Theory of elastic π3He scattering

Nuclear Physis A375 (1982) 470-480 © North-Holland Publishing Company

THEORY OF ELASTIC ~r 3He SCATTERING Y . AVISHAI Physics Department, Ben-Gurion University, Beer Sheva, Israel

and T . MIZUTANI Institut de Physique Nucléaire de Lyon, Villeur6anne, France

Received 22 December 1980 (Revised 11 September 1981) Abetrod : The pion-three-nucleon system is investigated using coupled Schrödinger equations . The coupling between the four-body (aNNN) and three-body (NNN) systems is explicitly implemented by operators for emission and absorption of the pion by each nucleon . The only simplifying assumption is the separable form for amplitudes pertaining to pure potential scattering . A set of Amado-Lovelace type equations is derived, from which the amplitude for the reaction rr+'He-~ a+ 3 He can be evaluated . The integral equations involve intermediate integration over single relative moments so that subsequent numerical solution is within reach .

1. Introduction The importance of pion-nuclear reactions is now well established'), and should not be repeated . It led to impressive progress both on the experimental and the theoretical fronts . It has been realized that a key element for any theoretical progress is the understanding of the process involving a pion and few nucleons . In other words, one would like to start from the basic ingredient of ~rN and NN 3 interactions, and study systems such as -rrNN (-rrd), -rrNNN (~r He) etc. Then, every approximation pertaining to the pion-nucleus system could be tested on its application to the smaller systems. This obvious statement resulted an intensive investiga2-s) . tion of the ~rNN system by several authors The aim of the present work is to take one further step and to study the system of (at most) one pion and three nucleons . A complete theory 'of this system should be able to evaluate amplitudes for which there are either four particles (NNN~r) or three particles (NNN) in the initial and/or the final states . Specifically, we are interested first in the "two-body" elastic and rearrangement reactions vr - + 3 He-> ~rr - + 3 He ~r ° +t etc. vr + 3 He ~ (NN) + (-rrN) 470

(a) (b)

(1 .1)

Y. Avishai, T. Mizutani / ~r3 He scattering

471

where lab) is a correlated pair. In the present work we will concentrate on the amplitudes for the reactions (1 .1). The study of the reactions

~ + 3 HeHn+d

(a)

n+d->n+d

(b)

will be given in the near future (it is rather complicated) . We have recently discussed the formal solution pertaining to the theory of coupled aNNN-NNN systems 6 ), and applied it to the study of the absorption of ~r - by 3He at threshold') . The motivation for studying systems containing more than two nucleons is obvious : The deuteron cannot be considered as an ordinary nucleus, since it is very loosely bound. Furthermore, recent experiments s) indicate that pion absorption by two nucleons in a nucleus ("quasideuterons") is not the only mechanism. Hence, considerable theoretical efforts have recently been devoted to the study of the pion-three-nucleon system 9) . The novel element in the present work is the derivation of practical equations starting from basic theory . The complexity of this problem is caused by the interplay of many-body aspects and true pion absorption . We use the solution of the four-body (~rNNN) problem in potential scattering suggested by Grassberger and Sandhas' °) (GS) but due to the coupling with the three-nucleon system this is not enough. Nevertheless the final equations are similar to those obtained by GS in form as well as in complexity . They couple only two-body amplitudes and hence involve integration over one relative momentum coordinate, a fact which is essential for numerical integration . In this respect, the present work represents a substantial improvement over the formal results of ref. 6). Stated more clearly, the central result of ref. e) [see eqs. (13)-(15) therein] is a set of connected equations for amplitudes with three-body initial and final states (such as ~+N+d) whose input is to be determined from another set of equations [eqs . (A .14) therein] . Thus the results of ref. 6), cannot be used in any practical sense . On the other hand, eqs. (3 .14) below have the standard form of the Amado-Lovelace equations (with non-diagonal propagator matrix) which are numerically solvable . We also point out that the formalism developed here has only little overlap with the work of ref.'). There, we have used DWBA as suggested in ref. 2), to find an expression for the absorption amplitude describing the reaction ~r + 3He -> N + N + N which is valid strictly at threshold. Here, on the other hand, we obtain exact equations which are valid at eneriges at least up to the threshold for second pion production . In sect . 2 we briefly recall how to arrive at equations for three-body amplitudes, relying on the explicit derivation of ref. 6) . These equations form the starting point of sect . 3, in which practical equations (in the sense discussed above) are constructed. The final equations, e.g . (3 .14) have the usual Amado-Lovelace structure which is familiar from potential scattering three-body problems . The paper is then concluded with a brief summary.

