Nttclaas Pbysia A171 (197~ 447-457; ©NortiY-Xo!land PttäirlFhto Co., MvtaidaasNot to be tapa+oduosd by photoprint as mia~ofllm without wrhtm ysrmission ttom ths Dublhiher

THE LOW-ENERGY PION-NUCLEUS INTERACTION

CLIVE ROGERS t

Inttitrt,/w 7TttorttLrdu JCtrnphyaik, Unioersititt KmLtrtahe, Gtrntmty

Received 6 October 1975(Revised 4 May 1976)

Abstract : We discuw thelow~netgy pion-nucleon intetactiôn within the framework ofthe optical model .We find that a consistent expansion ofthe aN amplitude in mommttmn isrequired in orderthat thelow-energy pion-nudeua interaction is physically compatible with a simple oompositie picture of ttienuclear weltering process. Tbia introduces an energy-dependent oompo~nt into the usual localpart of the Eriaon-Eriaon optical potential stemming from the s;N a-wave effective range.The nine of this component is estimated and the corresponding pion-nudeua effective tsngn a~volume parameters are determined for uniformly dense nuclei. Finally a modified extrapolation tothe bound state energies û oonaiderod in the light of these large effective ranges.

when q(r) and a(r) are given by

1. Introdactioe

Thelow-energy pion-nucleon interaction is well known to be dominated by s- andp-waves. The incorporation ofthis fad into a theory ofthe low-energy pica-nucleusinteraction has taken place over the last twenty years and is well documented t-a) .Since the original p-wave recipe of Kisslinger giving a non-local term in the opticalpotential it has bcen most notably modified by Ericson andEricson. They incorporatethe effects of oorrelations and the real absorption ofthe pica. The optical potentialused in their and other analyses of the low energy pica-nucleus interaction has theform

2~tV(r) = q(r)-Va(r)V,

ao(r) _ -4n~Pi 1~oP(r)+iPz t Im Cop2(r)].

Pi = 1+ m,

PZ = 1+ ~ .

t Supported by the Deutsche Fora~ungagemeinschaft .447

418

C. ROGERS

The difference between neutron and proton densities is neglected in the aboveequations. The identification used above for q(r) and a(r) are the forms used inpicnic atom applications . In this case bo, ImBo, co and Im Co are real constantsassociated with the elementary interaction. Principally with the neglect of oorrela-tions, the constants bo and ca are related to the s-wave scattering lengths and thep-wave scattering volumes, respectively

60 = ~(a1+2a3~

~0 - ~~33+~31+~13+a11) "

Finally, the form of the imaginary part is derived from the absorption of the pion ontwo nucleons .As mentioned above the identification of q(r) and a(r) given here is appropriate

for the description of picnic atom strong-interaction energy shifts. Thephenomeno-logical application ofa potential with the form given in eq . (1) has also been pursuedathigher pica energies s-') . However all these applications have been concerned withonly dif%rential and total cross sections . An essential difference between theseapplications and the picnic atom applications is that the parameters entering intothe optical potential are now determined so as to include the energy dependence ofthe elementary aN amplitudes.Apartfmm the zero-energy interaction and the application at below and near the

(3, 3) resonance very little is known about the pica-nucleus interaction at lowenergies (<30 Mew. The interaction in this region is more difficult since the picakineticenergy is comparable to the nuclear binding energies . In particular the validityof the effective range expansion has not been tested . There exists at present only afew measurements and certainly no systematic study has been undertaken .Other theoretical work in this area indicates a possible sign change from repulsionto attraction due to the growing influence of the non-local interaction ~.A recent work 1~ has considered a similar modification to the normal optical

potential description in the case of the carbon nucleus. The present paper supple-ments this work with a description of the variation ofthe resultswith atomic numberA and an estimate of the modification to the strong interaction energy shifts .

2. Pion-oacleas eftecäve range exp~oa

On the basis of the previously described optical potential we shall consider thedescription of the low-energy pica-nucleus interaction in terms of effective rangetheory . Specifically we consider the determination of the two parameters (generallycomplex) entering the effective range expansion, i.e . scattering length and effectiverange.

