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It .1 r v in . - ' I Hi... . 1|Q'I . .. 3.. . .II.\ ‘5'! I ‘ v.|| I: (.1- I n“ Muwlq\\:1s.1.1.-_ uhu.\\..h\ -0." ,\..\ .\ .Q fil‘ \ll $5 .t LNI ‘1‘ (L nix L .. r\l| .\ mt. . 0.1 O.|. . 0" .vbuh ' I. .1 .‘2 v. a 'i I . O I“. 5 Mil} . r | . ‘ I 4n!).\1\ .‘u‘ l‘ I- 0‘ I . 1 gnawHtvuAthtJJ.‘ “n . .1“..h¥1 1 “-03th 131...“... rflufiuvf... - it... . 1 AIM...” . .. - t q u K —\ l I LIBRARY Michigan state L University I This is to certify that the dissertation entitled A STUDY OF NUCLEAR PION ABSORPTION presented by HAITOOK SARAFIAN has been accepted towards fulfillment of the requirements for PH.D. PHYSICS degree in HUGH MCMANUS Major professor Date 8-10-1983 AW MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771 MSU LlBRARlES ” RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. A STUDY OF NUCLEAR PION ABSORPTION BY HAITOOK SARAFIAN A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1983 ABSTRACT A STUDY OF NUCLEAR PION ABSORPTION BY HAITOOK SARAFIAN Various aspects of pion-nucleus interactions are stud- ied. The S-wave pion optical potential is studied up to second order investigating effects due to the modification of the pion propagator. That is, the ordinary off-shell- pion propagator is replaced by one which includes the effects of the average field generated by other nucleons. S and P wave pion absorption in a finite nucleus and in nuclear matter are studied. Emphasis is on a rescat- tering absorption mechanism which turns out to be equivalent to the quasi-free deuteron model. The results are improved by including medium effects in the pion propagator, as in the S-wave scattering case. The possibility of nuclear opalesence and pion con- densation phenomena are discussed. The isospin dependence of the P-wave absorption in He isotopes in the resonance region is explained by the rescattering process. Finally the question of the region in which absorption takes place and the problem of overlapping sources is discussed. To my entire family Acknowledgments Above all I wish to express my gratitude and appreci— ation to my thesis adviser Professor Hugh McManus for his support and continuous interest in the progress of this work. Special thanks are due to Professor Hiroshi Toki, who was always available for discussions. He was a great help in clarifying points and showed me the way out of long and messy calculations. I would also like to take the opportunity to thank the ,following, for their indirect, although crucial, involvement in the preparation of this work: Professor J.S. Kovacs, my general graduate school adviser, for financial support through out my years of graduate study. Professor A. Galonsky, for giving me my first taste of real research. To me, the Department of Physics and Astronomy of MSU provided a unique opportunity not only for fruitful teaching as a graduate assistant, but also for interaction with the leaders of my research area through weekly ” Wednesdays " cyclotron seminars. The financial sup- port for my attendence to an AIP meeting in Bloomington Indiana in Fall 1981 is highly appreciated. Special words of thanks are due to the staff of the National Superconducting Cyclotron Laboratory computer facilities, since most of the long computations of this work were done on the z 7 , VAX 11/780 systems. Last, but not least, I'm grateful to my wife Aida, iii iv whose patience and understanding eased many of my diffi- culties and disappointments, for her advice served always as a ray of hope in the darkest moments of our life together. Special thanks also to my two daughters Nanaz and Nenette for their faith in me, which helped me to work harder. TABLE OF CONTENTS LIST OF TABLES 0.00.00.00.00...OOOOOOCOOOOOOOOOOOOOO LIST OF FIGURES 0.0.0.000...00...?OOOOOOOOOOOOOOOOOO Chapter Chapter 2-1 2—2 2-3 Chapter 3-1 3-2 3-3 Chapter 4-1 4-2 Chapter I INTRODUCTION ....................... II EN SCATTERING .. S Wave IIN Scattering ................... ITA First Order S Wave Optical Potential ................................ TIA Second Order S Wave Optical Potential ................................ Modification of ItA Second Order S Wave Optical Potential ............ ..... III PION ABSORPTION .................... 5 Wave Pion Absorption in Nuclear Matter ................................... S Wave Pion Absorption in Finite Nuclei ................................... On the Angulaer Distribution of ( 7T , 2N ) .............................. IV P WAVE PION ABSORPTION ............. P Wave Pion Absorption ................... Isospin Dependence of Pion Absorption by Nucleon Pairs in the He Isotopes ..... V AN ALTERNATIVE TREATMENT OF P WAVE PION ABSORPTION IN THE OPTICAL. vii viii l 6 11 14 21 21 34 39 47 47 59 Appendix MODEL A REPRINTS vi TABLES IV.l LIST OF TABLES PAGE Pion absorption cross section in several channels as a function of pion momentum k are compared, where l is the angular momentum of the incoming pion. These numbers are normalized to the (T,S,L) = (0,1,0) to (T',S',L') = (1,0,2) transition at k = 0.5 as indicated by * .................. 62 v.1 A comparison of the forms of A-h optical potential labelled " Microscopic " with the Kisslinger type labelled "Macroscopic "‘ 000......OOOOOOOOOOOOOOOOOOOOOO 72 vii LIST OF FIGURES FIGURES PAGE II.1 Scattering of a pion by a nucleon .......... 6 II.2a Scattering of a pion by meson cloud ....... 7 II.2b Scattering of a pion by nucleon core ...... 7 11.3 Three dynamical channels of pion-core interaction ............................... 8 II.4 Second order S wave pion scattering ....... 12 11.5 The second order S wave pion rescat- tering mechanism with inclusion of medium effects ............................ 15 III.1 S wave pion absorption via two- body mechanism ............................ 23 III.2 The imaginary component of the S wave pion absorption parameter of nuclear matter vs. pion momentum. Dashed line corresponds to a rescattered pion energy of 0.5 h! . Solid line corresponds to a rescattered pion energy of 0.5 /‘ ...................... 25 111.3 The imaginary component of the two-body S wave threshold pion absorption para- meter of nuclear matter vs. LLEE parameter,A................................. 28 viii III.4 III.5 III.6 III.7 ix Nuclear medium polarization due to pion propagation .......................... 29 The imaginary component of the two-body S wave absorption parameter of nuclear matter with inclusion of medium effects for 43 = 1.5 vs. pion momentum .......... 32 The dispersive component of the S wave pion absorption parameter of nuclear matter with rescattered pion energy of 0.5/:- and medium effects for A= 1.5 33 S wave pion absorption cross-section of "He vs. pion momentum. The solid line on the scale of this drawing corresponds to the following three almost indistinguishable cases one 7! exchange with AK' 1200 Mev and no medium effects 0° and medium >. a u one 7? exchange with effects at A = 1.5 one 1t exchange with A": 1200 Mev and medium effects at A = 1.5 Diamonds correspond to one pion exchange with A": 700 Mev and with and without medium effects at A = 1.5 ................ 36 The imaginary component of the S wave pion scattering length of "He vs. pion 111.9 111.10 IV.l IV.2 IV.3 momentum. Dashed line corresponds to one Tl exchange with A"; an and no medium effects. Dashed—dot line corresponds to one pion exchange with. ,AIF 1200 Mev and medium effects for A = 1.5 . Solid line corresponds to one It exchange with A": 1200 Mev and no medium effects. Dashed-dot-dot line corresponds to one 7! exchange with Art: 700 Mev with and without medium effects for li= 1.5 o is the experimental value for threshold pion [Ref. III.7] ......................... The four possible S wave If+rescattering absorption mechanism ...... ................ Schematic ( If, 2N ) reaction ............ The direct and crossed channels of the P wave pion rescattering absorption mechanism with intermediate one 7‘ exchange .................................. The direct and crossed channels of the _P wave pion rescattering absorption mechanism with intermediate e exchange .................................. The direct channel of the P wave pion rescattering absorption mechanism with intermediate fl'+ ( exchange and the 38 40 44 48 49 IV.4 IV.5 xi medium Effects 00.....0......0.00.0.0.00... 52 The imaginary component of the two- body P wave pion absorption parameter of nuclear matter vs. pion momentum. Dashed-dot line corresponds to one It exchange with An: 1200 Mev and no medium effects. Dashed line corresponds to 7T+ 6 exchange with A"; 1200 Mev and A9: 2000 Mev and no medium effects. Dashed-dot-dot line corresponds to one 11‘ exchange with A"; 1200 Mev and medium effects at /‘= 1.5 Solid line corresponds to ”5+6 exchange with A”: 1200 Mev and A? = 2000 Mev and medium effects at A = 1.5 Dashed-slab line corresponds to one 'fl exchange with Au= 700 Mev and no medium effects ......... ................ 55 The imaginary component of the two- .body P wave pion absorption para- meter of ”He vs. pion momentum. Dashed line corresponds to W+€ exchange with Aw= 1200 Mev and Ae==2000 Mev and no medium effects. Solid line corresponds to fl'+ P IV.6 xii exchange with A": 1200 Mev and Ae= 2000 Mev with medium effects at A = 1.5 ............................... 57 The pion absorption ratio __ do’(T=°) . R - 40' "a” vs. pion momentum. Dashed line corresponds to in? 1 and n'exchange alone with A”: 800 Mev. Solid line corresponds to all of the possible partial wave contributions ....... 63 The rate of the A + N -v- N + N process of “He at threshold vs. the cut off radius . ..... .... ........... . ...... ........ 71 The real component of the P wave pion-nucleus optical potential vs. pion kinetic energy. Dashed-dot line corresponds to the microscopic treatment (9) with P = 0.5 nuclear matter density. Solid line corresponds to the impulse approximation (Co) . Dashed line corresponds to the mac- roscopic treatment (f) with €== 0.5 nuclear matter density and.a‘= 1.5 ........ 75 The imaginary component of the P wave pion-nucleus optical potential vs. pion kinetic energy. Dashed-dot line corresponds to the xiii microscopic treatment (9) with e = 0.5 nuclear matter density. Solid line corresponds to the impulse approximation (co) . Dashed line corresponds to the mac- roscopic treatment (f) with €== 0.5 nuclear matter density and A= 1.5 76 The imaginary component of the two- body P wave pion absorption para- meter of nuclear matter vs. pion momentum. Dashed line corresponds to one 71’ exchange with An= 800 Mev. Solid line corresponds to N'+ 6 exchange with A": 1200 Mev and A? = 2000 Mev (standard set) Dashed-dot line corresponds to one It exchange with A": 700 Mev. Dashed-dot-dot line corresopnds to one If exchange with A“; 400 Mev .. ..... 79 The imaginary component of the P wave pion absorption parameter in nuclear matter vs. the cut off radius. Dashed-dot-dot line corresponds to one It exchange with A": 1200 Mev. ~Dashed line corresponds to one It exchange with An= 800 Mev. xiv Solid line corresponds to 77+? exchange with A“; 1200 Mev and We: 2000 Mev (standard set) . Dashed-dot line corresponds to one 7: exchange with AR: 700 Mev ........ 81 Chapter I Introduction Information about nuclear structure is usually ob- tained from nuclear reactions. The pion has become an important probe, with the production of high quality meson beams. Because it has zero spin, its interaction is sim- pler than that of spin l/2 or spin 1 probes. The pion has a small mass, mn = 140 Mev = 0.15 m” , so it is relativistic even at low energies. It can also be absorbed by a nucleus. In these two respects it is similar to a photon. However the pion strongly interacts with nu- clear matter, it also exhibits strong resonance behavior due to the formation of the A resonance in the P wave pion— nucleon channel. In this thesis the emphasis is on a particular mecha- nism for pion absorption, the rescattering mechanism where a pion is absorbed by a pair of nucleons, being scattered by one and absorbed by the other. There are two such mechanisms, the first being S wave pion scattering followed by absorption, and the second involving P wave pion scattering. The first is predominant at threshold, the second, which goes via the creation of a virtual A resonance, is the most important for pion kinetic energies > 50 Mev, including the resonance region. Here both It and f mesons are included in intermediate states. Experiments on pion absorption by the He isotopes in the resonance region, show that the ratio of absorption on T = 0 nucleon pairs to T = l nucleon pairs is very large, and this is explained in a natural way by P wave rescat- tering in the present model, thus justifying the old quasi- deuteron absorption model. However absorption parameters in nuclear matter cal- culated by fitting the parameters of the model to pion absorption by free deuterons generally give results which are too small as compared with experiment. One difference between absorption by a free deuteron and two nucleOns in nuclear matter is that in nuclear matter,the pion propagator is affected by the average field of the other nucleons, expressible in terms of the pion self-energy in nuclear matter. This is not a small effect, and occurs also in other physical processes. It could give rise to phenomena like pion condensation or nuclear opalescence. These processes do not appear to happen at normal nuclear densities. The strong momentum dependent P wave pion interaction with nucleons which would tend to give rise to such phenomena is damped by processes at short internucleon separation, usually parameterized by the Landau parameter 9', or equiv- alently by the Lorenz-Lorentz-Ericson-Ericson parameter,.J , ( LLEE ) , [Ref. I.1] which derives its name from an analogous process in the propagation of an electromagnetic wave in a polarizable medium. Non observation of nuclear opalescence in other experiments means the g'> 0.5 or A > 1.5 . The parameters of the theory are the coupling con- stants at the interaction vertices, the form factors Ag , A? at the vertices, which give the effective size of the nucleons or isobars acting as sources for the meson fields, and the LLEE parameter .A . This is all part of the. standard I? + (’ phenomenology involved in other areas of nuclear physics such as the excitation of spin-isospin flip states in inélastic proton scattering.i ' L The use of finite size sources, where the size is an effective one, and may be due to complicated prOcesses at short distance, indicates that the calculation is likely to be invalid if the sources overlap, so the region of nucleon separation in which absorption occurs is investigated. The relation, if any, between source effective size and the bag model of hadrons is briefly discussed. Clearly an under- standing of the parameters 1% or g' , would involve understanding the properties of overlapping bags. A recent attempt in this direction has been made by Weise (Ref. 1.2]. Two nucleon absorption may not be the whole story. Recent experiments near resonance indicate that cluster absorption involving the excitation of two A '5 may be important at resonance energies for large nuclei. This will not be considered here. In any case it involves the same basic processes, which have been used to estimate the ratio of 4-body to 2-body terms. This dissertation is organized as follows. In Chapter II the S wave part of the optical potential for pions is discussed. Calculations are carried out up to sec- ond order, including nuclear medium effects in the second order term, but excluding absorption. The purpose is to investigate the origin of the strong repulsive nature of this part of the potential observed empirically. In Chapter III pion absorption by S wave rescattering is investigated for both nuclear matter and qu . The nuclear medium efflect is calculated, as are.the isotopic ratios and angular distribution of emitted nucleons. Some of this has already been published by the present investi- gator. In Chapter IV P wave rescattering is investigated for both nuclear matter and 0He . Nuclear medium effects are calculated, as is the isotopic ratio for emitted nu- cleons. The latter has also been published by the present author. In Chapter V the question of the region in which absorption takes place and the problem of overlapping 'sources is briefly discussed. A calculation of the absorp- tion parameter in an alternative form of the P wave optical potential .