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Y. Aviskai, T. Mizutani

/ Tr 3 He

scatterinA

2. Effective three-body equations In this section we present equations for the amplitudes describing reactions such as ~+N+d~ ~+N+d , a+N+dHN+N+N,

(2.1)

N+N+NNN+N+N . These equations will have a disconnected kernel . However, we are not interested in their solution, since they only serve as the starting point for sect . 3. Since the pertinent equations have been derived in ref. 6), it will be enough to quote the results here, with reference to relevant equations therein. Let a, b, c, . . . , denote four-particle three-body channels (such as -trNd, NNd, etc.) and N the three-nucleon channels NNN. The physical amplitudes Xab , Xa,v, XNb and XNN were shown in ref. e) to ôbey two sets of integral equations with potentials B given in eqs. (6.9) and propagators z defined in eqs. (6.Sb) and (6.8b) . The equations read explicitly Xab

= Bab +~ Bac~%Mcb + BaN~%~Nb ~

XNb

= BNb + ~ BNcTMcb + BNNT~Nb ,

XaN

= BaN + ~ BacTcXcN + BaNTIVU~NN r

c

(a)

c c

XNN = BNN +~ BNcTc~cN +BNNTNXNN r c

(2 .2) (b)

adopting compact matrix notation (in a, b, c, . . . , N channel space) in which B is a matrix [see eq . (6 .9)] and To is a diagonal matrix with elements (zo)ab = Sa bra [see eq. (6 .8b)], (TO) aN = (Tp)Nb = ~, (TO)NN = TN [see eq . (6 .Sb)] ; we rewrite eqs . (2 .2) as X = B +Bro X .

(2 .3)

They satisfy three- (NNN) and four- (-rrNNN) body unitarity, and implement correct antisymmetrization among the nucleons owing to the use of antisymmetric wave functions tea, ~~, . . . , Xs [eqs. (6.7)] . We now move on and use eq . (2.3) in the next section. 3. Effective two-body equations In this section we will arrive at our goal, and derive coupled integral equations for two-body amplitudes . The solution of these equations will result in the amplitudes for the reactions listed in eqs. (1 .1). However, due to the physical coupling to the NNN channel, the set of equations will involve other amplitudes, that is,

Y. Avishai, T.

Mizutani / -rr 3 He scattering

473

amplitudes with free nucleon and correlated two-nucleon pairs at the initial and/or final state. These latter amplitudes do not correspond to the physical processes pertaining to eqs. (1 .2) as can be easily verified . They appear as scaffolding which is not needed to be known but must be evaluated in order to obtain the physical amplitudes of eqs . (1 .1) . Except for the separable. ansatz for three-body amplitudes no other simplifying assumptions are made, and the final equations (3 .13c) or (3 .14) involve only relative momentum between the two clusters, and hence they are of the same nature as standard Amado-Lovelace equations (except that the propagator matrix is not diagonal). Thus, they are amenable to numerical treatment. We start by writing the matrix B of eq . (2 .3) as a sum of 13 matrices, 3 B=Bo + ~ (Bx,+BQ,+BN,+Bnt,), i~l

