/-meson production in a pA reaction Eq. (5) can be written in the following form:

d3q T ~ ( k , q ) G o ( E ~ , q ) W ' ~ ( p , q )

TF°(p,k) =VP°(p,k) + ~

1

q - ~

P

J d3kl V ~ n ( k , k ' ) G n ( E ( k ) , k')VP~(p,k ') ,

(21)

where the first term has the same meaning as in Eq. (6) for the case of pion production. The NNrl vertex is given by [20]

HNNo = --ig,l~NYS~Nqb,7 •

(22)

Unfortunately, the value of the coupling constant g,7 is quite a matter of debate. For example, the analysis of the N N interaction [21] leads to g2~/41r ..~ 5 while the analysis of photoproduction data [22] suggests g~/47r ~ 0.4. The different values for the coupling gn are highly model dependent, depending on the number of secondary mesons employed [23] and their coupling to the nucleon as well as to the N*(1535). In some boson exchange models [24,25] - describing the NN ---* NNrl process - the inclusion of the p meson exchange diagrams leads to a significant increase of the total cross section. On the other hand, the boson exchange analysis in Ref. [23] found only a minor contribution from the p meson for the process pp -~ pprl and almost no contribution for pn ~ pnrl. Thus, at the present stage of knowledge, we discard explicit contributions from vector-meson exchange diagrams and assume their contribution to be included in a proper redefinition of g,r In our calculations of ~7 production on nuclei we will thus use both extreme cases to check the sensitivity of the results obtained to the different (model dependent) choices of g,. While the first term of Eq. (21) corresponds to the mechanism where the outgoing r/ meson is directly emitted by the incoming proton, the second term describes the two-step process where the r/production proceeds via the formation of an intermediate pion in the first proton-nucleus interaction. The third term represents the process where the produced r/ is rescattered or reabsorbed in the final state. We describe this effect by using the full r/-nucleus propagator

G O ( E ( k ) , k ') =

1

E ( k ) - E(k') - Vnn(k, k ' ) '

(23)

where Vno is the r/-nucleus optical potential which will be defined below. The r/-nucleus interaction proceeds basically via formation and subsequent decay of N*N -j pairs in the nuclear medium [26]. The imaginary part of the self-energy of the N* (1535) resonance - discussed below for the case of the (7r,~7) reaction - can be calculated within the framework of standard many-body theory using the known values of the free width and coupling constants (cf. Ref. [26] ) whereas the corresponding real part - due to lack of

B. Krippa et al./Nuclear PhysicsA 592 (1995) 539-560

552

information on the elastic N*N scattering - can only be parametrized in some way. In our calculations of the (p, r/) reaction the real part of the N*(1535) self-energy thus will be varied from +50 to - 5 0 MeV as in the work of Chiang et al. [26]. As is seen from Eq. (21) for the (p,r/) reaction one also needs to know the Tmatrix TO~" describing the process of pion-induced r/-meson production for secondary and higher order production steps. Here we use an extended version of the approach developed in Ref. [ 14] starting from the Lippmann-Schwinger equation for T °'r,

T°'r = vwr +

E

V~'G~PV~' ~r +

71~

lTrl'rtl'7"r~"P'rP~" - "o "

7rt

v Pc":P PG

+ S_, rl p

E

(24)

"rP

s

t

¢

In Eq. (24) we have used the relation TW0'G~ = W '7 G '7 , where G ~' is the full r/nucleus Green function. The sum in Eq. (24) includes all possible intermediate mesons (with different isospin states). We note that in general one should include the summation over all possible intermediate nuclear states, too. The first term in Eq. (24) corresponds to the mechanism of pion-induced r/production without distortions of the pion in the initial and the r/meson in the final state; the second and third terms include the effects of pion and r/-meson distortions, respectively, and the fourth term takes into account the distortions of both pion and r/meson simultaneously. In momentum space Eq. (24) reads as follows:

T'~(P'k)

=

,

V'7~(P'k) + (2zr)------~E

/

d3q V'7"r(q'k)Go(E'r(k)'q)T'r~r(P'q)