Since there are two conventions concerning scattering lengths we shall usethroughout the "particle physics" conventionand define theeffectiverange expansion

PION-NUCLEUS INTERACTION

449

for s-waves by the relation

q cot So(q) = 1/a+~rq2 + . . ..

(3)For later convenience when we compare with the nN system we also indicate the re-lationship to the inverse of this expansion or the s-wave K-matrix .~o(q) :

~o(q) _ ~q°(q) = a+bq2,

with 6 = -~aZr.

(4)

We shall describe b as the volume parameter. For the moment we shall consider theeffective range theory for a short-range potential only and leave until later modifica-tions due to long-range forces, i.e. Coulomb or vacuum polarisation effects.The determination ofthe effective range expansion parameters will be described in

terms of the simple analytic model provided by a uniform density nucleus. We shallfor the present neglect consideration of the non-local part of the optical potential .This can be justified by remembering that the s-wave picnic atom energy shifts aremainly determined by the local part oftheoptical potential . Furthermore, this shouldalso be the~for the s-wave phase shift just above threshold . It might be remarkedhere that we use in the uniform nuclear model an energy-dependent local opticalpotential so that we do not encounter the pathological properties noticed in ref. is) .We shall represent the nuclear density as being constant out to a radius R. The scat-tering length and effective range can then be represented analytically

where Ra is the wave number inside the nuclear medium at K = 0, i.e . Rô = - U1+iU2, and Lo(R) is the logarithmic derivative of the wave function at the nuclearsurface for K = 0; U, and U2 are the constant repulsive and absorbtive reducedpotentials experienced by the pion.Onemay notice at once that one obtains the well-known text-book result that for

an impenetrable sphere a = -R and r = 3R. This limit is obtained by lettingUl -+ oo with U2 = 0. On the other hand the limit Ul -~ 0 with Uz = 0 gives theless well-known result a = 0 and r -. - oo . These two simple results show that theeffective range passes through zero as one varies the height ofthe repulsive potential.

In fig. 1 we show the variation of the effective range r and scattering length a withthe siu of the repulsive real part of the nuclear potential. Only the real parts areindicated. For parameters comsponding to the Ericson-Ericson potential which do-scribes well the picnic atom energy shifts and widths the real part of the effectiverange is negative. This may be seen diroctly by oomp~aring the real part oftheeffectiverange with a typical value of the scattering length (-0.46 fm for 1zC) .

1a = LR~R), with Lo(R) _~,~ (5)

2R3 2R2 (1 +R/a)~ sin (21QoR) (6)

450

C. ROGERS

R~Iku*~~

IFfitl 0

20

Wig . 1 . Shows the variation ofthe real parts of the s-wave effective range, ra, and itcatttring length, a o,with the height of the repulsive real part of the pion-nucleus optical potential . The parameters of theuniformdensity nudearmodel correspond to "Cwith R = 3.1 fmand the absorptive potential determined

from ref. 3).

80TORN

80

)rg. 2 . Shows the variation of k cot d o(k) with the pica kinetic energy, T, obtained by the numericalsolution ofthe Klein-Cordon egnation with both the load and non-local parts ofthe Eriaons' opticalpotential . The figure reproduced from ref. ~ cort+etponds to parameters determined in ref. 3) except for

the absorptive parameter which is taken to be Im Co ~ 0.07 p-s .

From these simple considerations one may conclude that as regards the local con-tribution of the Ericson-Ericson potential, with parameters determined from picnicatom data, the real part of the pica-nuclear effective range is negative . This is axn

PION-NUCLEUS INTERACTION

451

to be a consequence ofthe essentially repulsive nature of the local part of the pion-nucleus optical potential which is required to repro_ duce the small (and negative inour convention) pion-nuclear scattering lengths . Furthermore this result is un-changed by the introduction of the non-local part of the optical potential . In thisconnection we reproduce fig . 1 from ref. ~ as fig. 2. The figure shows the behaviourof Re(K cot 80) for the Ericson-Frricson potential with parameters from the workof Krell and Ericson as used in their analysis of picnic atoms 3). This curve indicatesclearly, due to its downward slope at low energies ( <30 Mew, that the real part ofthe effective range is negative The behaviour at higher energies stems from thegrowing influence of the non-local contribution to the optical potential .We consider this result to be physically implausible on the basis of a simple single

scattering picture of the pion-nucleus amplitude. The factual basis of this argumentrelies on the fact that the aN effective range is positive and large. With the numericalvalues given in ref. i'), the values being specifically