(3 .1)

and explain the structure of each one of the matrices on the r.h .s . of eq . (3 .1) . As far as clustering considerations are concerned, we have six, four-particle threecluster channels so that together with the channel N of three nucleons we have seven channels, which we arrange in the following order [~Ns(NtNz)] [~Nz(NtNs)] , , L ~rNt (NzNs)] , [NNN] [dZN,N3] , , [OsNiNz] , ,

~d tNzN3]

,

(3 .2)

where d; is an isobar of the pion and nucleon N;. Hence the matrices B in eq . (3 .1) are 7 x 7 matrices . We shall illustrate the form of these matrices by figures in which a three-point vertex corresponds to either isobar dissociation or a pion emission, whereas four-particle lines between two vertices correspond to a free propagator .

b Fig. 1 . (a) (BP )ia, (b) (Ba,)is, (c) (Ba,),e and (d) (Bo,)i, . In all figures solid lines represent nucleons, a dashed line stands for the pion, parallel solid lines are rnrrelated (NN) pairs and wavy lines represent rnrrelated (aN) pairs.

Thus, fig. l a displays (Bo)tz, while figs . lb and lc give (Ba,)ts and (Ba,)se, respectively . The element (BQ,)~a is depicted in fig. (ld) and represents a clustering which is typical of problems involving more than three particles. Finally, the last column of B contains terms with three nucleons in the initial state. Typical elements are B1,, Bs, and B~~, which are represented by figs . 2a, b, c respectively . Most of the difficulties in the present problem originate from the existence of the (so-called) "peculiar terms" (third element in fig. 2a). They are similar in

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Y. Avishai, T. Mizutani / a 3He scattering

~-3 ~_____`_2 .f2~.3)i --- ~L I a Fig. 2. (a) B l ~, (b) Bs~ and (c) B~~.

3 ~-3 ^^~~*"^^+r-2 ~ I b

3 ~,tl-3).12--3) I

The heavy line represents all ME forces including OPE.

structure to (Bo,)ta in fig. ld, but the complication is due to the fact that the peculiar terms together with the "normal terms" (first and second elements in fig. 2a) appear at the same place in the interaction matrix B. On the other hand, the term in fig. ld is the sole occupant of Bta . In other words, the element Bt, (fig. 2a) contains elements with different clustering properties pertaining to the same physical process. This situation is typical of problems involving true pion absorption and more than two nucleons, to which we referred in the introduction . The regular terms (all terms of figs . 2a, b, c, d except the third term in fig. 2a) are collected in the matrices BN where e.g ., the elements (BN,)t7, (Blv,)s~, (BN,) ~.t2~3) I a

2 ,~~ I b

3 I

e

d

Fig. 3. Elements of the matrix B N , .

are described by figs . 3a, b, c, d respectively . The peculiar terms appear then in the matrices BM,. As an example, (Bnt,)t, and (Bn,,,),t are typified by figs . 4a, b. It should be clear by now how each one of the matrices of eq . (3 .1) are constructed. (Evidently, we have given the non-zero elements at and above the main diagonal . The elements below the main diagonal are obtained by interchanging initial and final states .)

Fig. 4 . Elements of the matrix B,,~, .

We have thus explained the cluster decomposition of B according to eq . (3 .1). For later purposes, we will also need to discuss the corresponding t-matrices TP, TA TQ TN; and TM and Green functions l'P, I'a l'Q l'N, and l'M, which are obtained from (matrix) Lippmann-Schwinger equations with potentials BP, B,, BQ BN; and BM respectively, and with To as the free propagator . In general, the t-matrices will have also diagonal elements in addition to the non-diagonal ones which the potential matrices B have . In the discussion below we shall also give the separable approximation for the above t-matrices . Thus, if y denotes any one of the clustering indices p, a;, rr;, N, or M;, we have Ty = By + By TOTy --- ~1 ;,dy Ay , Îy = ro+TOTyTO

(3 .3)

Y. Aoishai, T. Mizutani / -rr 3 He scattering

475

0

Fig . 5 . (a)