1 / d 3q V~n (p,q) G,7(En (p),q)V~'r(q,k) + (--~-~)3 +E

f d3qd3qt(2cr)--'----g-Vnn(P, q)Gn(E,7(P) , q)V'r'7(q, ql )Go(E'r(k), qz )Trr'r(ql ' k) ,

(25) where k and p are the momenta of the initial pion and final r/meson, respectively. The momenta q, ql correspond to the intermediate mesons; the sum in the second and fourth term of (25) runs over the different isospin states of the intermediate pion while T "r~ is the operator describing pion rescattering and Go is the pion propagator. The differential cross section for the (zr, r/) reaction then is given by =

dO

(47r) 2

Et2otP

jTO'rl2.

26)

Here EA, EB and Etot are the energy of the target, final nucleus and total energy, respectively, and ~ represents an average over the initial spins and sum over final spins. Introducing the partial wave expansion for Eq. (25) one can get the expression for the partial amplitudes of the (zr, r/) reaction as

B. Krippaet al./NuclearPhysicsA 592 (1995)539-560

553

- l E f dq q2Vj~"~(p,q)Go(E~(k), q)T~jt'~(q, k) Tit,7,r(p, k) = Vjtn~r(P, k) + (27r)3 + ~ +z

1

f dqq2Vj~n(p, q)GO(E,~(p), q)VtTn(q, k) f ~dqdqlq2q~ Vjntn(q,k)Gn(E(p), q)VtTn(q, ql )Go(E,r(k), ql )Tfffl~r(ql,k) (27)

After a quite lengthy but straightforward algebra we finally get for ~ ITm~l~

Z

LTO~[z= E

L ) £ ' ] / ( - ) c ' + c I % ( c o s 0,7)F(l, L, L', j,j', Ji, Jr) TLjT~,j, ~,(47r)3/2

F ( l, L, L', j, j', Ji, Jj) = ( LOL'OllO)2U(IL'jJi; Lj')U( ILj' Jy; L'j),

(28)

where U(abcd; if) is the standard Racah coefficient, cos0 n is the scattering angle of the outgoing r/meson, and L = (2L + 1) l / / f o r abbreviation. The first term of Eq. (25) is related to the t-matrix of the elementary process of r/ production in the rrN reaction as Vn~ = (B;

rlltia; 7r),

(29)

where A and B are the target and residual nucleus, respectively. To calculate the first term of Eq. (25) we have made a factorization approximation in which the elementary amplitude is calculated at an effective Fermi momentum (p) = - [ 2 k + (A - 1 ) Q ] / 2 A , where A is the nuclear mass number and Q = k - p is the momentum transfer. The details of the calculations for the matrix element (29) are given in Ref. [ 14]. We have used the elementary amplitude which - in the case for an interaction on the free nucleon - coincides with the amplitude proposed by Bhalerao and Liu [27], i.e.

t(q,q t, W)

-

g,Tgrr (2W)l/2

W

-

g( q') g( q) Mu. + iFlv. + Eu*"

(30)

In Eq. (30) g,~ = 1.301, g,z = 0.769 are the coupling constants for the ¢rNN* and N*NrI vertices, respectively; g(p) = A2/(AZ+p 2) with A = 1.2 GeV; W is the invariant energy, MN. = 1535 MeV and FN. = 150 MeV are the free mass and width of the N*(1535) resonance. Finally, 2N* represents the self-energy of the N*(1535) resonance in nuclear matter; its real part describes the N* mean field or mass shift while its imaginary part accounts for the modification of the N* lifetime at finite density due to real and virtual scattering processes [26]. The actual determination of EN- will he discussed below. We treat initial and final nuclear states in the framework of the single-particle shell model as before. The typical momentum transfer for the (Tr, 77) reaction is about 250300 MeV/c. We note that in the matrix element V'7." the upper and lower components of the target nucleon spinors do not get mixed because the interaction does not contain a Y5 matrix. Therefore, the use of non-relativistic wave functions is quite well justified for the (rr, r/) reaction, contrary to the ( p , ~ ) and (p,r/) processes where a matrix Y5