( A -h model ) is also presented. REFERENCES FOR CHAPTER I 1.1 M. Ericson, T.E.O. Ericson, Ann. Phys. 36 (1966) 323 1.2 W. Weise, Phys. Lett. 117B (1982) 150 Chapter II ”N SCATTERING The kinematical region of interest is pion kinetic energies of g 220 Mev . In this region, the 7ZN scattering amplitude consists of at most two partial waves, L = 0 and 1. P wave ( L = l ) scattering, dominates due to the formation of the intermediate resonance state, A» , in spin and isospin 3/2 channels. The 5 wave ( L= 0 ) channel, although small, is important at low energies. In this section S wave KEN scattering is studied. 2-1 SWave TIN Scattering S wave pions in principle could be scattered by the meson ( pion ) cloud surrounding a nucleon and also by the " bare " ( core ) nucleon, 7f ‘\ ‘X \\ Figure 11.1 Scattering of a pion by a nucleon or equivalently l f ” N I 7r// | ,/ // C1 ~ \\ : fl>*\\\ q H N a b Figure II.23 Scattering of a pion by meson cloud Figure II.2b Scattering of a pion by nucleon core Figure II.2a shows the scattering of the incident pion by the pion cloud and Figure II.2b shows the scattering by the core. The contribution of Figure II.2a is small and may be ignorerd [Ref. II.l] . But part 2b itself, is :omposed of at least three dynamical channels. Figure 11.3 Three dynamical channels of pion—core interaction The channel in Figure II.3a, due to the exchange of the scalar 6' meson, contributes to the isoscalar part of lfN scattering. Diagrams 3b and 3c are due to vector- meson ( and anti-nucleon N. exchange and describe the iso- vector part. For low energy scattering it is sufficient to use a phenomenological interaction [Ref. 11.2] Hm... gland). $312.” exam <2-1-u A _\ where 4' is the nucleon field operator and 4; and It A ( 33¢? ) are the pion field and its conjugate operator re- spectively, and ‘2 is the nucleon isospin operator. This interaction is parameterized through two scattering lengths, A1 and A;_which are given [Ref. II.3] 1.: 0.003 1'. 0.001 (2-1-2) 1;. 0.05' :t 0.00! By using these two scattering lengths the S wave )tN, 53, phase shift in isospin 3/2 and spin 1/2 channel up to 50 Mev and S“ phase shift in isospin 1/2 and spin 1/2 channel up to 250 Mev pion's lab kinetic energy are well reproduced. To improve the 83, phase shifts for higher energies, energy dependent scattering lengths are proposed [Ref. 11.4] . T . J- l,= 0.003.} 0.0344 __ .. c.0058 CC.) ’4- } . T (T a (2-1-3) A ___o,os'_. __ 0.0334 _ -0.os'8 _) where T is the pion's kinetic energy in the laboratory sys- tem. 2-2 ETA First Order S Wave Optical Potential Adopting the multiple:SCattering approach [Ref. 11.5] to study the ITA scattering problem, the basic process would be the scattering of an incident pion via single bound nucleons. To lowest order [Ref. 11.6], the optical potential lj't , is parameterized in the following form OP 10 0) 2w UOPt = _. 41c bee (2-2-1) cu where b is the first order S wave scattering parameter 0 and. e is the nuclear density. From the Hamiltonian Equation (2-1-1), the MN scat- tering amplitude t‘w , in momentum space for this process is m o A A. t -..-.- .25 E 2A. .. c(u+u') _j_:_- t-t] (2-2-2). ’ o I ‘ A 0 where L) is the rescattering pion's energy and t 18 the pion's isospin operator. The Fourier transform of the expectation value of this operator between initial and final nuclear states expressed in momentum space in the elastic scattering channel i.e. ‘J = ‘0’ results in the spatial representation of the optical potential 20(1sz alt-[Aifi-J-fi-Aatgfv‘ffl] (2-2-3) In nuclear matter and / or symmetric nuclei, where the proton and neutron densities are identical, Equation (2-2-3) would reduce to the more compact form 11 ll =._£EE. 2") ”Pt /‘ A‘io (2-2-4) By comparing this equation with Equation (2-2-1), the S wave scattering parameter becomes,in terms of the fitted scattering length A. , (a) (3;. A, (2-2-5) .9; f From this result one may conclude that the first order S wave scattering parameter b:, in the kinematical region of interest is independent of incident pion energy, and since it is negative, it leads to a repulsive optical po- tential. But since A. is very small, the first order S wave potential would not have any significant effect. Hence, higher order S wave effects should be considered. 2-3 71A Second Order S Wave Optical Potential 1n the second ordei S wave 71A multiple scattering process, the incident pion gets the chance of being scat- tered twice by two different nucleons. The incident on shell pion gets scattered by one of the nucleons, propagates off shell through the nucleus and is rescattered by a second 12 nucleon.This process is shown diagramatically in Figure 11.4 N N n:-a-—» -——££~— -—-JT IV AI Figure 11.4 Second order S wave pion scattering Applying the interactive Hamiltonian Equation (2-1-1) twice in this second order diagram in the’elastic channel, the scattering amplitude becomes U” 1 4* 2. a .1. A’ 9 (2-3—1) 3 a! . where 6““‘(3 1‘“ 1:; 1 is the off-shell propagator, 30 + -0 I 1 k is the pion momentum, and t‘ and B are first and second nucleon's isospin operators. Fourier transformation of this amplitude gives the optical potential operator (1) git? a. .1on t= IT e Jk (2-3-2) P can)! 13 where the 3?: F, 4-; is the relative separation vector of the two nucleons. The nuclear matrix element of this operator involves the exchange part of the two body density operator q 0' t[ J,(KFIE‘?1') J :3 ___L__ ..—.—. fl] (1‘) (r1) .. .. eel A‘A'd) ’6 n- 11 KF]3-?1I 6 . C (2 3 3) . . a where A 15 the atomic number of the nucleus and P". t . . . and F3- are the spin and Isospin exchange operators 6 ' A’ A1 .. .o ) (2-3-4) t l—m' A1 -|. . . KF = 1.4 fm ‘ IS the Fermi momentum,. and (J 15 the nuclear 3 density. For nuclear matter i.e. (’ = 2/3112 Kr and for threshold pions the following second order S wave optical potential parameter results (1.) be 3 - —6— (F (Ari-3):) (2-3-5) ‘F Above threshold, analytical calculation is not pos- silble and integration has be carried out numerically, [Ref. 11.7] . One may compare this result with the first order (1) (I) .- parameter Equation (2-2-5). The ratio of b. / h0 = 3 14 shows the significance of second order effects. Identical signs of Bt)and B:,stress the coherent nature of these two effects in generating a stronger repulsive optical po- tential, which is needed for better fits to 8 wave pionic atom level widths. 2—4 Modification of IA Second Order S Wave Optical Potential Even restricting the multiple scattering to second order, the method of the previous section is an approxi- mation to what actually happens. Although only two nucleons get involved actively in the second order rescattering process and the rest of the nucleus acts as a spectator, it does have an effect. That is the off shell rescattered pion does not propagate between two isolated nucleons in vacuum, but in a medium which exerts an average field on the pion. Usually this effect is refered to as the pion's " Self Energy " TT , [Ref. 11.8] . It should be added explicitly to the pion propagator 604,?) = -1. -5 (2-4-1) 2 a]. q 7. 1. 1 f+k -wz f+k~ -u +77 Diagramatically the self energy contribution is de- picted in Figure 11.5 N+ 15 4,; E->—__LE_@J_?_,_ 72' Ni in Figure 11.5 The second order S wave pion rescattering mechanism with inclusion of medium effects In brief, [Ref. 11.8), the off shell propagation of the pion in nuclear matter is considered as the excitation and de-excitation of an infinite number of virtual isobar- hole states ( A.- h ) . The complete pion self energy is obtained by summing the chain of A.-h diagrams con- nected by the reduced isobar-hole interactions. Analytically it is given by mm = nun/1.. 1 mm (“‘2’ 3 k1{-1(k) W n; is the self energy of pion due to a single A -h excitation I! fume; 2 1 J. _'_ (2-4—3) mm: ”5417.) k {umf (m-r n.) with the energy denominators 16 D,=ma-wn-o (2-4-4) D; = 7114-101" +w mA and m” are the A and nucleon masses respectively, the RNA coupling constant fit,“ f4?! = 0.32 [Ref. 11.9] and K' and. a: are the incident pion's momentum and energy, 6 is the nuclear density and ,(nfk) is the monopole form factor, 4"(k): AI-l‘z/A1+k1_gt (2-4-5) A, is the cut off mass parameter [Ref. 11.3] and A is the LLEE parameter [Ref. 11.10]: its range should be confined to l < 1‘ < 2 , [Ref. 11.11] . This takes ac- count of, among other things, the short range correlation between nucleons. 117k) can be written as ”(k)=_k2P{;(k) (2—4-6) Inclusion of this in the propagator means 1 a." 1 1. 1 . ~ - K»): (1 Pf")and, as at threshold to ‘l‘ , In. - 1. this means approximately that '/ku"_._p- '/k"1(“‘P) i.e. I 17 the propagator is amplified by a factor '/l-P Putting in the numerical values, at threshold, P: 1-19A*0,33). Note that if A was small, '/l—P could be very large. Performing the calculation, the second order 8 wave scattering parameter then becomes [Ref. 11.12] (x) z b =-__ _(41+ll:.:_'_) l— anPKF -'.. o It 7-: 1 PI 540-13105)" ] (2 4 7) Since P carries all the information about the nuclear medium, the overalfi many body contribution to the problem, i.e. the III-F> factor, amplifies the strength of the (H scattering parameter bo . The second term in Equation (2-4-7) simply shows the effect of the form factor. By extending the above formalism beyond pion thresh- old, we have shown that the second order scattering para- meters become (1) 1 Reb.__ ‘ 'Pko’ReI_ 87”“? ] (2-4—8a) 1 [‘F1("P)(A"-l'7‘) (l); 1 Into .-. - _‘_1. If. 1 ImI (2-4-8b) u l‘ I-P where 18 2. x, Rel' _.. an]: . J 1,“ 0M 1,. (2.4.9...) an 0° 2 1 z . K; IMI _-. 911K I J. (kFx)( 5‘2““ )dx (2—4-9b) o X By choosing An = 1.2 Gev and A = 1.6 , which is the appropriate value to use for pion absorption in nuclear matter [Ref. 11.13], discussed in detail in Chapter III, the combined first and second order S wave scattering parameter at threshold becomes - 0.066/;' which is about 2.5 times stronger than the original value given by B:,+ gu’without medium effect . This extra repulsive strength in the optical potential gives better fits to level shifts of pionc atoms. At this stage we would like to justify the validity of the sharp resonance approximation which was used to construct the pion self-energy. That is, the direct channel's energy denominator D. , given by Equation (2-4-5) , does not include the isobar's width. The exact D, should read D-‘z‘m -7?) -3. ~£r ‘ ‘ a A u I)" 1 A (2 4 10) where 3 .-. :2. (__ in“); _’..k (2-4-11) 19 and }( is the pion's momentum [Ref. 11.14] . At low energies, say Tn ‘5 50 Mev , the maximum value of (”6/3 = 0.02/a , which by comparing to the real part of D can be neglected. At higher energies the isobar's width 1 becomes larger. Consequently, for more accurate calcu- lations, one should take it into account. II.l II.2 II.3 11.4 11.5 11.6 II.7 II.8 11.9 II.10 II.11 II.12 II.13 II.14 20 REFERENCES FOR CHAPTER II J. Hamilton in, High Energy Physics , Vol. 1 and in, The Theory of Elementary Particles, Oxford 1959 A.E. Woodruff, Phys. Rev. 117 (1960) 1113 M. Brack, D.O. Riska, W. Weise, Nucl. Phys. A287 (1977) 425 J. Chai, D.O. Riska, Phys. Rev. C19 (1979) 1425 K.M. Watson, Phys. Rev. 89 (1953) 575 L.S. Kisslfhger, Phys. Rev. 98 (1955) 761 K.S. Stricker, Ph.D. Thesis MSU (1980) S. Barshay, G.B. Brown, M. Rho, Phys. Rev. Lett. 32 (1974) 787 e.g. 'G. Epstein, D.O. Riska, Zeit. Phys. A283 (1977) 193 M. Ericson, T.E.O. Ericson, Ann. Phys. 36 (1966) 323 E. Oset, W. Weise, Nucl. Phys. A319 (1966) 477 H. McManus, D.O. Riska, Preprint MSU (1980) D.O. Riska, H. Sarafian, Phys. Lett. 953 (1980) 185 M. Ko, D.O. Riska, Nucl. Phys. A312 (1978) 217 Chapter III PION ABSORPTION Introduction As pointed out in Chapter 1, the distinct feature of a pion which distinguishes it from other probes, is its absorption by a nucleus. A pion cannot be absorbed by a free nucleon. It can be absorbed by a bound nucleon, but more likely by a cluster of nucleons, which contains at least two nucleOns. Experimental data [Ref. 111.1] in the kinematical region of interest indicates that pion absorption in a ty- pical light nucleus often results in two-nucleon emission. So a two-body microscopic absorption model is investigated. The IIA scattering amplitude in the pion's energy region that we are interested in containes only two partial waves i.e. L a 0,1 . So, the two-nucleon absorption pro- cess is studied separately for S and P wave pion rescat- tering. In the next section details of the S wave absorption are presented. 3-1 S Wave Pion Absorption in Nuclear Matter True absorption by S wave pion rescattering is para— metrized by an Optical potential term'quadratic in the 21 22 nuclear density [Ref. 111.2] , which indicates absorption via a two-body mechanism, so .1 . 