P

SA ),

the form factors for 3He-~N+(NN) process . (b) Propagator for a(NNN) intermediate state .

where A,, is a row matrix of form factors and d,, is a two-cluster propagator . Clearly the form A td,.A,, is the separable approximation for T,.. The structures of A,, and d,, are given for y = p, A 1, rr,, N, and Mt in figs . 5-9, respectively. Their explicit algebraic construction is explained in GS. 0

0

Fig . 6 . (a) Form factors for the dissociation of the correlated (NN~r) rnnfiguration . (b) Propagator for the intermediate state N (NNa).

From the inspection of TM it is clear that it can be constructed using the algorithm of GS (developed there for amplitudes such as TQ,). Notice that unlike the case in TN nucléon Nt is not "passive" since in the course of the iteration it emits and absorbs the pion. This is denoted by the small rectangle on the nucleon line in the propagator . Also, the two nucleon propagator d M, is very "meager" and 0 a

Fig . 7. (a) Form factors for the dissociation of (NN) and (Nrr) pairs into (NN)Na or (N-rr)NN configurations . (b) Propagator for the intermediate state (NN)(aN) .

composed only of the repeated dissociation and recombination of d;. Therefore, we marked it with two parallel lines with nothing in between to distinguish it from d N, of fig. 8. The same is true for the vertex which is here denoted by a square .

o

b

Fig. S . (a) Form factors for (NN)-~ (NN)rr, (NN)-. N(aN) and (NN) ~ NN . (b) The (NN)N propagator .

We are now in a position to solve the "hyper" Schrödinger equation B)I ~G) = 0 starting from

(T~' -

t

This we do by defining the full Green function and the generalized Grassberger-

Y. Avishai,

476 r---

0

0

T. Mizurani / -rr 3He

0

0

scattering

0

a

Fig. 9. (a) Form factors for N(NN)-. (NN)N-rr and (NN)N-~ NNN. (b) N(NN) propagator. Sandhas operators (GGSO), namely r = (r0 ~ - B) -1 = rasab +raUabrb r

ar b = P~ ~i~ Qi~ Nn Mi r

la)

(3 .5)

Uab = r0 1 (1-sab)+ ~ TcTOUcbe c~a Concerning eqs. (3 .Sb), two comments are due. First, it is evident that most of the GGSO Uab do not correspond to physical transition operators, since not all the corresponding interactions Ba, Bb represent true asymptotic states interaction . In fact, operators with a ~ b = M;, N; (that is a or b or both) do not pertain to any physically observed amplitudes . On the other hand, the interactions B~, B,,; and BQ, do correspond to asymptotic states, so that Uab with a n b = p, a;, ~; might correspond to physical amplitudes . However, the "channels" .1 ; = N; (N;N k-rr) cannot be considered as physical channels since physically the pion can also be absorbed . There is no asymptotic state with the clustering a;. As usual, all the non-physical operators will be retained in the equations as scaffolding, upon which the physical operators Uab (a n.b = p, Q;) are built. The second remark about eqs. (3 .Sb) is the observation that these equations are disconnected even after iterations . In fact, it is easy to establish the following facts : T,,,roTN, is disconnected (fig . l0a) ,

(a)

To,raTM, is disconnected (fig . lOb) ,

(b)

TN,TpTM, contains both connected terms (fig . lOc) and disconnected terms (fig . lOd) .

(c)

(3 .6)

Except for the symmetric products, all other products TaroTb with a ~ b are connected.

0

Fig. 10 . (a) Disconnected term in T,,,roTN eq . (3 .fia). (b) Disconnected term in Tv;roTn, eq. (3.6b). (c), (d) Connected and disconnected terms in TN;roT, eq . (3 .6c) . The cure of this disconnectedness problem is feasible after one adopts the separable approximation for eqs. (3 .Sb) . Keeping in mind the first remark after eqs . (3 .Sb) we now define amplitudes and driving terms Xab s ~n aI T0 Uab~%O Inb) _ la .b=P~Ar~~i.Ni,Mi~ Zab ~ sab(AalrOhb)

(3 .7) (3 .8)

Y. Aniskai, T. Mizutani / a3He scattering

a

b Fig. 11 . Z,

e

Zoon ZP N,

and

477

d

ZnM, .