ß, +2ß3 = -0.15 f0.06,

a1 +2x3 = -0.06f0.06,

where

Re lô;~ = a+ßqZ+ . . ,the effective range for the nucleus would be expected to be

Re r x 74Z-1p-1 ,

(8)where an empirical estimate of the nuclear scattering length 'a) has been made .

3. Modiflcatioo of tLe optical potentialOn examination of the derivation of the pion-nucleus optical potential given in

ref. ~) we sce that the reason for the negative effective range resides in the way wetreat the energy dependence of the elementary pica-nucleon amplitude . Since inref. ~) the main aim is the description of picnic atom data the s-wave and p-wave~N amplitudes are treated in the scattering length approximation. Hence ifwe nowapply this potential at.non-urn pica kinetic energies the energy dependence of thenN amplitude is not correctly includod . No criticism is intendod of the applicationofthe scattering length approximationto describe picnic atoms. Wemaythen includethis energy dependence, at these low energies, by retaining contributions to secondorder in the momentum k. This requires the inclusion of the nN s-wave effectiverange. Assuming isospin and parity invariance, we may write for the c.m. nNamplitude:

lo(g', ~= lo+h= . s+I2Q .s+13(t . =Xa' s~

(9)

45 2

C. ROGERS

whereR = g X g'/Ig X g'h

K = Igl = Ig'I "Ifwe are nowto expand theamplitudej~(g', ~ to second order in the momentum Kone must include 1rCj contributions fromjo andjl derived from the s-wave effaxive~Se"

Derivod from these considerations the low~nergy isoscalar elastic aN amplitude is

f(K', K) = bo+doK2 +coK~ cos 8,

(10)

the parameter do being related to the s-wave volume parameters by the relation

do = ~i+2ßs)"In order to construct a potential from the above amplitude we need to know theamplitude both offandon the energy shell. This maybe scen from the usual multiplescattering equations for low-energy pious in a nuclear medium Z) . The first term inthis series corresponds to the single scattering ofthe pion from one nucleon, andtheequivalent optical potential, expressed in momentum space, is

Here S(q) is the nuclear form factor for momentum q

~9) =Ja~~'P(r~,

(13)

and we consider nuclei with equal numbers ofprotons and neutrons andthe densitydistributions.Of course, writing the on-shell amplitude as

bo +cog ~ g',

we regain eq . (1) with the neglect of oorrelations and absorption of the pion on twonucleons . In the application we present belowwe consider thed~ term as an energydependence . Hence K2 ' is related to the total energy both on and off shell,

K2 = EZ-ps.

This choice is, of course, equivalent to the procedure employed at higher energieswhere the parameters bo and co are determined from the aN phase shifts at theparticular energy of interest. The only difference in our present ~ocedure is thechoice at low energies of parameterising the energy dependence with an effectiverange expansion. In this sense our procedure represents a conservative choice.

Before beginning to discuss this work we note that an alternative procedure mightbe the continuation :

PION-NUCLEUS INTERACTION

453

which when combined with the other terms in eq. (10) allows one to derive thefollowing potential in configuration space :

where

2pV(r) _ -4aA[bop(r)-(co +do)Vp(r)V-~dodp(r)].

(14)

Returning to the uniform density model of the nucleus we mayestimate the effectof this energy dependence if we neglect the contribution from the non-local term .In fig. 3 we show the variation of .aC-t(q) with energy for the cases do = 0 anddo = 0.05 - s . It is clear that for such an energy dependent potential that thedefinition of the effective range must now be changed. For such a quadraticvariation of the potential with K,

UE _ - 4adoPtP(r)~

(l5)