1w UOP" = ._ (“£340 (3-1-1) Here a: is the incident pion's energy, 6' is the nuclear density and B0 is the absorption parameter. On the other hand from the multiple scattering pion-nucleus series expansion the optical potential may be written as =+ (3-1-2) E -H°+¢ where the Optical 'potential (k'IU/k> is usually denoted by U, and (k’lV/k> is the nuclear matrix P" element of the elementary single pion-nucleon interaction operator. The pion momenta are K and A" . By inserting a complete set of nuclear states 2;”)(«Flai in the second order optical potential and splitting the nuclear Green's function into its dispersive and absorptive parts it becomes on U .-.- .1. 2': 7} _1_ .. °nS(E- u_E )JT. (3-.l-3) on .0. {1M3 {[E¢+U-E{ ‘ c+ ‘f {‘ Here 7;; is the matrix element of the two-nucleon pion absorptive Operator between the nuclear states i and if of energy E; and Ef respectivly and .11 is the nuclear 23 volume. For a Hermitian absorption operator Equation (3-1-3) beomes —- J- _‘_ 2. [_L_ - t‘7l5(E£+u-Ef).-I(7‘i{l (3-1-4) A {’6' EC+w-E+ : By comparing Equations (3-1-1) and (3-1-4) we get 1. RC‘BO: - L0 _ i.- ’73:, ‘P chzfl if“ Ec+‘-’-E,t (3-1-5) I’MB. ___ u f [Ty/J 5(E‘.+u-E{) ne‘n {an 1 In most of the 5 wave calculations [Ref. 111.3—4] the absorption operator is based on the diagram given below hl+ 11V . n:->-__4Lhn_-§;_--—‘r N) h Figure 111.1 5 wave pion absorption via two-body mechanism Here the S wave pion is scattered by one of the nucleons and absorbed by the second one. The scattering vertex can be described by Equation (2-1—1) and the absorption vertex by 24 the usual Hana Hamiltonian density [Ref. 111.5] A *4... =1.../,« if J. $ 1,. (3-.-.) By applying the standard Feynman rules, the absorption operator becomes T__—--.' 375 f ' 0" L l c _t' 11 (me'fle‘itc) (3-1—7) UN“ 1 l + The calculation of the matrix element of 7' is straight forward. Two cases are considered. The first is nuclear matter, wfiere the Fermi gas model is used to cbn- struct the initial pair wave function [Ref. 111.3]. The second deals with finite nuclei, where harmonic oscillator wavefunctions are used to construct the initial pair wave function [Ref. 111.4] . In both cases, the final pair nucleon wave functions are described in terms of out going plane waves, that is, distortion is not included. For threshold pions, nuclear matter calculation leads to Re B0 = 0.008/:4 and 1m ab = 0.02/74 [Ref. 111.3,6] . The energy dependence of the latter is shown in Figure 111.2 , and the energy dependence of Re 8 is given in [Ref. 111.3]. As compared to the value Obtained from pionic atom level widths, 1m B; = 0.042/: [Ref. 111.7], the absorption value is too small by a factor Of two. 25 MSU-83 -346 O.|2 , , .1 0.10 “0.08 ‘1 3 (130.06 E 0.04 0.02'— _ 0.00 1 J 1 1 0.0 0.4 0.8 I.2 I.6 2.0 q(fm" ') Figure 111.2 The imaginary component of the S wave pion absorption parameter of nuclear matter vs. pion momentum. Dashed line corresponds to a rescattered pion energy of 0.5 a: Solid line corresponds to a rescattered pion energy of 0.5 f‘ 26 We try to improve the result [Ref. 111.8] . By the same method as used in Chapter II we introduce the effect of the nuclear medium on the pion propagator. A Also the medium affects the absorption vertex, so that it is renormalized [Ref. 111.8] . These modifications can be taken into account by simply replacing the Yukawa one pion exchange potential 7,05) = (1+ fifiwl-flfi/flr <3-i-a) by the following integral 1 a: . K3.) Merck); 20:) 7,0 r)... 4 J1 ' n a). <3-1-9> 1 1 1 7V" o k +33% +71%“ where the vertex renormalization is given by (7“) = l/([_ 34 No(k;/k2{:(k)) (3-1-10) In both Of the above equations the pion's effective mass 11 . . is I“: V/‘l-‘J . The Integrand of Equation (3-1-9) contains a regular pion monopole form factor given by Equation (2-4-5) . Based on these modifications for threshold S wave 27 pions, the variation of the absorption parameter 1m B0 is studied as a function of the LLEB parameter i.e. ,A . The results are shown in Figure 111.3 . In this figure it is quite interesting to note that by choosing A1 = 1.6 - 1.8 which is exactly in the region of the preferred range of .A [Ref. 111.9] , we get the proper enhancement which we need to match the pionic atom data. Also this result agrees with the absorption parameter which is used in phenomenological IZ'A scattering optical potentials. A detailed study of the pion's self-energy reveals’ the possibilities of an interesting physical phenomenon. To make the point clear, one should recall the regular static OPE potential between two nucleons in vacuum, i.e. a. fl (aiJ;7(igli) 71¥Jkl (3-1-11) 4(7): '(fmw lf“ where the coupling constant is given in [Ref. 11.9] and q] is the pion's momentum transfer. Replacing the vacuum by a Fermi sea, the pion polarizes the medium. Polarization of the medium can be viewed as the virtual excitation of nucleon-hole as well as isobar-hole pairs car- rying the pion's quantum numbers, shown in Figure 111.4 28 MSU-83-348 O.|O I I I 0.081— €0.06... 5} m _. EO.O4_ 002- 0.0 1 1 1 0.0 I.O 2.0 3.0 4.0 x Figure 111.3 The imaginary component of the two-body 5 wave threshold pion absorption parameter of nuclear matter vs. LLEE parameter ,A 29 P-— -- --d Figure 111.4 Nuclear medium polarization due to pion propagation The polarization is determined by the strength of the P wave pion-nucleon coupling, the density 6 of the medium and by the strength of the short—range correlation accompa— nying the driven OPE interaction. The modification can then be written as (Inn) :. Vn(q)/e(q) (3-1-12) where the mesonic polarization effects are summarized in terms of the diamesic function 617) . If, in a much sim- plified picture, the repulsive short range correlations are I z I 1 -A.a1 parametrized by a (3? )( C.'C ) 5(1‘) where g' is related to the LLEE parameter .1 introduced in Chapter 11, g' =— A/3 30 e(q)_ .1. ['3' _ “MN/r)17—T‘f.]7l(7 g) (3-1-13) where Y(fi.(’) contains the information about each individual nucleon-hole or isobar-hole excitation, represented by a single bubble in Figure 111.4 . The size of the diamesic function is essentially determined by the competition between the attraction from OPE and the short-range repulsion controlled by g' . '/é(9) becomes maximum, at about 7= 3,“ and for g' = 0.4, and at regular nuclear matter density, (’3 0.17 fmJ,it becomes infinite. The singularity of 5(1) at 3 3r comes from the strong amplification of the pion's field. This amplification corresponds to the existance of a large number of pions, and nuclei should then undergo a phase transition. Nuclear structure would be reordered such that the ground state would have a strong pion-field component and long—range ordering of the nucleon spins would occur due to the spin dependence Of the pion-nucleon inter- action. However the empirical value of g' is = 0.6 - 0.8, corresponding to A 3 1.8 - 2.4 and condensation is avoided at nuclear matter density. A nucleus might, however, undergo a phase transition at higher density [Ref. 111.10]. In heavy ion collisions, where creation Of exotic Objects with densities higher than regular nuclear densities 31 are possible experimentally one might see the phase trans- ition phenomena. In Figure III.5 we have shown the behavior of 1m Bo with inclusion of the medium correction at .A a 1.5 vs. pion lab momentum. This shows the overalf many-body effect in S wave pion absorption. The curve is plotted for t0'al/2/‘. By comparing with Figure 111.2 one sees that not only does the many-body effect enhance the threshold value of 1m Bo , but it gives less variation with energy. In Figure 111.6 the effects of the medium correction at .A = 1.5 on the dispersive component Re B, of the ab- orption parameter is shown. There is no noticeable ampli- fication at low energies. To conclude the discusion of the S wave pion absorp- tion in nuclear matter, from a two-body mechanism point of view, we should talk about the ratio of pion absorption in two different possible initial isospin channels. To be more specific, for instance one could think of 11"absorption on initial np or pp nucleon pairs. The interesting quantity is R‘ = (K "Pd-h") (3-1-14) (n'PP —-"P) The observed ratio for the above quasifree processes is about 10 . The most SOphisticated calculation based on the two-body absorption mechanism [Ref. 111.6] predicts Rs 8 3 . The same work [Ref. 111.6] predicts 1m Ba 8 0.03QF 32 MSU-83-349 0J0 j I I I 0.02 ~— -‘ l J J l 0.0 0.4 0.8 '__l.2 l.6 2.0 q(fm') Figure 111.5 The imaginary component of the two-body 5 wave ab- sorption parameter of nuclear matter with inclusion of medium effects for A = 1.5 vs. pion momentum. 33 MSU-83-347 0.10 , , 0.08r- .0 o m 1 -ReB( W4) .0 o f 0.02- 0.00 l l 0.0 0.4 0.8 |.2 q(fm") Figure 111.6 The dispersive component of the 8 wave pion ab- sorption parameter of nuclear matter with resct- ted pion energy of 0.5‘p. and medium effects for .A a 1.5 . 34 which is close to the parameter used to fit pionic atom level widths [Ref. III.7] . Therefore [Ref.111.ll] , one conclusion could be the involvement of more than two nucleons in the absorption process. 3-2 S Wave Pion Absorption in Finite Nuclei Motivated by these results, we have studied the ef- fects of the pion self energy in S wave pion absorption in finite nuclei. Here, harmonic oscillator functions were used to construct the initial nucleon pair wave function and plane waves were used for knockout nucleons. Since the medium effect is calculated for nuclear mat- ter, the absorptidh rate is averaged over the nuclear density, i.e. r_—.. “norm: (3-2-1) To be consistent we set 4\= 1.5 and for convenience, instead of the absorption parameter we studied the absorp- tion cross-section. The absorption cross-section is _z D. 0': anf/k 1“ (TN NEH-mg) . (3-2—2) 0n the other hand the imaginary component of the scattering length is 35 Ina: _fl; (3-2-3) M? where the absorption rate l‘ is .1 anrr}: IT“! 3(EJ+Q-E;) (3-2—4) By combining the above three equations we get 0' :1 _‘LE. IMO. (3‘2’5) K Hitherto we have used for the vertex form factors Equation (3-2-5) the standard value .A = 1200 Mev , obtained from fits to ‘HL deuteron absorption. At this point we in— vestigate the dependence of the S wave absorption both on medium effects and the vertex form factor. The results are shown in Figures 111.7 and 111.8 , which show the absorption cross-section and the imaginary part of the S wave pion scattering length for ”He as a function of the pion incident momentum, for different choices of the vertex form factor parameter 1‘, and with and without medium effects. As in Chapter 11, for calculating medium effects we take A = 1.5. Referring to Figure III.7, absorption cross-section versus pion momentum, we have 2 curves. The top curve, 36 MSU-83-343 ‘ 1 I I 20- _, E :5- __ 7; 13 b" 10— .. Figure 111.7 S wave pion absorption cros-section of “He vs. pion momentum. The solid line on the scale of this drawing corresponds to the following three almost indistinguishable cases one 71' exchange with A; = 1200 Mev and no medeium effects one 7: exchange with A: = a and medium effects at A = 1.5 one 1: exchange with A“ = 1200 Mev and medium effects at A = 1.5 Diamonds correspond to one pion exchange with An = 700 Mev and with and without medium effects at A . 1.5 . 37 solid line, corresponds to the standard case .A = 1200 Mev, no medium correction and the bottom case, emphasized by diamonds, corresponds to /\ = 700 Mev, no medium correction. Thus we see that the absorption cross-section is extremely sensitive to the vertex parameter 1‘ , i.e. the size of the source, < fly/1; (Ia/A . Actually the top curve, solid line, also corresponds to the standard case A. = 1200 Mev, with medium correction calculated for A 1.5 , which is indistinguishable on the scale of the drawing from the case without medium cor- rection. The same is true for the bottom curve. Thus also for A = 700 Mev, the effect of the medium correction, for .K = 1.5, is neglbgible. Thus medium correction effetts for a small finite nucleus, i.e. "He , do not lead to an amplification of the absorption, unlike the case of nuclear matter, where medium effects gave an amplification factor 3 2. The difference between the two cases can be explained as the low effective density given by the folding procedure (3-2-1) (surface absorption). Figure 111.8 shows the same result for the imaginary part of the scattering length. The case of A = 1200 Mev, no medium effect, solid line, andp ‘A = 1200 Mev, medium effect for A = 1.5, dashed-dot line, are just distinguish- able from one another on this Scale. The experimental value at zero energy is also shown. It agrees with the atandard case A = 1200 mev, i.e. source radius, ‘=' 0.4 fm, with or without medium effect. It is clear that 1‘ = 700 Mev, i.e. 38 MSU-83-352 0.I0 l T I T 0.08“. -—4 006— ,,/” _ 1 '—' //// .— oo4r” _ gab—bffi'pO—‘w ‘- —- — — ‘1‘ ‘.‘. —1 0.02: .Ht‘ 4, 0.00 L 1 J L 0.0 0.4 0.8 I. 2 LB 2.0 q(fm") Figure 111.8 The imaginary component of the S wave pion scattering length of 9He vs. pion momentum. Dashed line corresponds to one 7: exchange with Ax= 0° and no medium effects. Dashed-dot line corresponds to one u: exchange with An: 1200 Mev and medium effects for A = 1.5 . Solid line corresponds to one 1!: exchange with At: 1200 Mev and no medium effects. Dashed-dot-dot line corresponds to one u? exchange with Aut= 700 Mev with and without medium effects for .A = 1.5 O is the experimental value for threshold pion [Ref. III.7] . 39 source radius 2 0.7 fm, gives a result that is much too small to agree with experiment. Referring back to Figure 111.7 we notice that the S wave pion absorption cross—section falls smoothly with in- creasing pion momentum, i.e. shows a typical l/u- behavior. So since the cross-section has an appreciable magnitude, 3 15 mb, only up to incident pion energies of 3 20 Mev, S wave pion absorption is confined to low energies. P wave rescattering absorption quickly becomes the dominant mode, and will be discussed in the next chapter. 