554

B. Krippa et al,/Nuclear Physics A 592 (1995) 539-560

must be included in the interaction density due to the pseudoscalar nature of the mesons produced. As mentioned above, in order to calculate ReXN* one should know the N * N elastic scattering amplitude which, however, is not known at present. We thus carry out the calculations for the different limits Re£N, = 0, 50 and - 5 0 MeV as in Ref. [26] and discuss the sensitivity of the results to the value of R e X N , . The method of calculating I m X N , is described in detail in Ref. [26] and also employed in our previous work [ 14]; we here only briefly discuss the main idea of this technique, i.e. the free width of the N* resonance is modified due to nuclear interactions of the resonance of particle-hole type in the nuclear medium. The contributions from these terms are evaluated using the Cutkosky rules [28[ and by a calculation of the Lindhard functions within the Fermi-gas model. Furthermore, as in case of the (p, 7r) reaction, we neglect the off-shell part of the pion propagator and use a Glauber model to include the effect of pion rescattering (cf. Ref. [ 14]). We again describe the r/ rescattering in the final state by using the full B-nucleus propagator defined by Eq. (23). Here the B-nucleus optical potential can be related to the corresponding polarization operator via H " = 2 w V "~ where H " is calculated along the model suggested in Ref. [22] as H n = g2po [ W - M ~ , + i F ~ , / 2 - i Im SN- + Re£N -- Re£N, ] - 1.

(31)

In Eq. (31) P0 is the normal nuclear matter density (P0 = 0.17fm-3), W is the corresponding invariant energy and ReXN is the nucleon self-energy which is approximated by R e i n = --50 MeV. Instead of using the full B-nucleus Green function one can alternatively use a Tmatrix in the B-nucleus channel to describe the effects of r/rescattering. However, it is easier from a technical point of view to use the r/-nucleus Green function since here the pole in the propagator is shifted in the complex energy plane and the corresponding integral no longer exhibits a singularity in the physical region. We note that the r/meson - contrary to the pion case discussed above - is of low energy such that the on-shell approximation for the corresponding propagator (cf. the approximation of Eq. (17)) is no longer valid. In Fig. 8 we show the cross section for the reaction 13C(~'+, 7/)13Ng.s' for the initial pion kinetic energy of 490 MeV where ReXN, was assumed to be - 5 0 MeV. The cross section in i x b / s r is shown as a function of the ~/ C.M. angle. The dotted curve in Fig. 8 corresponds to the calculations where only the first term of Eq. (25) is included, the dashed curve is the result of a calculation including the first and second term of Eq. (25) and the solid curve represents the results of the full calculation including both pion and B-meson rescattering. One can see that both pion and r/meson distortions are quite important and give non-negligible contributions. We note that r/ and pion rescattering somewhat compensate each other, but the net effect is still quite significant. The overall cross section falls off rapidly with increasing momentum transfer reflecting the natural decrease of the nuclear formfactor.

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

~

1

'

I

'

I

'

I

'

555

I

130(~,q)13N

"C" "--. 0,1

..{3 "0

0,01 full calculations ........

:.~,

1st and 2nd terms of eq.(24)".. ', .

. . . . . . . . . . . . 1st term of eq.(24) 0,001

i 0

,

L 10

,

, 20

,

i 30

Oc.rn.(deg)

,

x

"... _", L

40

,,"l 50

Fig. 8. Differential cross section for ~7-meson production in the reaction 13C(,/r, l'/)13Ng.s, versus the C.M. angle of the outgoing r/meson for 490 MeV initial pion kinetic energy. The full line is the result of calculations including all terms of Eq. (24), the dotted line is the result of calculations with only the first term of Eq. (24) and the dashed line corresponds to the calculations including the first and second terms of Eq. (24).