Then eqs. (3.Sb) go over into Xab

= Zab + ~

c~a

(3 .9)

Zac~+Mcb

The detailed structtue of Zab is given by figs . 11-14. This is a straightforward manipulation since the vectors Aa, Ab, . . . are already defined in figs . 5-9. We see

Fig. 12 . Zr,rr

ZA,on Zr,Qr Zr,r,,

(two disconnected terms), Zr,Ni and Za,M,. (Z,,~, ~ 0 for i ;f j.)

immediately that all Z's with i ~j are connected. Therefore, if we arrange the two-chister channels in order p, ~,~tNtMt, 2, 3, and write eq . (3.9) in matrix form then the disconnected part of the interaction matrix Z (namely Zd) will have a form of three block matrices . Specifically [with X = {Xab}, Z = {Zab}, d = {daSab}], X = Z +ZdX,

(a)

(3.10) N a t rrt t .M~ = At x x ~t N1 x x ~ ~x X , Ml I

Fig. 13 . ZuiNn Z~,,, and Zo,M,

(Zo, = Z~~ s 0 for i ~j) .

Y. Avishai, T. .Mizutani / a; Ht .scatt~rinR

478

Fig. 14 . ZN,N, (two connected terms) ZN ;,,~~ (connected and disconnected terms), ZN;,~, and Z,,,,,,,.

where the crosses are the disconnected terms as shown in fig. 11-14. Notice that in (N t , Mt) only the disconnected part Of ZN,na, is taken. As is shown in fig. 14b, ZN,M, has also a connected term which belongs to Z` . The above considerations imply that the disconnected amplitude Xd (to be defined below) is also built of block matrices . Together with Xd we define Mtöller operators ,(ld , .R d and the (non-diagonal) propagator matrix ©. Thus Xd = Zd +Z ddXd ,

(a)

Due to momentum conservation and the structure of the disconnected terms Zâb (k, k', E) (k, k' = final and initial relative moments) one has z Zâb (k, k', E) = S(k - k7Zâb (k , E) , Zâb(k2, E)=~

J

z d z Lnat(9 Z)(E - 9 2- k ) tdbi(RZ)] q d9

(a)

(3 .12)

(b)

[see for instance fig. 14b] . Thus, eqs. (3 .lla, 3 .11c) are algebraic equations. For example, eq . (3 .llc) means (3 .11c') 2 z z ®ab(k Z , E) = sab[d(k , E)]aa +~ [d(k , E)]aaZâ~ (k , E)®~b(k 2, E) , which can be easily solved . We have already noticed that unlike the propagator matrix d, the full propagator matrix ® is not diagonal . This does not pose any principal difficulty and in particular, it does not add integration coordinates. The solution of eq . (3 .10a) is now straightforward, that is, we treat it as a twopotential (Z ° +Z`) problem and find