We see immediately that the energy dependence provides us with an effective rangeexpansion with a positive and large effective range. We present the effective rangesand volume parameters and also the corresponding scattering lengths for differentisospin-zero light nuclei in table 1 . One notices that the effective range can be con-siderably greater than the nuclear radius for the lighter nuclei . This is also true forthe imaginary part ofthe effective range. Indeed the variation with atomic number Ais revérsed in comparisonwith the scattering length. This is easily understood in termsof the simple estimate we presented at the beginning. That estimate furnished uswith a variation inversely proportional to Z, eq . (8), which was a direct consequenceof assuming that the volume parameter, b, was proportional to Z.The values presented in table 1 cannot be taken to be quantitatively correct due to

the neglect of the non-local contribution . We believe, however, that qualitativelythese results will remain unchanged for a realistic calculation . A comparison withref. t~ indicates that the imaginary part of b may be most notably modified in this

T~aLe 1The scattering lengths, effective ranges and volume parameters of different isospin-zero nuclei

Nucleus Scattering lengtha (~)

Effective ranger (~)

Volume parameterb (~')

4He -0.18+10.06 16.45+i16.30 -0.40-~0.076Li -0.26+80.04 9.57+1 4.91 -0.38-iU.07'°S -0.41 +iU.09 5.09+t 4.32 -0.58-i0.16"C -0.46+i0.13 3.87+f 4.14 -0.68-~0.21'4N -0.55+è0.16 3.04+i 3.65 -0.75-i0.25' 60 -0.63+d0.15 2.49+! 2.53 -0.69-i~0.24'°Ne -0.76+l0.18 1 .81+i 1 .92 -0.75-~0.27

454

C. ROGERS

~9610

Fig . 3 . Shows the variation ofk cot b°(k)with the pion kinetic energy for the modified local pion-nucleusoptical potential (a) in the case ofa uniform density nuclem . Curve (b) is the corresponding result for theunmodified local pion-nucleus optical potential . The cimes which apply to the'~C nucleus correspondto the following effective ranges and scattering lengths : (a) a° ~ -0.485+10.129, ro = 3.87+14.14 ;

(b) ao = =0.485+10.129, r° ~ _4.34-11 .99 .

5

10

15 EIMev120

Fig . 4. Shows the variation ofd ~ ( .7C~.' - .~t",~� ,)/ .7C~,' with energy for the K-matrix expansion (a)and the et%ctive range expansion (b). The exact solution ~.' corresponds to a local potential without

absorption and with parameters corresponding to'°B .

connection. Finally it may be remarked that due to the large values of the effectiverange the effective range expansion may not be appropriate to describe the lowenergy pion-nucleus interaction .

It would appear from comparison between eq . (3) and eq . (4) that if the effectiverange is large amore valid descrüption would be provided by a Kmatrix expansion.To illustrate, we show in fig. 4 the percentage difference between the exact solutionand the two respective low-energy expansions as a function of energy for a uniform

PION-NUCLEUS INTERACTION

455

nucleus. Here as previously done the non-local contribution has been neglected to-gether with the imaginary part ofthe potential. We see that the Kmatrix expansionis considerably better than the effective range expansion being in error by about4.5 ~ at E = 20 MeV. This has to be compared with 9 ~ for the effective rangeexpansion. These numbers apply to a boron nucleus. It is clear that a more detailedcalculation is required but we feel that the Kmatrix expansion should be used inplace of the effective range expansion in this low-energy region .

4. Modiflcatioe of 1~aemrn's expa~ion

As mentioned previously the modification of the effective range theory due tolong-range components in the pion-nucleus interaction has not been considered .The modification of effective range theory due to the Coulomb field and its relationto the shift in the bound state energies was first treated by Trueman '3). The furtherinclusion of vacuum polarisation effects has bcen treated in other contexts byHeller 1 `) .We shall in the present section be content with considerations ofthe effects of the

Coulomb field. We shall show that the Kmatrix expansion provides an alternativecontinuation to the bound state poles in the Coulomb field.