In Fig. 9 we, furthermore, compare the results of the full calculations for different N* self-energies: Re£N, = 50 MeV (solid curve), R e 2 m = 0 MeV (dashed curve) and R e a m = - 5 0 MeV (dotted curve) for an initial pion kinetic energy of 490 MeV. From Fig. 9 one can see a significant difference between the results of calculations with different values of R e i N , ; this is quite natural since we calculate in the vicinity of the N* resonance pole where small changes of the pole position can lead to significant modifications of observables. In Fig. 10 we furthermore show the cross section for the reaction 13C(zr+,*7)13Ng.s. at an incident pion kinetic energy of 520 MeV. The solid, dashed and dotted curves have the same meaning as in Fig. 8. From Fig. 10 one can see that as in case of the lower kinetic energy of the pions the calculated results are quite sensitive to the value of ReEN*. We observe that the cross section has a maximum roughly in the forward direction. The actual values should be large enough to be measured at present machines with high-energy pion beams (e.g. at KEK). In general (from Figs. 8-10) we find that the (,r,r/) reaction at threshold is a useful tool to provide information on R e E m which is practically unknown at present. In order to draw more definite conclusions one needs to extend our present model, e.g. by including a full microscopic treatment of the propagation of the N*(1535) resonance

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

556

C ('/l:,q) 13N

~-~ 0,1 "0

0,01 .

.

.

.

.

Rez N.=0MeV .

.

.

.

.

.

.......

",.

R e Z N, = 5 0 M e V i

0

',

ReZ;~. = - 5 0 M e V

.

I

20

"'- J

ec.m.(deg)

I

40

60

Fig. 9. Differential cross section as a function of the C.M. angle of the outgoing ~7 meson at the bombarding energy of 490 MeV for different values of R e £ ~ , (1535). Solid, dashed and dotted curves correspond to the results of calculations with Re2;N* ( 1535)=0, 50 and - 5 0 MeV, respectively.

in the nuclear medium while simultaneously taking into account high order corrections to the width and mass of this resonance. On the other hand, experimental data on the (zr, r/) reaction in the threshold region would be very desirable, too. We now turn to the production of r/mesons in the 12C(p, r/) 13N reaction. The details of the calculations are practically the same as those for the (p, 7r) reaction which could be shown to provide a reasonable reproduction of the experimental data at 800 MeV for ~2C and 160 targets. So we discard the lengthy presentation of more technical formulas and directly go over to the presentation of the results. The solid curves in Fig. I la,b are the results of the full calculations at 1 GeV (and 0.8 GeV) proton energy using the extreme values for the coupling constant g2~/4zr = 5 (a) and 0.4 (b), respectively. The dashed curves in Fig. 11 show the results of our calculations using only the first term in Eq. (21), i.e. the direct production process. From Fig. l l a one can see that the main contribution stems from the first term of (21); the second term gives sizeable corrections whereas the contribution of the third term is very small. However, in the other extreme case g2n/4~- = 0.4 the role of one- and two-step reactions is opposite (cf. Fig. 1 lb). Due to the weak coupling of the r/ meson to the nucleon the production essentially proceeds via secondary pion-nucleon inelastic scattering.

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

10

I-

I

~..

130('n:,q) 13N

I

557

I

1

o) ..Q '-O 0,1

0,01

~NN~

=_.050MeMVev

....... 0,001

i

0

,

"'",~

',

Rer.N.= 50 MeV t

20

,

i

40

,

i

60

ec.rn.(deg)

,

i

80

'l

100

Fig. 10. Same as Fig. 9 for 520 MeV kinetic energy of the incoming pion.

The cross section is quite low even in forward direction for g2n/47r = 5. Of course, definite conclusions about the accessibility of the corresponding experiment can be drawn only in the context of the concrete experimental set-up. It might be more informative to study exclusive ~/production with light nuclei like B or Li. In this case one can hope to obtain a larger cross section due to a smaller momentum transfer. The results obtained in Ref. [29] indicate that in this case reactions on nuclei are even more model dependent due to the medium modifications for the corresponding propagators and vertices. When decreasing the energy of the incoming proton to 800 MeV the cross section drops dramatically as shown in Fig. lla. Close to threshold at Tp = 610 MeV the differential cross section finally decreases to about 0.4 pb/sr even in case of g2n/4¢r = 5. We note that the results obtained for the (p, 7) reaction are not sensitive to the value of Re2/v. (for g2n/4~r = 5); the cross section in forward direction increases only by about 5-7% when changing ReEu. from - 5 0 to zero.