3-3 On The Angu‘lar Distribution of (' 71' , 2N ) Experimental data for low energy pions indicates that knocked out nucleons are emitted back to back and have an isotropic angular distribution. In this section we study the angular dependence of the scattering cross- section for a typical ( RI, 2N ) reaction, assuming S wave absorption only. The calculation is carried out for qu . For an incident R" beam, we have the following four different ab- sorption possibilities, 40 * N’ a} n‘ ’C-' ——>—-— _. _ .. __* P #71 TI 4? a b . . f , Figure 111.9 The four possxble S wave R7 rescattering absorption mechanism The scattering cross section is 2.. if. s an: [1;] (3-3-1) JD. where 7;; is the matrix element of the transition ( absorption ) operator between the initial and final 41 nucleon pair states. The operator is given by Equation (3-1-7). Referring to Figure 111.1 the symmetrized amplitude can be rewritten in the following form 1': (c'+c‘)+t' + (828331... (8.8“) t <3-3-2) I 2. where the C’ and C are the usual iSOSpin operators of the first and second nucleon respectively and (+) stands for the spherical component of the given isospin operator 1 «if? aim/,3 A. #7,; with [ (3-3-3a) egg 8m 8'; a") 9 -agmqfiua'ta“)? = ~inuN/l-3A1 f/;; no: r): 7(04-0)’: (3-3-3b) Here OJ and oj'are the usual Pauli spin operators of the first and second nucleon respectively, ; is the incident pion momentum. The scattering lengths A. and 1‘ are given in Chapter II, the coupling constant {mm and the pion effective mass /H* are given in [Ref. 111.5]. The modified Yukawa function ‘%(ffi)is given by Equation 42 (3-1-8). The CM coordinate of the two nucleons is A A A R = 1/2( r. + r1 ) and the relative separation of two nucleons A _ A A r - rI - r1 Since we are interested in low energy pions, the above amplitude can be approximated. This approximation is a trivial one, that is ' c‘é _ , r“ ,_. (3-3-4) $m(i1.r)-0 , CKP(L?-R)—I As the pion brings in no appreciable momentum, the center of mass motion remains unchanged so we only have to consider the relative wave function of the two nucleons. Equations (3-3-3a) and (3-3-3b) become I 2 I. | A A.) . = {my a. in: win . 361)°$03‘ii°"<3-3-Sa> *1 $ 0 A. .3“ b .‘a t :: {mm 0.3 )il' /J; x(/ar)(u+w)(a+a- ).1(os(i1-t93-3—Sb) To calculate Ti" it is more convenient to label the initial and final nucleon pair states in terms of total isospin 'T and spin 5' quantum numbers. That is 10:17.39) , and If>=/Tf.s{> . Hence for' [5% pm, pp ) i.e. Figure III.Sa,b non- vanishing spin isospin matrix elements become 43 712,- z - (3-3-6) 1 T;»;| (6+6? (at. 851% n, 7.6:.) and for It( nn, pn ) i. e. Figure 111. 9c, d the non vanishing spin isospin matrix element becomes Ts! I 1 ' -1. T62. I“: < fh’s‘: ’(C 4C) (ff-0 )Ihn, 363°) (3-3-7) In this special channel only one term survives so we look at this case. The complete form of Equation (3-3—7) is Ta. I T'-.".’ (3-3-8) Tiff :8 (P0; g‘2'l (C {13"}... 6'1““, ;:=O> Since the initial nn pair is in the singlet spin and triplet isospin state, it's spatial wave function is even. We confine ourselves to orbital angular momentum L = O A harmonic oscillator basis is used to construct the spatial wave function, i.e. .. A (Alt): E1091?” (3-3—9) 44 where n = l = m = 0. We use plane waves to describe the spatial components of the final nucleon states and since the final pn pair is coupled to an isospin and spin triplet state, the anti- symmetrized spatial wave function is n . . f x. .. #(r): I Os J (k, y c“ 7 (It!) (3-3-10) f fl ) ‘i d ‘g'fl:? ‘5'; where k; is the relative momentum of the two ejected nu- cleons, shown in Figure 111.10 “b qFW 0»: Figure 111.10 Schematic ( 71, 2N ) reaction Codi??? is expanded in terms of the partial waves (+) e Co:(' ‘8) uni a) J (1 r) *(‘1 f) (3-3-11) {:1- = , .1 7 *7 i 4 Ian .&M 45 where (+) means only even values for l are allowed. By setting 1 = 0 in Equation (3-3-11) , appropriate for low energy pions, conservation of angular momentum confines the final angular momentum in Equation (3-3-10) to l = l . With these approximations the scattering cross section i.e. Equation (3-3-1) becomes * A 2' I(/' Jk{) thn‘kf)’ (3-3-12) where the radial transition matrix element 1 is defined as ‘° <3-3-13) 1(ffk;)= a} J,( "W Hi1" >1 (It) Rutdrad“ -1 here 2° surpLOr-l) and for He we used 0= 0.493 fm . ‘ 2. The polarization information of the ejected paired nucleons is carried by Ylmnlfl . Therefore, by summing over all the possible polarization directions we get dd” I at a .._— = —— 1‘ , ’l (3-3-14) L0. to l f ‘5, where the N contains all the energy independent factors. Any anisotropy in scattering cross section i.e. Equation (3-3-14) is contained in the radial matrix element 1 , which depends on 3+ . But conservation of energy and momentum give no dependance of k on a< and the emis- .; sion is isotropic. III.1 111.2 111.3 111.4 III.5 111.6 III.7 III.8 III.9 111.10 III.11 46 REFERENCES FOR CHAPTER III e.g. M.E. Nordberg, K.E. Kinsey, R.F. Burman Phys. Rev. 165 (1968) 1069 M. Krell, T.E.0. Ericson, Nucl. Phys. B11 (1969) 521 J. Chai, D.0. Riska, Phys. Rev. C19 (1979) 1425 G.F. Bertsch, D.0. Riska, Phys. Rev. C18 (1978) 317 Nucleon-Nucleon Interaction, G.E. Brown, A.D.Jackson The Niels Bohr Institute, Nordita F.Hachenberg, H.J. Priner, Ann. Phys. 112 (1978) 401 J. Huffner, Phys. Rep. 21C (1975) l D.0. Riska, H. Sarafian, Phys. Lett. 958 (1980) 185 E. 0set, W. Weise, Nucl. Phys. A319 (1966) 477 S.0. Backman, W. Weise, Mesons in Nuclei Vol. 3 D. Gotta et al, Phys. Lett. 112B (1982) 129 Chapter IV P WAVE P I 0N ABSORPT I 0N 4-1 In this section we concentrate our study on P wave pion absorption in a nucleus. Free )[N total scattering cross sections show a res- onance peak at about 200 Mev pion lab energy, with an ap- proximate width of 110 - 120 Mev. This well established resonance is interpreted as the formation of an unstable intermediate resonance state in spin and iSOSpin 3/2 chan- nel i.e. the A. resonance. Based on this fact, P wave pion should be absorbed by a cluster of at least two nucleons by the process I To indicate the two-body absorption mechanism the optical potential is parameterized in the following form [Ref. 1v.1] 0 6c ‘6‘ (4 1 1) where C. is the P wave pion absorption parameter. The imaginary component of C is related to the absorption rate r. by Equation (3-2-4) via [Ref. IV.2] 47 48 I." Co _ w __ r (4-1-2) WI: k‘f‘fl where I‘ is the incident pion's momentum and [I is the nu— clear volume. Calculation of the absorption rate per unit volume i.e. (In is based on the absorption amplitude constructed from the following two diagrams, Figure IV.1 MA. I u N +11 A A x-O- wL—J——-r .. 1, .4 .. Figure IV.1 The direct and crossed channels of the P wave pion rescattering absorption mechanism with intermediate one [c exchange where Figure 1V.la,b are refered to as the direct and cros- sed channels. The direct channel has an obvious physical meaning, the crossed channel is interpreted as the destruction by an incident real pion of an intermediate 4. which is formed already in the target nucleus by exchange of a virtual meson. The formation of the intermediate A. resonance on one 49 of the nucleons has a direct effect on the energy dependence of the absorption. Im Co vs. pion lab energy has a resonance shape. A The A decays not only into IN , (A—v'KN ) but also into (9N , (A... e N ) . Both of these two modes should be included in the construction of the absorption am- plitude. So the following processes are added, Figure 1V.2 N fimR/wi” Ira—Ni V A A lfiib-ma) w 9 u‘ 4N ~A 1N Figure 1V.2 The direct and crossed channels of the P wave pion rescattering absorption mechanism with intermediate (3 exchange To construct the absorption amplitude in addition to Equation (2-1-1), we use the Hamiltonian densities, *1 A Hm“: fl/f ’7 § . 9+4? 3' X 4. (LC. (4-1-3a) 50 ”(NA Lean/am 1+6. (1736):)! .1 he. (4-1-3b) ”(NM :2 31’4" (14.x) X+$+( 3X6)€.?§X (4-1-3C) l where /41C = 0.55 ,‘K = 6.6 and = 6 G /5 (1+K) 9? 96A gf’ r as predicted in the static quark model [Ref. 1V.3] with . 1 RNA coupling constant *4/41! = 0.32 . Here g and X are the nucleon spinor and isospinor A .1 and g and x are the A vector spinor and vector iso- E A . spinor. The pion isovector field operator is 43 and the . . . .x rho_vector meson isovector field operator is e , the mass of the pion and nucleon are denoted It and m respectively. By applying the standard Feynman rules for Figure IV.1 the absorption amplitude becomes . " 8.72 _ _ T=c $— -‘7‘I-5% {man/f. ”All“; ;(k:+/“‘°' ,1.) I: (4 l 4) 4 A (A, .31 I;l 7:: -51qu Wrxt2] where K}, and (0' are the momentum and energy of the rescat- tered pion. The (+) component of isospin operator 5115 defined t:,é(t:+€t;) with an analogous definition for that of the (£532, . The energy denominator D is 51 _ ‘ (4-1-5) where k1 (4-1-6) 1». w. m A... 1 here m is the A mass, and the A '5 width is given by A Equation (2-4-11) . For (a meson exchange i.e. Figure 1V.2a the absorption E amplitude becomes T=_‘-.). _L 3__f_3 ‘fod I _1: ~‘ 1 «:2. if 3M 1) k,‘+m;.u" (4+7) d. ‘L k: [1; 81-513’xa‘)(818‘)+]]. '1: Spin and isospin matrix element calculations of these operators between initial and final paired nucleons are quite complicated. The results are given in [Ref. IV.4] However, for K and (3 mesons, there are the following relations ‘f-WYoU-fr) gage an‘Fr) (4-1-8a) §n )3 (If?) __... .. EC lib»? I") (4-1-8b) where E; and E? contain all the kinematical factors given in [Ref. IV.4] and the effective masses of pion and rho mesons 52 ‘3?"‘7 are given by lf=¢ftud and 1»? =J'Me 4" . Thus ( exchange adds to It exchange for the central part, but reduces the tensor part of the interaction. The nuclear medium effect is shown diagramatically in Figure 1V.3 . ”i TC...__1,J L Al .14 Figure 1V.3 The direct channel of the P wave pion rescat- tering absorption mechanism with intermediate 7t'* I exchange and the medium effects The calculation is similar to that of the S wave. That is, the calculation is divided into two categories. The first is P wave absorption in nuclear matter [Ref. 1V.2,4] . The Fermi gas model is used to construct the initial paired nucleon wave function and plane waves are used to describe the final knockout nucleons. In the second category we deal with P wave pion aborption in a finite nucleus which is discussed later in this section. The results for the absorp- tion parameter 1m Co for nuclear matter, using various form 53 factors, with and without medium corrections are shown in Figure 1V.4. Details of numerical analysis reveals that the trans- ition of paired nucleons coupled to L = 0 to L' = 2 chan- nel, via the tensor interaction dominates. So, for the sake of simplicity only the contribution of this channel is shown in Figure IV.4 . Monopole form factors i.e. Equation (2-4-5) with standard cut off mass parameters i.e. A'-= 1200 Mev and A€==2000 Mev were used. These were obtained by fitting to experiments on the absorption of pions by deuterons. They correspond to a nucleon source size of 0.25 fm for f Vs and of 0.4 fm for K's . For threshold pions 1m Co is 0.043/‘ . The value of 1m Co estimated from pionic atom level width varies between 0.036/i‘to 0.07/17‘ [Ref. IV. 5-7] depending on the LLEE parameter. By confining our calculations to the standard cut off mass parameter i.e. parameters used in Itd_a-pp [Ref. IV.8] and by comparing curves..._...., )t,exchange only,and.....- , 7F+ ( ex- change,of Figure 1V.4 one can see that the destructive inter— ference between 71and f exchange in the tensor part of the interaction is essential in obtaining a reasonable value of 1m Co. It is interesting to note that curve———u— Figure IV.4, that is combined 7r and (’ meson exchange with the standard cut off mass parameters, can almost be reproduced by 7r ex- 54 Figure 1v.4 The imaginary component of the two-body P wave pion absorption parameter of nuclear matter vs. pion momentum. Dashed-dot line corresponds to one It exchange with A" - 1200 Mev and no medium effects. Dashed line corresponds to K'+ 6’ exchange with Alr' 1200 Mev and Ae= 2000 Mev and no medium effects. Dashed-dot-dot line corresponds to one fl' exchange with AI== 1200 Mev and medium effects at A.= 1.5 . Solid line corresponds to fl’+ 6 exchange with A; . 1200 Mev and Ac 2 2000 Mev and medium effects at .A . 1.5 . Dashed-slash line corresponds to one n' exchange with A“. a 700 Mev and no medium effects. 55 MSU-83-340 2.0 l.6 |.2 0.8 Saws. _, ~ ~ _ _ _ q — _1\|M.11h:11aiiiml _.\\m\\ x . X‘s-‘3 .\ I\\\ o ‘3‘ \.\\\. \’\\\ I \o .‘I‘ o\.\\IIo v\t \ ll \.. . \olliuuul.‘ a\ \\ ..\.. .\\. »\ f../ .../l, / .I. . . // 1 1| I. .10., lilo/o // / col..l._’ - ./. s / . ..l. /. /. / l I. 4 /o /\/ 1 /;///, ../ /C 5. _ _ _ _ r _ . _ t _ _ _ . _ MW LL01 nu nu nu m w w m. 2 MW 8. 6. m. 2 0. 2. 0. L ... I .... 0 0 O .0. o ¢.>H «havoc 0.0 0.4 q(fm") 56 change alone with a reduced cut off parameter, Ax= 700 Mev, curve—_L—J——L.Figure IV.4 . This corresponds to a nucleon source size of 0.7 fm for It's , which is closer to the physical size of the nucleon, and that given by the Cloudy Bag Model of Thomas et al, [Ref. IV.9] To include nuclear matter effects the central and tensor components of the OBE potential i.e. Equation (4-1-8a,b), are replaced by the following integrals ‘9 1 a? j ”uh/x”) ”H‘sjn‘h) dk Ir/‘Clflfl 0 ft, k1- “01+ 71"“, y;()fiI)-—a— (4—1-9) where the vertex renormalization (Wk) and pion's self- energy N“) are given by Equations (3-1-10) and (2-4-2) respectively. By comparing curve-u——-, standard parameters with no medium effect, and-—,standard parameters with medium effect for 1:: 1.5, we conclude that the many-body correction in— creases 1m Co at all energies. The P wave pion absorption calculation for qu proceeds as in the S wave case. The result is shown in Figure IV.5 . The general feature of 1m Co vs. pion momentum is similar to the be- havior of ‘Im Co in nuclear matter. The dashed curve shows the result, without medium correction, for the standard 57 MSU-83-353 I2 I r F I H |.0 0.9 r 0.8 '5 0.7 g 0.6 g 0.5 0.4 0.3 0.2 0.! 0.0 -” 1 J J L 0.0 0.4 0.8 'l.2 |.6 q(fm"") Figure 1v.5 The imaginary component of the two-body P wave pion absorption parameter of "He vs. pion momentum. Dashed line corresponds to 3+ 9 exchange with .Ar - 1200 Mev and hf = 2000 Mev and no medium effects. Solid line corresponds to n + 9 exchange with Mr - 1200 Mev and Ac - 2000 Mev with medium effects at ).- 1.5 58 parameters. The effect of including the medium corrections for the same parameters is shown by the solid line. The overall amplification due to the many-body effect is quite similar to the amplification in the nuclear matter case. So one concludes that, contrary to the case of S wave pions, the medium correction does have an appreciable effect even in a small finite nucleus, for P wave absorption. To conclude we have calculated S and P wave rescat- tering absorption and the second order S wave repulsive contribution to the optical potentials using parameters fit- ted to pion absorption on the deuteron. These give, in general, too small an effect. However, values are amplified when we take into account an effect not present in the deuteron, the self energy of a propagating pion in the average nuclear field. In fact the pion interacts so strong— ly with the nuclear medium that this amplification could become extremely large ( nuclear opalescence , pion con- densation ) but for the damping casued by two nucleon correlations and other effects parametrized by the LLEE variable A. or Landau parameter 9' . For g' = 0.4 , A» = 1.2 nuclear opalescent effects would be prominent. However g' is usually estimated to be 9' = 0.65 - 0.85 with A = 1.9 - 2.5 . We used A = 1.5 so we probably have a slight overestimate of medium cor- rections. The results are the same for P wave absorption, whether we use the conventional I + (’ exchange with nu- 59 Ill cleon source size 0.4 fm, or R? exchange alone with source size 2 0.7 fm . However the S wave effects, which depends on pion exchange alone would become negligibly small in the latter case. 4-2 Isospin Dependence of Pion Absorption by Nucleon Pairs in the He Isotopes The P wave rescattering mechanism also predicts the ratio of the absorption of a pion on a pair of nucleons in a T = 0 state to that on a pair in the T = 1 state, which is the. only channel for the process 115-41 ( pp )4» pn . This ratio R = “T'°)/U(T'U was measured recently by Ashery et al [Ref. IV.10] in experiments with 165 Mev 7t’ and It. on IHe and “He . They found that R 3 50 , with large errors, i.e. absorption on two nucleons in a T = 1 state is extremely improbable. The absorption amplitude is the sum of two terms, one from the direct, the other from the crossed channel, shown in Figure IV.1 . The amplitudes are A .5' 4+; A+ 7; g {mm/l. {nub/r 03.11, t: ..