which elastic the the correlated (3potential (3 conclusion, first to this absorption amplitudes already isYvL can same coupled present determine (k, work itaof the (total work one Let that scattering (k, by great we we be [reaction theory, kHere, reads cross peculiar the ismanner )the amplitudes k') solved either is discussed (NN) mention =The are mentioned is Itherefore eq theory angular we NN arNNN-NNN analogous one deal ZaL denote one the for section (dropping input We Yâc, the (3®ab(kZE) already second since have pairs problem obtains is aspect matrix (1 the by has usual of as hope extension provides concerned, that (k, (driving momenturrt, standard work pure here aconnected, to shown initial are it at are is does the to kind Avishai, kset is evaluate is "static" that threshold, relevant an the inversion afrom enriches not )now determined that with elastic the coupled physical to (potential +effect system of extension and not several E-dependence), how is numerical terms do diagonal is methods appearance of quantum aeq the non-static Twithin coupled affect the deuterons "non-static" all before GS and scattering Mizutani/ only of parity, to the (3 algebraic or set Summary In dk which amplitudes the interesting and final arrive true and scattering) the Padé of X~, from principle, three-nucleon rrof to kThen the Z's one integration to n2 numbers propagators) the Yakubovski's integral isospin) problems driving clusters fprces pion will is of n3He inelastic namely at realm of eqs only, (e approximant cl equations the can present two the tractable correlated eq suffice elements absorption scattering for (3 content there four-body get fig The terms partial-wave we kinds equations and (3 (k, of Therefore, in X~ elastic with oftheory (pion-production) numbers theories can la) kproblem which appearance let (3 should for our =equations extension of )®ca(k and For depend concerning of pair, nucleon which Yom use L, scattering correlated In eqs equations on problems the ause with to the L' itpractice, Therefore, = pertaining obtained after be elastic decomposition include beyond )Then In the to binvolves SIN (3 only Yar one amplitudes of number =some no Iofdisconnected for determine denote p, partial-wave the physics experiment difficulty the (NN) these on integration will the the are scattering and elastic channels however, the from way, Faddeev (kpartialrelative ato as amplitheory follow solved A those pairs kinds , then static usual long kWe true > are the r )of nd 2in,

Y.

.

479

Eq. .13c) momenta decomposition, variable. diagonal in eq. .13c) YaL :bL'

. ...

.

r

cf

which Notice as In ones wave tudes ;vL'

.

:bL'

.13b)

r

~ r c.dL' J

n

Zap ;~~-

rr

.13c)

. .8),

. .

;bc'

n (3 .14)

. .

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.14).

. .11c') . .

4. In of determining solving there present equations. pion in nowadays . The have Although absorption Another The NN . solution of Finally, scattering (Faddeev)

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. . .

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480

Y. Avishai, T. Mizutani / a37ie scatterinq

One of us (Y .A .) would like to thank T. Ericson at CERN for numerous discussions and warm hospitality during the summer of 1980. The other one (T.M.) would like to thank the INP at Lyon for their warm hospitality during the fall of 1980. References 1 2 3 4 5 6 7 8

9

10

See e.g. A.W . Thomas and R. Landau, Phys . Reports S8 (1980) 123 T. Mizutani and D.S . Koltun, Ann . of Phys . 109 (1977) 1 A.S . Rinat, Nucl . Phys . A287 (1977) 399 A.W . Thomas and A.S . Rinat, Phys . Rev. C20 (1979) 216 Y. Avishai and T. Mizutani, Nucl . Phys . A326 (1979) 352; A338 (1980) 377; A352 (1981) 399; and Orsay preprint I PNO/TH-80-25 Y. Avishai and T. Mizutani, Phys . Rev. C22 (1980) 2492 Y. Avishai and T. Mizutani, J . of Phys . G6 (1980) L203 Basel-Karlsruhe Collaboration, SIN 1980, Contributions to the Berkeley conference on nuclear physics and to the Eugene conference on few-body problems ; J . Franz et a/., Phys. Lett. 93B (1980) 384 ; G.R. Mason et al., Nucl. Phys . A340 (1980) 289 M. Locher and H.J . Weber, Nucl . Phys. B76 (1974) 400; H.W . Fearing, Phys. Rev. Cll (1975) 1210 ; Cll (1975) 1493 ; C16 (1977) 313 ; A.M . Green and E. Maqueda, Nucl . Phys . A316 (1979) 215 ; V.B . Belyaev, A.L . Zubarev and A. Rahimov, J. of Phys . G6 (1980) L49 P. Grassberger and W. Sandhas, Nucl . Phys . B2 (1967) l81