5. Summary and conclusion In this paper we have developed a model in momentum space to describe the exclusive production of pions and ~/ mesons in proton-nucleus collisions in the energy region above the A resonance while accounting for all kinds of hadron distortions. The

558

B. Krippa et aL /Nuclear Physics A 592 (1995) 539-560

a)

"-..

.c 0,1

TLab= 0 750

.

800

8

~

i

=

850

900

1000

950

qc.m (MeV/c) b)

•

i

,

~

'

i

,

i

,

0,01 U)

t"o

"6 "1o •

TLab=lGeV

0,001 i

750

8OO

850

i

J

900

950

1000

qc.re (MeV/c) Fig. 11. (a) Differential cross section for the reaction 12C(p,*l)13Ng.s.. The solid curves correspond to the full calculations of Eq. (21) and the dashed curves represent the results with only the first term of Eq. (21) for bombarding energies of 1.0 and 0.8 GeV, respectively. The *l-nucleon coupling g2~/4¢r = 5 was used. (b) Saree as in (a) but for the value gg/4~r = 0.4. nucleon wave functions for the bound and scattering states have been obtained by solving the Dirac equation with corresponding scalar and vector potentials, whereas pion distortions have been treated in the framework o f the Glauber approximation since the pions appearing in the amplitudes are o f high energy. The relativistic nature o f the nucleon wave functions turns out to be very important as the typical m o m e n t u m transfer involved in the (p, or) and (p, ,1) reactions is in a region where the lower component o f the nucleon bound state spinor can no longer be viewed as a negligible correction. We have calculated the differential cross sections for l°B, 12C and 160 targets at bombarding energies o f 800 and 489 M e V with the residual nuclei

B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

559

left both in the ground and exited states. A reasonable agreement between the results of our calculations and the experimental data at 800 MeV has been obtained for pion production with the residual nuclei in their ground states. The effect of pion distortions were found to be important and to provide a significant improvement over calculations with only pion plane waves. This finding is in agreement with the analysis of the (p, ~) reaction in Ref, [9] where the pion-nucleus rescattering has been incorporated via a pion self-energy. Furthermore, taking into account the proton distortions leads only to moderate changes of the angular distributions. For transitions to excited final nuclear states a reasonable agreement could only be obtained for those final states that can approximately be treated as single particle excitations of the ground state with the core remaining inert. These reactions show a typical fall off with increasing momentum transfer. For nearly isotropic angular distributions the effects of configuration mixing should be substantial such that the use of the "core plus nucleon" model is no longer adequate. Whereas our studies on the exclusive (p, or) reactions can be regarded as a test for the assumptions of our model we have, furthermore, calculated the differential cross sections of the (p,r/) reaction for different values of the rlNN coupling constant go on lec at the energies of 1, 0.8 and 0.61 GeV. The differential cross sections turn out to be quite small even for the largest reasonable coupling constant g~/4~r = 5 and for ~7's emitted in forward direction, Nevertheless, our calculations might be well used for respective experimental setups at COSY or CELSIUS. In addition we have calculated the cross sections for the (Tr, r/) reaction which is closely related to the process (p,~/) for low gZo. The results obtained are found to be quite sensitive to the value of ReXN. assumed. Thus especially the (rr, r/) reaction at threshold can be used to extract the information on ReXN* in the nuclear medium. Finally we note that the present approach is quite general and can be applied to a couple of other reactions. For example, it would be interesting to consider the production of A hypernuclei with an outgoing K + meson in the final state or to apply our formalism to the case of 7r- production,

Acknowledgements We gratefully acknowledge many helpful discussions with R. Shyam, who also supplied us with the relativistic spinors used for the evaluation of the matrix elements.