‘n,a:.) Sy": .5401?“ . (4-2-la) $.12?” + 127-3 a a _‘I A A 4 Tb = fawn/l“ {mu/f- 61-h! C“ 3'1““) 8 ° kn 1;! @1136) .n‘ .. a (4-2-lb) . Saki” +"‘='r9‘ 60 where the coupling constants {mm and RNA are given in the previous sections. The 6‘ and 5‘ are the usual Pauli's nucleon spin and isospin operators, the S and T. are the transition spin and isoSpin operators of A isobar [Ref. IV.11] . GD and G: are the direct and crossed channel iso- bar propagators, 6 = ”A-..” 151/3M -1..- l"A (4—2-2a) 1:. .1 6C: ‘MA- m 4p‘l/JMA 4w (4-2—2b) If we take the pion orbital angular momentum relative to the nucleon pair to be lm= 1 , we can easily see that the direct channel gives no contribution to absorption by a T l nucleon pair in a state of relative angular momentum L 0 , and hence by the Pauli principle S = 0 . The inter- + mediate state A,n , then has spin parity Ju= l . It could have isospin T = l or 2 but as it has to decay into two nucleons, only T = l is allowed. So the final state I + . NN has to have ‘1 = l , T = l , i.e. S = l . L = 0 or 2 and T = l . But such a state is forbidden by the Pauli principle. 50 the only contribution comes from the crossed channel. -Similarly the only contribution to absorption from 61 a T = 0 initial pair state comes from the direct channel. J0'(TaO) .' QB don-d) 9: A. I as first pointed out by Then for [N'= l , Chai and Riska [Ref. 1V.4] when discussing the ratio of final nn to np pairs produced in the capture of pions from the 2p level in pionic atoms. This gives a ratio about = 200 for 165 Mev pions, and is independent of any mechanism except the assumption that absorption is a two step process, proceeding through the formation of a A . It remains to be seen whether other pion partial waves, :3 = 0 and 2 contribute to the small JO'( T = 0 ) cross section. To simplify the calculation, we used only pion exchange with a cut off parameter Auw= 0.8 Gev instead of II + f exchange with A“; 1.2 and AI = 2.0 Gev as in the standard model, as we have already seen that this gives equivalent results in the calculation of the absorp- tion rate. The results are given in Table IV.1 . 62 mmm n r' s' 1' a hulens 10 15 10 15 T' 0. S ‘ l. L " 0 0 l l i 0.027 0.40 2.9 9.7 7.3 1 l 0 2 l‘ 4.9 21 $2 30 2 1 1 3 00% on L0 42 16 T- 1. s- 0.1. - 0 0 1 1 1 0.008 0.086 0.44 1.1 0.63 1 0 1 2 0.076 0.17 0.24 0.26 0.26 2 1 1 3 0.002 0.023 0.14 0.40 0.26 TABLE IV. 1 Pion absorption cross section in several channels as a function of pion momentum k are compared, where l is the angular momentum of the incoming pion. These numbers are normalized to the (T,S,L) = (0,1,0) to (T‘,S'.L') = (1,0,2) transition at-k = 0.5 as indicated by * This shows that 1n = l is indeed dominant in the T = 0 channel, but that the other partial waves are more im- portant, except at threshold, in the T = 1 channel. Adding up the contributions from all three partial waves we find the ratio is reduced to R = 40 at 165 Mev consistent with the experimental results [Ref. 1V.l3], as plotted in Figure IVOGO 63 MSU-83-350 0 58 172 T1r New I l R = dq(T=0) dq(T=l) 2001- ». ... I \ I \ I \ I I ‘\ / 1£7:| I \ I \ I \ 100 - ’ ‘ .. I \ \ all ~_____~‘~‘ l 1 ‘CL() l.() 2L!) " [rT‘1i] Figure 1V.6 T A The pion absorption ratio R = j:fr::j vs. pion momentum. Dashed line corresponds to in: l and It exchange alone with An,= 800 Mev. Solid line corresponds to all of the possible partial wave contributions. IV. IV. IV. IV. IV. IV. IV. IV. IV. IV.10 IV.ll IV.12 l 2 3 5 6 8 64 REFERENCES FOR CHAPTER IV E. Oset, W. Weise, R. Brockmann, Phys. Lett. 82B (1979) 344 G. Miller, Phys. Rev. C16 (1977) 2355 R. Rockmore, E. Kanter, P. Goode, Phys. Lett. 77B (1978) 149 C.M. Ko, D.0. Riska, Nucl. Phys. A312 (1978) 217 P. Haapakoski, Phys. Lett. 48B (1974) 307 J. Chai, D.0. Riska, Nucl. Phys. A329 (1979) 429 J. Huffner, Phys. Rep. 21C (1975) l L. Tauscher, W. Schneider, 2. Phys. 271 (1974) 409 C.B. Dove, Ann. of Phys. 79 (1973) 441 D.0. Riska, M. Brack, W. Weise, Nucl. Phys. A287 (1977) 425 G.A. Miller, A.W. Thomas, S. Theberge, Phys. Lett. 91B (1980) 192 S. Theberge, A.W. Thomas, G.A. Miller, Phys. Rev. D22 (1980) 2383 D. Ashery at al. , Phys. Rev. Lett. 47 (1981) 895 G.E. Brown, W. Weise, Phys. Rep. 21C (1975) 1 H. Toki, H. Sarafian, Phys. Lett. 119B (1982) 285 Chapter V AN ALTERNATIVE TREATMENT OF P WAVE PION ABSORPTION IN THE OPTICAL MODEL 5-1 We would like to mention another way of incorporating P wave pion absorption in the optical model, i.e. the A -h model. previously we parameterized the absorption as a term in the Optical model where Co was calculated via the absorption rate, Equation (3-2-4) . This gave a resonant term in the optical potential adding to the resonant behavior without absorption. Both terms are due to the formation of the intermediate A resonance, so it may be better to display this explicitly ( A -h model ) . The P wave part of the optical potential is written without absorption, .16.:th = V 101'ng (5-1—2) where . 4“- um'c.(> _., -91.; [fj'f %(O)+ 041-80] 15-1-31 65 66 and 5'. 5m (5—1-4) 04 (U):- €(V)/UR -0 _- Here 0'" is the isobar width given by Equation (214-11) due to the process AapTl'd-N . In the presence of nuclear matter [3". is modified 1) by Pauli blocking in the nuclear medium ff,“ ... Q9,“ 2) by true absorption A+ N -o- N + N . This second pro- cess depends on the nuclear density. Its effect is to increase the width by an additional width, the absorption width (3;: nclfl/(‘o so in this formalism the effect of the nuclear medium is to change Ff?" , Qm—vQQru-ifie‘rt So UAIN’ .... U‘UI’) , where (74(0)): (”fl/“1"“ .54; (game 4. Hem/f.) (5-1-5) and U 4 *1 1 ‘3 a r’ 4 Au m . .7. I //. V): 4‘“) +UA(-u)]V (5+5) The basic task is then to calculate I: i.e. the rate of the process A+ N —)- N + N . As we have already seen, the dominant process in absorption goes from, for It? , a ( A'Hn ) intermediate state with quantum numbers T = l L = 0 S = 2 which goes via the tensor force to a final state of two protons with 67 quantum numbers T'= l L'= 2 S'= 0, so we consider only this process. The calculation is done for «He replacing a proton by a df+ isobar with the same spatial wave function, and the standard " 17+ f " model is used for the process. The transition rate is run 1:: at: -E,.-.,;IT,/’ (5-1-7) {6 For pion exchange the vertices are given by KN Sf ”=4/rijgfoav €$¢ (5-1-8) £1r~A=f7lu ‘7 a» I. $31“; kc. (5-1-9) where 4‘ , 4) , ‘A, are the nucleon, pion and isobar's fields and C and T are the nucleon isospin and isobar's transition isospin operator [Ref. v.1] , and coupling con- 1 1 stants are f /4It = 0.08 and ,C‘M'lt = 0.32 The transition operator is then 3 4+ .3 A.’ Tum.) N") = {{7/‘1/J/unl'Spa"!Z(x)+5uz’lfl)]7:.éa(5rl-10) i 1 1 where f =¢f 44' is the pion effective mass and I a D = 1/2/‘ is the resacttered pion energy, 0' is nucleon A - spin and S is the isobar transition spin operator [Ref.V.l]. 68 Only the second term is important yaw: (H- 3A... 3&8) VJ!” (5-1-11) 51.; = 33,.r0'),-r_3,.a (5-1-12) The matrix elements of the isospin operator give , _J_‘ T (T ({{WTJ 7,. ”a, “39%) = "” :17'3H1H1.’ (5-1—13) 31, '1 T ..." i t } gun; ngxiu any ’1 V1 T while for the spin operator 7;! I ‘f (3‘ _:;. A (S {1)0 Sniff-1);): J)? yéhl‘r) (5-1-14) where M is the 2 component of the initial spin state. For “He the harmonic oscillator basis wave functions are exfl-ivfiz) , cxfl-iIVV-R) , for each " nucleon " so that initial pair wave function behaves as exr(-vR1)eq(.$9r1) , and the final pair wave function becomes 410:1): [if 9“? (‘.Z°a°5(£}‘?) here CM and relative co- 1 ordinates and momenta are defined as A ‘ a .1 A A 3 R={(r,+rz) akl+kl I (5-1—15) -. .- a ? .3 a r :- rl"_ r7- k{32‘:. (In-Jet) To handle the final state relative wave functions, 69 expansion (3-3-11) is used of which only the l = 2 term gives a contribution. To calculate the rate, the 5- function giving con- servation of energy is _ ‘ . ‘11 z 1 2‘. an: 3.44)., M3 { 5154m.n.f)545an;* (5-1-16) where E; = -B , B is the nucleon binding energy and m is the nucleon mass. The integration yields . 2 rl’-’- 378' I log, (5—1—17) where .I.1;is given by an . ¥ (5—1-18) I,” j J1(l:*r))$[lbr)exp(—Lvr1)rzdr Y 7t_______ where K; = Jhlf-B) , and rc is a cut off in the radial integration to take account of short—range corre- lations. We now put in form factors at the vertices. To simplify matter we take half-dipole form factors V (5-1-19) ¥(?1)=(x1_f1/x1+71) 1 at each interaction vertex instead of the regular dipole form factors, but keeping < r1'> the same, i.e. A = yJSA. These form factors modify the radial integral 70 ”31J2(h*f)yl (f¥r)¢‘Pl-—VT)YJV+ (5-1_20) ;.'“J:6 .11“! r))I ”whirl-L w )r Jr .. K‘I J.(k,.r)y,mmpt— m )r J: V" and also I; gains a multiplicative factor (A-7P4fit {I} e - exchange is taken into account in the same way as in Section (4-1) . Figure V.l shows the behavior of I] vs. cut off radius r for the standard parameter set. I] is rela- c tively insensitive to this parameter up to r c 1 fm. We conclude that short range correlations are not crucial, with the standard form factors, to the eval- uation of .Fi . The value of F; 3 55 Mev . This is in order of magnitude agreement with the empirical value of r, g 100 Mev. Medium corrections would presumably increase these values, in better agreement with empirical values, but we have not done this calculation. It should be mentioned that to get agreement with experiment using the A - h model, a more complex non— local expresion has to be used, where the single particle potential seen by the .A in nuclear matter, including a. strong spin-orbit interaction, is important. A comparision of the forms of this simplified .A - h optical potential labelled " microscopic " with the Kisslinger type labelled " macroscopic " , is shown in Table v.1 . 71 um a: 5 mo mmuoOuQ AE‘VS .mamcmc uuo uau as» .m> paonmouzu z+z¢z+ V a...» no can.» on... H.> uuammh P _ _mmummunmz 72 d o «tunaufsha a \ Mao . a . any I \ .qmwnflm... «Sen «1933. Z w T35... a .3 a... II?||II 5 o q HIIIIIIJ IHIIIIIJP I N I) // \ / Inn u— .R Emvcmcuoe >Oonuozu occ oco Emwcmcome xoooaozu ocm mco co mEcou cw _mmucouoo .muvuoo ocu mo covuuu.couoEmLma mo mEcou cw _m_ucouod _mUwuoo on“ mo comum~mcmuoEmLmd .m 0 In flko UR: \V 3.... nus? 3 .34 canto zuocmu an _m.ucmuod _momudo o>mz d cmoco cuocm~ an _mmucmuoo _momuoo o>mx a .N .n .20 gusauca 3 K «a km to k u 9....0 a R I a skufljéutw..- u 3.9153 .6 s u . Q hllllllulllfl ‘ I \ I \ . / \ I/ H N Emcmmwu mcvxo__ou ocu co uwwcw ommca Damon >__mu.aoumocu.E noun—30.00 m, xuwucono 06mm och uo mELmu c+ o coquMLmo mcwtouumum com—unclco.Q o>mz a .. uwooumoco*2 omaoomocumz uraoumocumz : Dw__mnm_ oa>u Lomcw_mmvx ocu cuwz : u*ooumocu*2 : Uo__oom_ _m.ucodo _oovuoo.+.¢. mo mELou mCu mo com¢cmoEoo < —.> w4m m4m

\ :2 C1??" /° ‘\>\'-‘ .. - ‘-\ E ./ \\ "' 0.3t- *fi 0.2.. I 0 .l‘_ // —‘ ”3"“ I I 1 1 _j__ - O 50 IOO ISO ZOO 250 T7r( Mev) Figure v.3 The imaginary component of the P wave pion-nucleus optical potential vs. pion kinetic energy. Dashed-dot line corresponds to the microscopic treat- ment (9) with P = 0.5 nuclear matter density. Solid line corresponds to the impulse approximation (Co) Dashed line corresponds to the macroscopic treatment (f) with €= 0.5 nuclear matter density and A= 1.5 77 3) The original M.I.T. bag [Ref. v.4] has a radius of =: 1 fm and makes the nucleon very ' soft " . Nucleons interpenetrate easily, but remain in quark clusters of three. Inside a nucleus these clusters of three quarks‘ behave like a nucleon of mass 0.75 m which, it is claimed, is seen in e e' p experiments in nuclei. A disadvantage is that six quark bags are not very massive, so that the repulsive force seen in N-N scattering experiments must be due entirely to the large, := 2 fm, non locality in the interaction arising from quark dynamics. The point is that if nucleons are bags of radius R then, when their centers are a distance r = 2R apart, the bags begin to touch and quark—gluon dynamics rather than meson exchange begins to be the predominant interaction. Actually the percentage overlap increasese only slowly with decreasing distance between centers. At r = 1.2 R the percentage overlap is =:12% . At this point we might expect the bag to collapse into a six-quark bag, as the quarks are relativistic and can rearrange the distribution instantaneously. We could take this to be the upper limit for a critical distance, rc below which the meson exchange model would be invalid. The lower limit for rc we could take empirically from nucleon: nucleon scattering as the hard core radius a 0.5 fm. The form factors used in our calculation would correspond to nucleon size if the root mean square radius given by the parameter A g—-"’bag radius R. The meson exchange model might be expected to give the 78 dominant contribution to absorption if most of the contribution comes from that part of the radial integrals from distances r 3 re . Ignorring this for the moment, Figure V.4 shows the P wave absorption parameter Im Co as a function of pion momentum for 4-cases, pion exchange only with form factor parameter A = 800 , 700 and 400 Mev, corresponding to root mean square radii of 0.6 , 0.7 and 1.2 fm respectively, and the standard " ix + f " model with An: 1200 Mev, or root mean square radius of 0.4 fm. It is seen that R‘ exchange only with I\ = 700 Mev, or radius of 0.7 fm, agrees with the standard model. However n - exchange alone with 4‘ = 400 Mev, i.e. radius 1.2 fm gives much too small a result. So if the M.I.T. bag radius is correct, nothing comes from meson exchange and everything is due to quark-gluon dynamics. However, A = 700 Mev with radius 0.7 fm, corresponding to the cloudy bag model is 0.x. The stsndard model is consistant, naturally, with G.E. Brown's little bag model. Now let us look at the effect of a lower cut-off due to overlapping sources. Figure v.5 shows Im Co at threshold as a function of the lower cut off radius rc in the radial integral. For the case of interest, pion exchange alone with A» = 700 Mev or 800 Mev, and the standard model, the It results are quite insensitive to a cut off radius up to 0.9 fm, and so consistant with the cloudy bag model, but Ill r c - again not with the M.I.T. bag radius. We conclude that as far as the P wave absorption is 79 MSU-8323M T I I I NU-btflbiflmm Figure V.4 The imaginary component of the two-body P wave pion absorption parameter of nuclear matter vs. pion momentum. Dashed line corresponds to one 7: exchange with An= 800 Mev. Solid line corresponds to 7t+ P exchange with A“: 1200 Mev and AP = 2000 Mev (standard set). Dashed-dot line corresponds to one u: exchange with Au= 700 Mev. Dashed-dot-dot line corresponds to one 72: exchange with A; = 400 Mev. 80 Figure v.5 The imaginary component of the P wave pion ab- sorption parameter in nuclear matter vs. the cut off radius. Dashed-dot-dot line corresponds to one u: exchange with A; = 1200 Mev . Dashed line corresponds to one If exchange with M: = 800 Mev . Solid line corresponds to 7: + (’ exchange with A3,: 1200 Mev and he = 2000 Mev (standard set). Dashed-dot line corresponds to one n: exchange with AI. = 700 Mev . 