In the presence of a Coulomb field one may define a quantity similar to the K-matrix which wasdefined in eq . (4) (the suffix Cdefines strong-interaction quantitiesin the presence of the Coulomb field)

~c 1(q) ~ CZq cot 8ô(q)+ßi(q~

(16)where ~ = 2aac/(eZ*~-1) is the Coulomb penetration factor and a = ß/2q is theCoulomb parameter with ß = -2~Ze2 negative for an attractive Coulomb field.The function f(q) which is often encounted with Coulomb wave functions is

I(q) _ -Y-ln a+ ~

Z2

z

(1~

As mentioned at the end of the last section we expect that a K-matrix expansionshould provide amore appropriate description ofthe low-energy pion-nucleus inter-action. We shall hence consider the expansion

~c~4) = ac+b~q~+ . . ,

instead of the effective range expansion as used by Trueman. We may then use theabove expansion to determine the location of the booed state energies specified bythe conditions

cot bô = ~

(18)

q -i iQ,

where Q = ß/n(l+x~

(19)

456

C. ROGERS

The quantity x expresses the shift in the presence of the Coulomb field due to thestrong pion-nucleus interaction from the purely Bohr energy levels . The locationsof the bound state poles .are given by the equation

hereBis the Bohr radius and ~is the psi function . Regarding the quantity xas smallwe may, retaining terms to order x°, write

_ 2

-ac/B+(1/nz~c/B3x

n 1-2dac/B+(4/n3+2d/na~~/B3'

(21)

whered = In n -1 /2n-SY(n) .Assuming further that d~lB and bclB3 are both less than unity the energy shift

may be written

dE�

4

ac

bc

10 4d ac bc

(

3 a

E~_ - n~-B +

naBa+ {n3 + ns

~B Ba

- j2d +nJ \B~

c 2

- ~n + ne l \B31 +3rd order termsl .

(22)

In comparison with the expansion derived by Trueman we notice in addition to thefirst-order term (4/nß/B we obtain in addition the contribution -(4/n3~c/B3.The ratio ofthe relative contributions to first order is then provided by the quantity

bc/Baac^' ZZ,

which also indicates that the modification will increase with Z2. An estimate of theabove ratio from table 1 indicates an effect of the order of about 3 ~ for 2°Ne. Wehave here assumed that the Coulomb modified parameters are not significantly dif-ferentfrom the values given by table 1 . Such an argument should certainly be suffi-cient for our present purpose ofshowing the continuation to the bound state poles isonly affected to a small extent by our modification of the optical potential. Use ofthe 1~Gmatrixexpansion further from threshold, ofcourse, maycertainly be preferred .

We should like to thank Professor H. Pilkuhn for suggesting this problem and forhelpful discussions at various stages . we should also like to thank ProfessorM. Knellfor bringing to our attention ref. ~. We have also profited from a discussion withDr. c. wilkin .

References

1) L. S. ICisslinger, Phys. Rev. 98 (1955) 7612) M . Eripon and T . E . O . Ericson, Ann. of Phys . 36 (1966) 323 ;T. E. O . Ericson, Int . School of Physiw Enrico Fetmi, course 38 (A~ic Press, New York, 1967)p . 253 ; Lectures at Hercxa Novi, Sept . 1969, CERN TH 1093

PION-NUCLEUS INTERACTION

457

3) M. Knell aad T. E. O. Ericaon, Nucl . Phys. Bll (1969) 5214) L. Tauacker and W. Schneider, Z. Phys . 271 (1974) 4095) M. Sternheim and E. Auerbach, Phys . Rev. Lett . ZS (1970) 15006) M. Knell and S. Harmo, Nucl . Phys. B20 (1970) 4617) E. H. Auerbach, D. M. Fleming and M. M. Sternheim, Phys. Rev. 162 (1967) l6ß38) J. Reiner and P. Huguenin, Int. Seminar on pion-nucleus interactions, Strasbourg, 19719) M. Ericaon and M. Knell, Phys. Left. 388 (1972) 35910) S. Barmo and H. Pilkuhn, Phys . Lett . B, to be published (1976)11) G. HShler, H. Jakob and R. Strauss, Nucl . Phys . B39, (1972) 23712) J. Hßfner, Phys. Reports 21C (1975) SR13) T. L. Trugiman, Nucl . Phya . 26 (1961) 5714) L. Heller, Phys. Rev. 120 (1960) 62715) Coupling Constants Compilation 1975 (to be published)16) L. R. Schenk, CERN TH 1428 (1971)