References I 1] B. Hoistad, Adv. in Nucl. Phys. 11, eds. J. Negele and E. Vogt (Plenum, New York, 1979) p. 135. 121 E Soga et al., Phys. Rev. C 24 (1981) 570. 131 J.J. Kehayias et al., Phys. Rev. C 33 (1986) 725. 141 S.E. Vigdor et al., Nucl. Phys. A 396 (1982) 61,

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B. Krippa et al./Nuclear Physics A 592 (1995) 539-560

[5] P.W.E Alons, R.D. Bent, J.S. Conte and M. Dillig, Nucl. Phys. A 480 (1988) 413; R.D. Bent, EW.E Alons and M. Dillig, Nucl. Phys. A 511 (1990) 541; R. Brockmann and M. Dillig, Phys. Rev. C 15 (1977) 361. [6[ B.D. Keister and L.S. Kisslinger, Nucl. Phys. A 412 (1984) 301; M.J. lgbal and G.E. Walker, Phys. Rev. C 32 (1985) 556. [7] E.D. Cooper and H.S. Sherif, Phys. Rev. Lett. 47 (1982) 818; Phys. Rev. C 25 (1982) 3024. [8] R. Shyam, A. Engel, W. Cassing and U. Mosel, Phys. Lett. B 273 (1991) 26. [9] R. Shyam, W. Cassing and U. Mosel, Nucl. Phys. A 586 (1995) 557. [10] M.V. Barnhill 1II, Nucl. Phys. A 131 (1969) 106; M. Bolstedi et al., Phys. Rev. C 10 (1974) 1225; J.L. Friar, Phys. Rev. C 10 (1974) 955. [11] J. Berger et al., Phys. Rev. Lett. 61 (1988) 919. [121 L.C. Liu and Q. Haider, Phys. Rev. C 34 (1986) 1845. [13] L.C, Liu, Phys. Lett. B 288 (1992) 18. [ 14] B.V. Krippa and J.T. Londergan, Phys. Rev. C 48 (1993) 2967. [15] B. Lopez-Alvaredo and E. Oset, Phys. Lett. B 324 (1994) 125. [ 16] W. Cassing, G. Batko, U. Mosel, K. Niita, O. Schult and Gy. Wolf, Phys. Lett. B 238 (1990) 25; W. Cassing, G. Batko, T. Vetter and Gy. Wolf, Z. Phys. A 340 (1991) 51; W. Cassing, T. Demski, L. Jarczyk, B. Kamys, Z. Rudy and A. Strzalkowski, Z. Phys. A 349 (1994) 77. [17] R. Glauber, Lectures Theoret. Phys. 1 (1959) 315. [18] G.M. Huber et al., Phys. Rev. C 36 (1987) 1058; Phys. Rev. C 37 (1988) 215. [ 19] C. Alvarez, H.E Arrellano, EA. Brieva and W.G. Love, in: Proc. Int. Conf. on spin and isospin in nuclear rections (1991) p. 219. [20] R. Machleidt, K. Holinde and Ch. Elster, Phys. Reports 149 (1987) 1. [2l] G.E. Brown and A.D. Jackson, The nucleon-nucleon interaction (North-Holland, Amsterdam, 1976). [22] L. Tiator, Proc. 2nd TAPS Workshop, Alicante (1993). [23] T. Vetter, A. Engel, T. Biro and U. Mosel, Phys. Lett. B 263 (1991) 153. [24] J.M. Laget, E Wellers and J.E Lecolley, Phys. Left. B 257 (1991) 254. [25] J.E Germond and C. Wilkin, J. Phys. G 15 (1989) 437. [26[ H.C. Chiang, E. Oset and L.C. Liu, Phys. Rev. C 44 (1991) 738. [271 R.S. Bhalerao and L.C. Liu, Phys. Rev. Lett. 54 (1985) 865. [28] R. Cutkosky, J. Math. Phys. C 1 (1960) 429. [29] E. Scomparin et al., J. Phys. G 19 (1993) L51.