81 — _ Nvm umm .2m _2 m.> muzmflm 82 concerned , the standard model is consistant with the little bag, the calculation with F— exchange only is consistant with the cloudy bag model, but elimination of the 6 may not be consistant with other nuclear physics phenomenology. In particular it is not consistant with S wave absorption, where An = 1200 Mev is required to get any appreciable absorption. If the M.I.T. bag size is correct, then the parameters of the model have no direct physical meaning, they simply are a parameterization of a non-local inter- action in local form. The question of bag dynamics has not been directly ad- dressed by any one in this context. The calculations of Kisslinger and Miller [Ref. v.5] assume absorption of the pion by a nucleon to form a A , which then coaleses with a nucleon to form a six-quark bag. But as we have seen, formation of the A» fixes the quantum numbers of the in- termediate state, in this case the 6 quark bag, so that everything is undistinguishable from the standard model, the only difference is the calculation of the rate of the re— action A + N -> N + N via a 6 quark bag. This just changes the absolute scale and is parametrized anyway. In conclusion calculations of pion absorption coef—. ficients in nuclei, using the standard " I + f " two-nucleon rescattering model with parameters fixed from' ltd absorption gives results in reasonable agreement with isotopic ratio and angular distribution of emitted nucleons. To get the magnitudes right, nuclear medium effects on the 83 pion propagator must be taken into account. The theory is not complete as higher order processes have been shown to be important. From the fundamental piont of view, the model is consistant with the " little bag " model. It might be consistant with the cloudy bag model if another mechanism could be found for S wave absorption. If the M.I.T. bag model is correct, the model is purely parametric, containing no reference to fundamental processes. 84 REFERENCES FOR CHAPTER V G.E. Brown, W. Weise, Phys. Rep. 21C (1975) l G.E. Brown, M. Rho, Phys. Lett. 823 (1979) 177 G.E. Brown, M. Rho, v. Vento, Phys. Lett. 84B (1979) 383 V. Vento, J. Jun, E.M. Nyman, M. Rho, G.E. Brown, Stony Brook report (unpublished) G.A. Miller, A.W. Thomas, S. Theberge, Phys. Lett. 913 (1980) 192 S. Theberge, A.W. Thomas, G.A. Miller, Phys. Rev. D22 (1980) 2838 A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, V.F. Weisskoff, Phys. Rev. D9 (1974) 347 K. Johnson, Act Phys. Pol. B9 (1975) 865 A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn Phys. Rev. 010 (1974) 2599 T. DeGrand, R.L. Jaffe, K. Johnson, J. Kiskis Phys. Rev. D12 (1975) 2060 L.S. Kisslinger, G.A. Miller, " Pion Production and Absorption in Nuclei " 1981 (AIP Conference Proceeding No 79) 85 Appendix A ( REPRINTS ) Volume 1191). number 4.5.6 PHYSICS l.|-.'I”I'I- RS 2 V30 December 1982 ISOSPIN DEPENDENCE OF PION ABSORPTION BY NUCLEON PAIRS IN THE He ISOTOPES H. TOKI and H. SARAFIAN National Superconducting Cyclotron Laboratory and Department oj'l’ltysrcs- 45tr()nom_t', Michigan State University. East Lansing, MI 48824-1321. USA Received 28 June 1982 Revised manuscript received 7 September 1982 We calculate the relative absorption ratio ot‘a pion by 7' = I.) and .7 e l nucleon pairs In the He ISOIOpCS measured recentl} by Ashery et a]. Standard theory based on .1--isobai intermedmtc exctwions nut-cs Willi the .->.perimental observation that for energies around the resonance. pion absorption by :t f.-. l nucleon pm: i\ .tronply suppressed Very recently Ashery et al. [1] performed the (NI, 2p) and (n’ , pn) measurements on 3He and 4He at pion kinetic energy of T" = 165 MeV in order to ob- tain information on the cross sections for pion absorp- tion by T= 0 and T= 1 nucleon pairs. They found sur- prisingly large ratios for do(1r*, pp)/do(rr‘ , pn), on the order of~100. By counting the numbers of initial nu- cleon pairs with the isospin T= 0 and T= 1 in those nuclei, these ratios were expressed in terms of the iso- spin of initial nucleon pairs. It was then found that R E do(T= 0)/do(T= 1) z 50 [l] . If one estimates this ratio in terms of the isospin geometry for the forma- tion of A isobars, one findsR = 2 [1,2] . In this short note, we would like to demonstrate that our current understanding of the pion nucleon in- teraction and the A isobar formation provides a natural explanation for the observed large value of this ratio. In recent years, there have been a number of theo. retical derivations of the pion—nucleus optical poten- tial parameters from the pion—nuclear many body theory [3-11] . In particular, the imaginary part of the two body Optical potential has a direct relationship to pion absorption. The S-wave and P-wave pion absorp- tion mechanisms seem to provide a microscopic under- standing of the optical parameters derived from pionic atoms and pion—nucleus elastic scattermg cross sec- tions [12] . The S-wave pion absorption mechanism ac- counts for the experimental fall off of the 1rd -r pp cross section with increasing pion energy and becomes negligibly small above Tn 't 50 MeV. On the other hand. the I’M/ave pion absorption mechanism, in par- ticular through the A isobar intemiediate excitation. becomes dominant above Tfl -‘~ 100 Mev. Since our interest is in pion absorption in the reso- nance energy region, we concentrate on the P-wave ab- sorption process through the A isobar intermediate ex- citation alone. as depicted in fig. 1. The wavy lines de- note the spin-isospin dependent interaction which is usually described in terms of the pion and rho meson exchanges [13] . The T-matrix for pion absorption by the pion exchange mechanism is obtained by the Feynman graph technique in momentum space; 71k,.k2;ki=[f(k,)/ul(o,'k,)t,-D,(k1) x [f‘(1c,)/ir](s‘;4<,)-rg’GDU'An/Uo/u].sz-kr2A I mkil/“l(“1°ki)‘1'Dn(k1) >< [f‘(k,)/u](52'k1)'TzGC(PA) x [f(k),’u] 534:7“ + (l = 2). (1) Here k is the incoming pion momentum in the labora- tory frame. which after absorption gets distributed as k1 and [:2 among the two nucleons l and 2 involved in the process. 0.1 are the spin, isospin Pauli matrices and S, T the corresponding transrtion operators be- tween 3 nucleon and a A isobar. The nNA coupling 285 Volume 1198, number 4.5.6 to) .0! :cf 10! Fig. l. P-wave pron absorption mechanism wrth A isobar inter- mediate excitation. constantf‘ is related to the nNN coupling consrant f through I" = 2f (Chew—Low relation) and has the monOpole form factor f'(k) = [(A2 -- rrz)/(/\2 — k2)] f', where A is the cutoffmass. D" is the pion propagator and tr the pion mass. The A isobar propaga- tors for the direct (figs. la and 1c) and the crossed (figs. lb and 1d) graphs are ..[_ 2 .. GD(PA) -mA —m+PA/2mA — w- §rl. c;c(pA)-1=mA - m +Pi/2mA + .3, (2) where the A isobar mass and nucleon mass are denoted by m,3 and m, and the pion energy by to. We take the static nucleon approximation and therefore IPA!2 = Ikl2 for GA and IPAI2 = moo for CC- For the width of the A isobar. we take the empirical relation [14]. r: {0.47/[1 +0.6(q/ur3] tairfi (3) The pion—nucleon center-of-rnass momentum q is re- lated to the incoming pion momentum through q = (m/\/s_)k, where s is the s-channel invariant mass. The T-matrix [eq. (1)] together with a similar ex- pression for the rho meson exchange process can be further worked out using the technique developed in ref. [6]. A lengthy expression is then obtained for pion absorption in terms of two body transition opera- tors in the nucleon space. the details ofwluch will be published elsewhere [15] . In principle, many pion partial waves contribute to the absorption cross section; 1,,(pion partial wave) Sk°R (nuclear radius). However, for the sake of dis- cussion let us take the dominant partial wave 1" = 1, absorption ofwhich by a lS-orbit nucleon leaves the intermediate A isobar in the l'S-orbit [l] . 1n this case. the ratio R does not depend on any details of the mod- el parameters and is simply given by the ratio of the squares of the A-isobar propagators CD and GC of eq-(Z); 286 PHYSICS LIC'II'I’RS 23/30 December 1982 R = dotT= Orr’dntT: throw/6C1? (4) The calculated ratro usrng eq. (4) rs shown by the dashed lrnc in fig. 3. Because of the resonant behavior in the direct channel. the ratio has a peak at k % 2p (Tn *- 180 MeV). At the experimental energy, T" = 165 MeV [1} , this ratio is ~200. This behavior is understood as follows. White In = l pion absorption by a T: O nucleon pair rs allowed by the direct absorp- tion mechanism (figs. la and 1c). this is forbidden for a T= 1 nucleon pair. For the latter case, absorption of a 1,7 = l pion by a T: 1 pair (S= 0,1. = O)leads to T = 1,!" = I+ intermediate states. Those quantum num- bers are not allowed by the Pauli principle for the final nucleon pair. Instead, pion absorption by a T= 1 pair proceeds through the crossed absorption mechanism (figs. lb and 1d) with exactly the same matrix elements as the direct T= 0 absorption process except the A iso- bar propagators. A question to examine now is if other partial wave contributions in particular for T= l pion absorption are small enough to keep the ratio sufficiently large. One thing to note in this respect rs that absorption of 1,7 #— l forces the intermediate A isobar to move with a frnite angular momentum (IA 4b 0). This fact, together with the large momentum transfer (Aq ~ M) be- tween the two nucleons required for pion absorption [16] (short range process) tends to cut down to a great extent the sparral transition matrix between the initial and final states. In the pron absorption operator. there is spin-spin interaction as well as the tensor interaction [6] . The large momentum transfer requirement forces two nu- cleons to be 111 the short distance (r 1:. 0.5 fm), whereas the strong short range correlations prevent them from turning together. Thrs interplay makes the contribu- tion of the spin spin interaction negligibly small [16]. Therefore. we shall take only the tensor interaction terms in our calculation. Furthermore, the ratio R weakly depends on the angle between the outgoing nu- cleons and we choose the angle such that the relative momentum of the nucleon pair is perpendicular to the incoming pion momentum. The results with pion ex- change alone with a cutoff mass of A" = 0.8 GeV are shown in table 1. When this value of Afl is used in the 1r exchange model, it gives results similar to those of the standard (it + p) exchange model, where a larger value of /\W is used [16] . The results on the ratio are Volume 1198. number 4.5.6 PHYSICS l.l’.'l'Tl".l(S 23/30 December 1982 Table l Pion absorption cross sections in several channels as a function or pron rnorrrerrtum k are compared. where 1,, is the angular momen- tum of the incoming pion. These numbers are normalized to the ('7'. S. I.) = (0. 1.0) to (T'. S'. l 'l = (1.0. 2) transition at k = 0.5 u as indicated by ‘. initial 1,, r 5‘ 1 T‘O.S=1,L=0 0 l l l l l 0 2 2 l l 1 T=l,S=O.L=O 0 l l l l 0 l 2 2 l l 3 ———._.—..---..—.... .... not very sensitive to the choice of A" within the rea- sonable range. We show in table 1 the pion absorption cross sec- tions in several channels for 1,7 < 2 as a function of pion momentum k. The l" = O and 2 contributions are indeed very small as compared to the dominant one (I" = l, T= O -* T' = l)even up to large momenta con. sidered here particularly for the initial isospin T= 1 case. This is due to the small spatial matrix elements for ln = 0 and 2 as compared to those for 1,7 = l, typi- cally about 1/5 in the resonance region, and strong cancellation between the two competing terms for T o 58 .72 TwlMeV] er : do T=Ol : l» dorlT= l) l 200 “t l , t l I . r l ‘ l l I‘ ‘ 177:! ' r , . l l ‘ r l root- ‘. i L___ ...2... -- 1 L 0.0 LO 2 O k ("‘11 Fig. 2. The pion absorption ratio R = do(T= 0)/da(T = l) as a function of the pion momentum k (7’" = kinetic energy). The dashed line denotes the 1,, = 1 contribution only [eq. (4)] , whereas the solid line includes all. “the experimental data taken at different angles at T, = 165 MeV are depicted by the points with error bars. _---.—— O kl“) = 0.5 l.0 l.5 2.0 2.5 0.027 0.40 2.9 9.7 7.3 l‘ 4.” 21 52 30 (tone or l to 4.2 3.6 0.008 0.086 0.44 1.1 0.63 0.076 0.17 0.24 0.26 0.26 0.002 0.023 0.14 0.40 0.26 = l. Note that the explicit distinction between the di- rect and crossed A propagators CD and GC, which was neglected by K0 and Riska [6] , is important for this cancellation. The higher partial wave contributions (I,Y ,2 3) are smaller by an order of magnitude. The summed values for the T= 0 and T= l initial parts are then used to derive the ratio R as a function of the pion momentum. which is depicted by the solid line in fig. 2. The ratio is now down to 2:40 at T" = 165 MeV. The experimental results taken at different angles are also shown with error bars [1]. Finally as stated at the beginning, the S-wave pion absorption process (1” = 0) is important at low pion energies. Therefore. the calculated results have to be used with caution below T” k 100 MeV. The capture of pions from the 2p level in pionic atoms [17] might. however. be compared with the present result at 7‘" = 0 MeV [6.10]. Furthermore, we have been neglect- ing the P-wave absorption process through nucleon intermediate states. This is certainly small for T= 0 pion absorption [16] . but might give an additional contribution for absorption by a T= 1 pair. ln conclusion. we have demonstrated that our knowledge of the pion—nucleon interaction and the A isobar formation gives a natural explanation for the large ratio of the pion absorption cross sections by T = O and T= l nucleon pairs in the resonance region. The resulting ratio R = do(T= 0)/do(T= 1) shows only a slight resonance behavior. It would be interest- ing to measure this energy dependence and also the angular distributions for a T'= 1 pair, which should show an interesting variation with energy due to the interplay between the three (I, = 0 + 2) partial waves. 287 Volume 1198. nunrber 4.5.6 PHYSICS Lli'l‘l'liRS 23/30 December 1983 We are extremely grateful to G.F. Bertsch for intro- ducing us to these interesting experimental data and for enlightening discussions in the course of the work. Thanks are also due to G.E. Brown and W. Weise for stimulating discussions and to D. Ashery, E. Piasetzhy and JP Schiffer for experimental information. This work was supported by the National Science Foundation under grant no. PHY-80-176OS. References [1] D. Ashery et al.. Phys. Rev. Lett. 47 (1981) 895. [2] IN. Ginocchio, Phys. Rev. C17 (1978) 195. H.131. Jackson et al.. Phys. Rev. Lett. 39 (1977) 1601; 11.12. Jackson et al., Phys. Rev. C16 (1977) 730; R.D.Mc1(eown et al.. Phys. Rev. Lett. 44 (1980) 1033, R.D.Mc1(eown et 31.. Phys. Rev. C24 (1981) 211. [3] F. Hachenberg and HJ. Pirner. Ann. Phys. (NY) 112 (1978) 401. [4] G.F. Bertsch and DO. Riska, Phys. Rev. C18 (1978) 317. 288 [5] .l. Chai and I).(). Riska. Nucl. Phys. A329 (1979) 429. [6] C.M. Ko and DO. Riska, Nucl. Phys. A312 (1978) 217. [7] G.A. Miller, Phys. Rev. C16 (1977) 2335. [8] 13. Oset, W. Weise and R. Brockmann, Phys. Lett. 82B (1979) 344. [9] R. Rockmore. 1:. Kantcr and P. Goode, Phys. Lett. 77B (1978) 149. [10] K. Shunizu and A. 1-:ressler. Nucl. Phys. A306 (1978) 311. {11] DO. Riska and 11. Sarafian. Phys. Rev. C22 (1980) 1222. [12] K. Stricker, H. McManUs and l. Carr, Phys. Rev. C19 (19.79) 929. [131 1;. Oset, ll. Tokr and W. Weise. Phys. Rep. 83 (1982) 281. [14) Particle Data Group. A. Rittenherg Ct 31.. Rev. Mod. Plrys.43 (1971) $114. [15) 11.1'okiand ll. Sarafian, to be published. [16] ().\r'. Maxwell. W. Weisc and M. Brack, Nucl. Phys. A348 (1980) 388. and references therein. | 171 M.li. Nordberg. K.1-'.. Kinsey and R.L. Burman. Phys. Rev. 165 (1968) 1096. Volume 958. number 2 PHYSICS LETTERS 22 September 1980 THE EFFECT OF THE NUCLEAR MEDIUM ON S-WAVE PION ABSORPTION‘3 D.O. RISKA and H. SARAFIAN Department of Physics and National Superconducting C yclorron laboratory, Michigan State Universr‘ry. East Lansing, MI 48824. USA Received 1 April 1980 Revised manuscript received 2 June 1980 it is shown that the effects of the nuclear medium on the pron propagator and the renormalization of the «NN interaction. described in terms of excitation of Virtual isobar-hole pairs can enhance the predicted absorption rates for pions at threshold to values close to the empirical ones. Recent attempts at explaining the empirical rates for nuclear pion absorption at threshold in terms of simple two-body rescattering mechanisms have led to underpredictions of the order 20—30% [1 l . Similarly the predicted values for the main absorptive two-nucleon component of the S-wave pion-nucleus optical potential have also been considerably smaller than those suggested by optical model studies of pion atom data [1-4]. We shall here show that the effects of successive excitation of virtual isobar—hole pairs in the nuclear medium on the propagation of the rescattered meson, and on the P-wave nNN interaction, can enhance the predicted absorption rates by the amounts needed. The absorption of S-wave pions at threshold is usually described in terms of the two-body rescattering mecha- nism illustrated in fig. is. Here the pion undergoes an initial S-wave scattering off one nucleon, and then prep- . Research supported in part by the National Science Founda- tion. ‘ o b Fig. 1. Swan pion absorption mechanism without (a) and with (is) medium effects. agates through the nucleus until absorbed on a second nucleon by the usual P-wave nNN interaction. The pres- ence of the nuclear medium adds a self energy 1'1 to the pion pr0pagator and renormalizes the final P-wave interaction. These medium corrections are schematically indicated in fig. lb. For the pion self energy we employ the picture of Barshay, Brown and Rho [5] and Baym and Brown [6] which involves excitation of isobar—hole pairs. We shall use a similar description to obtain the «NN vertex re- normalization, following the suggestion of Rho [7]. in this picture the pion self energy is made up of the single isobar hole pairs connected by the reduced isobar—hole interaction (R). with the single pion ex- change contribution subtracted out, as illustrated in fig. 2. - The self energy contribution from the single isobar— hole pair is, with account of both time orderings of the A33 resonance, "o--%(%)’r’r%plb‘-.+o%)- Here I: is the pion momentum, p the pion mass and p Fig. 2. Model for the pion self energy n involving the reduced isobar—hole interaction R. 185 Volume 958. number 2 the nuclear density. In (l)fA is the usual nNA coupling constant (fa/4n ~ 0. 32—0 .35) [8] and f (k) the form factor associated with the «NA vertex. Finally the iso. bar energy denominators D are, with the pion energy being u/2 for threshold absorption, Dl=mA-rn-u/2, Dz=mA-m+p/2, (2) and with mA and m being the nucleon and A33-reso- nance masses respectively. Because of the low energy of the virtual pion we have not included self energy cor- rections to the isobar propagators in eq. (2). Although the free width ofthe isobar vanishes below threshold an additional width in the nuclear medium can arise g from coupling to P-wave absorptive channels below threshold. The complete self energy is obtained by summing the the chain of isobar—hole diagrams connected by the reduced isobar—hole interaction (R) indicated in fig. 2. The result may be expressed in the general form T 1 IE , 1- [no(k)/k2f3(k)lR(k) (3) where Ran‘) is the isobar—hole interaction in the iso- spin 1 channel with the single pion exchange contribu- tion removed, and multiplied by (f,.,/i.t)'2 (following the notation of Weise [9]). The model we consider for the nNN vertex in the nuclear medium is based on the isobar—hole chain dia- grams illustrated in fig. 3. The final isobar-hole pair is connected with the active nucleon by the reduced isobar-hole - particle-hole (Ah-ph) interaction in the isospin 1 channel R', from which the single pion ex- change contribution has been subtracted. The other isobar—hole pairs are connected by the reduced isobar- hole interaction R defined above. We define R' so that the explicit pion-exchange strength factor ffA/u2 has been divided out in analogy with Weise’s definition of R [.9] (I is the pseudovector nN coupling constant). ii a Fig. 3. Model for the aNN vertex factor F involving isobar- hole pairs connected by the reduced isobar—hole interaction R and Ah-ph interaction 11'. “(1‘): floor) Tr 186 PHYSICS LETTERS 22 September l980 With that definition, the medium renormalized nNN vertex factor may be written'as .T 1+ Ino(k)/k2f3(k)1IR (’0 R001" l-[Enid/“[2001RI") I‘(k) = Tr (4) We have not included here the exchange diagrams for the vertex normalization effect. These should be expect- ed to be small according to the discussion in ref. [7] With the results (3) and (4) for the self energy and the renormalized vertex factor l‘(k) for the nNN inter- action we can now consider the effects of the medium on the S-wave absorption process illustrated in fig. 1 . Assuming that the initial S-wave rescattering interac- tion can be described by a phenomenological zero-range hamiltonian. the two-body rescattering amplitude be- comes T_ i a. rte/3.0:) pm ”2+ k2 - w2+ n(k) .(2).k_ 12 X[h112—i7(1+-)(11X12)a]. (5) In this expression A, and X2 are the two coupling con- stants for the Sowave ltamiltonian [10] which have the . values h1= 0.003 and )‘2 = 0.050 as determined from the S-wave trN phase shifts. The simplicity of the eXpression (5) allows one to calculate the absorption rates for pions in nuclei at threshold using the expressions given in ref.'[l] for the case 11 = 0, 11= I. with the simple replacement of the Fourier transform of the pion propagator (Y1) in the radial matrix element expressions with the more general expression: - 3 - 2 “mm“ ofd k rltkrirtkirxtr k2 + in“ WC) To proceed we need explicit models for the isobar- hole interactions 120:) and R'(k) in eqs. (3) and (4). The simplest model is obtained by considering the inter- scrions made up solely by pion and p-meson exchange. if the short range correlations between the successive isobar—hole lines are taken into account to the minimal extent that they eliminate the 5 function components in the meson exchange interactions, one chains for R the result [9] Rumor) - is... trim — 2c, rim] . (7) (6) Volume 958. number 2 where fp(k) is the form factor associated with the pNA vertex and C9 the strength factor for the p-meson ex- change interaction: C},'=(ir,,.,/2m)2(u/fg)2 . (8) Here g‘m is the pNA coupling constant [8] . With the same “minimal” account of the short range correlations the reduced Ah—ph interaction R' can be cast into the form R’.....(k) = i6"... Uitk) + 26,7300] . (9) with the p-meson exchange coefficient (———-—-)(-—) ' Here g,” is the vector p-nucleon coupling constant and x the tensor coupling strength. The first (pion) terms in these expressions for R and R' represent the classical Lorentz—Lorenz-Ericson- Ericson effect [1 l] as derived by Barshay, Brown and Rho [S] for the self energy and by Rho [7] for the vertex renormalization. it was shown in ref. [9] that a more accurate treatment of the nuclear pair correlation function reduces these pion terms to a fraction of the value in eqs. (7) and (8), whereas the pomeson terms are only reduced by a moderate amount. Accordingly we shall neglect the pion terms in R and R' and incorporate the finite range corrections approximately into fp(k). Note now that 52 ..éafll‘m—ntt c, rm, This ratio is unity if the ratio of the nNA to nNN coupling constants is the same as the ratio for the pNA to pNN coupling constants as predicted in the static quark model. In that case 12' = R and thus the numera- tor correction factor in eq. (4) for R vanishes, and the simple vertex renormalization model of Rho [7] results. We shall use that simpler model here with the justifica- tion that recent studies of the resonant rescattering con- tributions in the reaction «*d 0 pp indicate that the (10) (11) n and p coupling constant ratios should be close [12,13]. In the quark model [8,9] C, - C; =- (xpul2m)2(l + «)2 (rt/r)2 ~ 2 . (12) with the values p'ven in ref. [14] for the p-nucleon coupling constants. Using these results and the substitution (6) we have PHYSICS LETTERS 22 September 1980 used the formula given in ref. [1] to calculate the param- eter 1m 80 for the two-body absorptive part of the 8- wave pion-nucleus optical potential. Taking the form factors ["(k) and fp(k) to have the usual monopole form 5 .. M) = -———-———A "' "'2 . A2 + k2 __ (‘02 with it being the mass of the exchanged meson and A = 1.2 GeV/c2 we obtain 1m 80 = 0.048 p“ in reasonable agreement with the empirical value 0.042 n“. Nate that with no medium corrections the corresponding result would be 0.020 p“ [2], which is too small by a facror of 2. The value A = 1.2 GeV/c2 for the pion form factor is the same that has been used in studies of fl+d -* pp [12]. On the other hand, in studies of. rr*d -> pp, A is usually taken to be much larger than 1.2 CieV/c2 for the p-meson vertices [12,13]. The use of the smaller value here simply serves to mock up (13) ‘ neglected finite range corrections to the nuclear pair correlation function as mentioned above. if A were taken to be 2.0 GeV/c2 for the p-meson form factor we would obtain the far too small value 0.016u" for 1m 80! The values for the residual isobar—hole interaction R given in eq. (7) were obtained with the schematic model of a step function for the nuclear pair correla- tion function. Using a more realistic model for the pair correlation function, Weise [9] found that the whole denominator factor in eqs. (3) and (4) could be well approximated by an almost momentum-indepen- dent expression of the form 11 k l-% -:—(2—) 712— , (14) I with K it < 2, depending on the specific details of the correlation function. in fig. 4 we show 1m 80 as calcu- lated with this simpler model as a function of the param- eter h (neglecting the pion form factor). Clearly Im 80 is a sensitive function of h. The desired value 0.04M“ for 1m 80 is obtained with h = 1.7, which agrees well with the “optimal” value A 8 1.6 found in recent com- prehensive optical model studies of piononucleus scatter. ing data [15]. (If the pion form factor (13) were in- cluded in the self energy the desired value for 1m Bo is obtained with h = 1.5.) The fact that the absorptive optical potential may be explained with a similar value for the Lorenz-Lorentz factor A as used in studies of 187 Volume 958. number 2 meow") (108 006 (104 CLOZ l 2 3 4 Fig. 4. The parameter 1m 80 describing the strength of the in2 term in the pion-nucleus S-wave optical potential. other aspects of the pion-nucleus interaction should, in our opinion, be viewed as strong support of the valid- ity of the present picture of the medium effects. The combined medium effects on the pion prOpaga- tor and nNN vertex enhance the predicted value for 111180 by roughly a factor 2. As this parameter repre- sents the absorption rate in nuclear matter, the medium corrections will be much smaller in light nuclei, which have large densities only in the central region. To illus- trate this we consider the absorption rate in ‘He at threshold. We calculate this as an average over the nucleon density p(r) for the 4He nucleus r= f d3rp(r) 1‘00) . (15) with 1‘00) being the result obtained by replacing the pion exchange Yukawa function in the explicit expres- sion for 1‘ given in ref. [1] by the medium corrected form (6). Using the harmonic oscillator model in (15) we find that the medium corrections enhance the cal- culated absorption rate by M30% for the tit-particle if h in eq. (14) is taken to be 1.7. Depending on the value 188 PHYSICS LETTERS 22 September 1980 of A in the pion form factor (13) the imaginary part of the scattering length for ‘He falls between 0.017ti‘1 (A,' = 1.2 CeV) and 0.034 p“ (A" = N) which should be compared to 1ma(exp) 2 0.03 if! . Without the medium effects the corresponding range of the calcul- ated scattering length would be 0.012 u“ <1m a < 0.021 p", which demonstrates the need for the medium effect. In conclusion we have demonstrated that the me- dium corrections to the pion prOpagator and WW vertex in nuclei enhance the predicted absorption rates by roughly the amount needed to explain the values extracted from pionic atom data. The pion self energy by itself induces a far too large enhance- ment which is moderated by the vertex renormaliza- tion. Both effects must therefore be considered to- gether. References [I] G.F. Bertsch and 0.0. Riska, Phys. Rev. C18 (1978) 317. [2] J. Chai and D0. Riska, Nucl. Phys. A329 (1979) 429. [3] F. Hachenberg and ii. Pirner. Ann. Phys. 112 (1978) 401. [4] K. Shimizu, A. Faessler and H. Miither, The pionic disinte- gration of the deuteron and the absorptive pion-nucleus Optical potential, preprint Univ. of Tllbingen (1979). [5] S. Barshay, G.E. Brown and M. Rho, Phys. Rev. Lett. 32 (1974)787. ' [6] G. Baym and G.E. Brown, Nucl. Phys. A247 (1975) 395. [7] M. Rho, Nucl. Phys. A231 (1974) 493. . [8] G. Epstein and 0.0. Riska, Zeit. Phys. A283 (1977) 193. [9] W. Weise, Nucl. Phys. A278 (1977) 402. [10] A.E. Woodruff, Phys. Rev. 117 (1960) 113. [11] M. Ericson and T. Ericson, Ann. Phys. 36 (1966) 323. [12] J. Chai and DO. Riska, On the reactions pp 0 drr‘, MSU Nucl. Phys. preprint 1979/203-3. NucL Phys. (1980), to be published. [13] O. Maxwell, W. Weise and M. Brack, Meson-baryon dynamics and the pp 00 dtr‘ reaction: (1') total and differ- ential cross sections, preprint, Univ. of Regensburg (1980). [14] G. H6hler and E. Pietarinen, Nucl. Phys. B95 (1975) 210. [15] J. Carr, K. Stricker and H. McManus, to be published. PHYSICAL REVIEW C VOLUME 22. NUMIIER 3 SEPTEMBER 1901) Effective S -wave nNN interaction in nuclei D. O. Riska and H. Sarafian Department of Physics. Michigan State University. East Lansing. Michigan 48824 (Received 21 February 1980) We show that virtual pion rescattering in a nuclear medium gives rise to an effective S-wave rrNN interaction Hamiltonian. The first few terms in the density expansion of the strength parameter are estimated and shown to be appreciable in nuclear matter. [NUCLEAR REACTIONS Pica-nucleus interactions: S-wave interaction.] Considerable, if inconclusive, attention has been given in recent literature to the "Barnhill" am- biguity in the S-wave uNN interaction in nuclei.M The ambiguity concern the str ngth parameter A in the Hamiltonian for t is interaction: Hufw’E-Vfi-Bw. (1) Here it and a are the nucleon and isovector pion field operators and V, a symmetrized gradient operator acting on the nucleon fields.‘ In (11f is the pseudovector pion-nucleon coupling constant (/'/4u-0.08) and u the pion mass. Straightforward nonrelativistic reduction of the Lorentz invariant pseudovector rtNN interaction Hamiltonian leads to A = u/Zm (m 8 nucleon mass). The unitary freedom inherent in this reduction leads to the ambiguity in x.’ The physical reason for this ambiguity is the lack of a definite treat- ment of the binding effects in a nuclear medium that force the nucleon under consideration off shell.“ Only in simplified boson exchange models is it possible to make definite predictions for A.“ Therefore it has been suggested that the parameto er be determined empirically by means of (p, it’) and (s’,p) reactldns interpreted with a single nu- cleon stripping model.m Apart from the obvious criticism that the simple stripping model may not be adequate, we shall in this work show that size- able density'dependent medium corrections to the effective‘one-body operator (1) are caused by vir- tual pion rescattering. Thus it will not have any universal value valid for a range of nuclei, and attempts at empirical determination of x will be futile. The main rescattering process that contributes to )1 is that involving two nucleons: an incident S-wave pion rescatters off a nucleon (which is elected) and is absorbed by a particle-hole pair, orit rescatters off a particle-hole pair and is absorbed by a final nucleon (which is ejected). This is illustrated in the diagrams in Fig. 1, which include direct and exchange terms. These processes take place in addition to "true" two-nu- cleon absorption processes in which two nucleons are ejected from the nucleus. . The diagram in Fig. He) represents distortion of the incident pion wave function and should not be included in the basrc trNN interaction, as the distortion can be treated with an optical potential. The corresponding exchange term in Fig. 1(d) should be excluded for the same reason. The am- plitude for the diagram in Fig. 2(a) vanishes in spin or isospin 0 nuclei because of the Spin-vector isovector nature of the rrNN absorption operator. Therefore only the remaining exchange term di- agram [Fig. 1(b)] contributes to the S-wave ab- . sorption interaction. In order to construct the amplitude correspond- ing to Fig. 1(b) we employ the phenomenological zero-range Hamiltonian x ...... -o -- ° ”Mateo-ow‘trfig-wrohw (2) to describe the S-wave rescattering vertex (1: = ago). In (2) the coupling constants x are de- / / a b 0-- .. , // // c (1 FIG. 1. Pion rescattering contributions to the effec- tive S-wave INN interaction. 1222 © 1980 The American Physical Society ll: i-.. ..-. I l I FIG. 2. Three-body contribution to pion absorption with double pion rescatteriig. termined from the S-wave pion-nucleon phase shifts to be A, 130.003 and x, =0.05.' This Hamil- tonian leads to a satisfactory prediction of the 8- wave pion- nucleon phase shifts at low energies and is used in the standard derivations of the first and second order S-wave pion-nucleus optical p n- tial. it has also been used successfully in the two- nucleon model for nuclear pion absorption,"m al- though lt implies a very simple direct all shell extrapolation of the pion-nucleon interaction. For the final absorption vertex we use the standard P-wave INN interaction. The two-body rescatter- ing amplitude is then a 5 'i; 0 , 0 7"”(19771? 1219.-.“?! xrm. (3) with it being the momentum of the exchanged pion. The isospin index of the initial pion is i. Taking the matrix element of 7' over the closed particle—hole line in Fig. 10)) yields 4 1', 12:7) T-i- L, (A,+x,)h,’fd’r, ( a u k,r (4) 135.. ..fi “I"; (2n)3 b’+ u’ 8 Here it, is the Fermi momentum and F= F, - F1. To reduce (4) to an effective one-body operator we take the O-range limit it - m, which leads to 8 Here p, and p, are the momenta of the initial and final nucleons, which appear after a partial inte- gration. The employment of the zero- range limit can be expected to overestimate the two-body con- tribution in Eq. (4). It is, however, a better ap- proximation than that which the static pion propa- gator in Eq. (4) would imply, because of the self- energy correction to the propagator which reduces the kinetic energy term. This point will be dis- cussed in detail in conjunction with the higher order corrections below. The amplitude .(5) can be generated from the in- EFFECTIVE S-WAVE INN INTERACTION IN NUCLEI I223 teraction Hamiltonian (1) with the density depen- dent parameter A”: 2 3 *u' ‘33- (M’IMJVE-(fi‘f) (MW). (e) In nuclear matter It, a 1.35 fm" and thus it" 210.08, which is similar in magnitude to that of the usual one-body Galiliean invariance counter term (A, = u/Zm 20.07) obtained from the rela- tivistic pseudovector Hamiltonian. The magnitude for A“ given in Eq. (6) is clearly an overestimate, as the effect of hadronic form factors and nuclear wave function correlations will tend to reduce it. Although the simple zero- range approximation used above will not permit a realistic estimate of these effects, a first estim- ate of the form factor reduction is possible. As- suming the pion-nucleon vertices to be'described by a monopole type form factor (A’ - it’l/(A'He’) the expression (1) ought to be multiplied by a iactox (1 - p’/A'). With A-i GeV/c’, this would lead to a ~51, reduction of the previous estimate for 1,, remains the same as that of A, Equation (6) represents the second term in a density expansion of the parameter A. The next term involves a secondary pion rescattering, as illustrated in Fig. 2. Here the dominant contri- bution will come from the P-wave component in the second rescattering amplitude. As the nucleon pole terms in this amplitude represent binding corrections to the two-body amplitude above, we only consider the intermediate states containing the A” resonance treated in the sharp resonance e i FIG. 3. Three-nucleon contributions with A” interme- diate states to the edectlve S-wave INN interaction. I224 D. 0. RISKA AND II. SARAFIAN 22 approximation. Considering both time orderings for the 15...,3 re- sonance intermediate state contributions there will then be a total of 36 three-body diagrams in which two of the final particles remain within the Fermi sea, compared to the four two-body diagrams in Fig. 1. After exclusion of all those diagrams that properly represent distortion of the incident pion wave func- tion, and the corresponding exchange terms, and all those that give no contribution in isospin 0 or spin 0 nuclei, we only need to consider the six diagrams in Fig. 3. To construct the scattering amplitudes for these diagrams we use the itNA coupling Lagrangian an? rr- same. (7) Here 5 and X are the spin _and isospin field oper- ators for the nucleon and s and §the corresponding 1 =-4m 9(mA-m) “2 (217)J x(61-E,)Z,-E,- 32,252- The same sum for the diagrams in Figs. (3c) and (d) give rise to an identical result. To complete the evaluation of the particle-hole line matrix elements for these diagrams [Figs. 3(a) and (b)] the expression (9) must be integrated over F, and F3, folded with the relevant three-body density ill—UL'Q'UL—l—u—IQH. (10) 4! 12,13, k,r,, The three-body density appropriate for the dia- grams in Figs. 3(c) and (d) may be obtained from (10) by the replacement 1"" - 1"". Finally, because of nuclear correlations the 6 function in the brack- et in Eq. (9) should be eliminated by the replace- ment k,’ - - p“. In order to obtain an approximate one- nucleon amplitude we again resort to the zero-range limit of the pion propagators in Eq. (9), i.e., u’ --°. In that limit the integrations become trivial and the combined result for all the four diagrams in Figs. 3(a)-(d) is . 32 k ' if,“ - - - =- J g . T 124319 ( l1 "I‘- m (Midge (p,+p')T,. (11) This amplitude can be obtained from the interac- tion (1) with Alum: 2 J , _ _ - - _ 8 ffa I d b; etll-irS-rzletta-(rz-r,’ operators for the A” resonance. From the decay width of the resonance one has {62/421 =0.35. For the isobar contribution to the general three- nu- cleon two-pion exchange diagram in Fig. 2 one obtains 534:, T=-4m’ 8 I]; a a (k12+ NW,“ #3) 9(ma- m) p3 X{E, -E,[2x,r,='- 21,64 x $5,143! - ()2, all) XIZAfi’ x r‘), - ix,(?’ x (F2 x?’)),]}, (8) with it, and it, being the momenta of the primary and secondary exchanged pions. The mass of the resonance is denoted by ”is- Carrying out the spin and iSOSpin sums over the closed particle-hole lines indicated in the dia- grams in Figs. 3(a) and (b) and taking the Fourier transform of the amplitude (8) yields 1 l k‘2+ “2 k22+ “2 +iiz’5‘-E.). (9) 128 f 3 u k B = —— .'_.A_ __ J 243”: (41! ) ’"A- ,n (1“) (A1 + A2) a (12) In nuclear matter A”, 20.024, which is roughly 30% of the corresponding value for A“. We finally consider the diagrams in Figs. 3(c) and (f). These can actually be summed to all or- ders in the internal isobar-hole propagators and then simply represent the self-energy corrections to the pion propagator in the two-body amplitude (4) corresponding to the diagram in Fig. l(b). They may thus be taken into account by the re- placement 1 1 ks+ns k2+u2+n6 (13) in Eq. (4), with IiA being the isobar-hole contribu- tion to the pion self energy”: =-L£(2£1 _et’___ 11‘ 4a 9 p.3(mA-m) ' (14) This self-energy expression is reduced by taking into account the isobar-hole interaction in the one- pion exchange approximation to” 1 n '* rig-1141-5 3,3) , (15) g; EFFECTIVE S-WAVE INN if the nuclear pair correlation function is taken into account to the minimal extent that it elimin- ates the o-function terms in the interaction. The main role of this self energy will be to improve the accuracy of the zero- range approximation used in the other many-body diagrams above. Since the magnitude of 11‘ in (14) and (15) is fir-0.618, it reduces the kinetic energy terms in the pion propa- gators to less than half of its unmodified value. We therefore feel that the zero-energy approximation should be accurate anough for a first estimate of the medium corrections to the S-wave IlNN vertex in nuclei. The net three-body correction to the parameter A is thus that given in Eq. (12), and it is again necessarily an overestimate as hadronic form factors and nuclear wave function correlations will reduce it. With the same argument as used i above to estimate the form factor correction to INTERACTION IN NUCLEI l225 A", we would find A,” reduced by at .least ~10£ due to the form factors. The relative smallness of A,“ compared to A" in nuclear matter suggests that higher order re- scattering mechanisms will be of little significance. in any case the large value- of 1,, compared to the “free nucleon" value A :- u/2m shows that the ef- fective one-body S-wave pion absorption operator (1) will depend strongly on the nuclear density. The interaction strength will thus vary from nu- cleus to nucleus, and there will be no possibility of determining a universal value for l by pionic stripping and knockout reactions. It appears that the only realistic approach to the S-wave pion-nu— cleus interaction will be to take into account its two- nucleon and many-body components explicitly. This research was supported in part by the National Sczence Foundation. 'M. v. Barnhlll 1n. Nucl. Phys. 513;. 106 (1969). 1M. Bolsterli. w. R. Gibbs. B. r. Gibson. and c. J. Stephenson, Phys. Rev. C _12, 1225 (1974). ’J. M. Eisenberg, J. v. Noble. and H. J. Weber, Phys. Rev. c 11. 1048 (1975). ‘li. W. Ila—M. Alberg. and E. M. Henley, Phys. Rev. c 12, 217 (1975). ’M.'§ohterii, Phys. Rev. Ci_s. 981 (1977). 'J. v. Noble. Phys. Rev. Lett. 3. 100 (1979). 'L. D. Miller and ii. J. Weber, Phys. Rev. c _1_7_, 219 (1978). _ 'M. Brack. D. o. Risks. and w. Weise, Nucl. Phys. A287, 425 (1977). 'éTTsertech and o. o. Riska, Phys. Rev. C13. 317 (1978). “0.0.111“; and H. stream. Michigan State Univ. Nuclear Theory Report No. CTN 203 /80-2 (1980). to be published in Phys. Lett. uS. Barshay, G. E. Brown. and M. Rho, Phys. Rev. Letty, 787 (1979).