able to account tor the r..irrow width* of the known candidates
(or bound states. Howe- , T , this problsn could be solved using
potentials including oíí-»liell effects and Imaginary parts
with strong energy 4etv-».*.cncot, associated with the closing
of .01:10 \nnihi I if ion channels due to lack of energy of th<:
virtual NN system.
Another possibility for the bound system is a diquomum, that is, a system composed by two qq pairu, related to the diagrams mentioned in bracket d of fig. 3 . The fact that these st.itos are exotic yields an explanation for the narrow widths, since those configurations are weakly coupled to nu-'son states. On the other hand, this vtry nuchanism m.ifces it difficult to explain the observed decay rates from
"atonic" to diquonlum states.
It Is important to note that the two possibilities for the bound state discussed above, namely that it could be a NN system or a diquonium are not mutually exclusive. The bound state can, in fact, be an admixture of these two possible states. T: e probability of such mixed states could be non-noqltqiblc when the binding energy is big, since in those casra the N and N would be close enouqh to allow t!«r tlircvt
in<ci'.i>-t ion of some of their constituent quarks and anti-'junks.
In the preceding discussion we have dealt with bounj st.itou, hut its quul 1 t.itivo aspects arc relevant also fur the study of resonant states.
9 9 pions. The state Xq is studied by means of the missing mass or these final pions.
The observation of spectral Y rays originating from the deexcitation of "atomic" to "nuclear" states also yields Information about bound states.
The existence of narrow bound state is discussed in two recent papers. In one of them, the observation of discrete Y rays indicates bound states with masses 1210,
1638, 1694 and 1771 MuV (RIC 83). In the other experiment (DKR 82) the analysis of the reaction pp • X ir~ In the forward direction his shown no evidence for bound states with misses between 1.6 and 1.B7 GeV/c* , for an experimental upper limit of 8 ub/ur . The lntcrpiutation of the experiment has been done by me,ins of the diagram below for tlio caso («0 (DOS 78), corresponding to a forward crous section of order of mb , that Is about throe orders of magnitude larger than the •qpexiMntal upper limit.
IUÜ
Fig. 8
This calculation has been redone recently for the caso f = 2 , producing a differential cross section of order of yb. Thus it is possible to conclude that the experiment does not exclude the possibility of bJryonium production with isospin equal to 1 and high angular momentum. It is worth pointing out that states with high angular momenta tend to be more stable due to the centrifugal barrier.
A detailed phenoraenologicil analysis of the reaction pp - X*TT~ for p momenta between 0.8 and 1.4 GoV/c has been made in order to evaluate the importance of diagrams of fiq. 9 , containing the A or A resonances (AMJ 80>.
s . . . S
TÍFig. 9
I n d i a g r a m ( i ( , thr> a m p l i t u d e tiN i s a s s u m e d t o b e d o m i n a t e d by th<; f o r m . i t i u n o f t h u A r e s o n a n c e - . D i a g r . i m (l>)>
o n th<> o t h e r I n i n d , e x j » r e ;;<•!> t h e - a:;:;uinijt i o n o f t h e d o m i n a n c e o f t h e p r o c o i V i «N b y t h e S . T h o s e h y p o t h e s e s <ir<; ri*inrjn.ible i n t h e c n e r ' i y r o i j l o n o o i i s I d / r i - i l . Th<: c.i l c u l . i t I o n h i» bi.-.'n rlnni. o r t q t n . i l t y f u r z>-t<> . n i ' i u l . i r nt'itiifiit um i n r l w MiJ s y ;l • -m.
Tlu- t i v i u l t f l o l d .• I n o l ::\\iM Lli.tt t h u c o n t i i l i u l i t m o l i l l . n i r . u n " . ' > f M i ) . ' ) , u f i | l l i I c I l i i j i u l I . H i t I i l l t i n ' | > l i >. I n , ' I l o l l c i , i . , . . ' M o l l
101
wlii-ii thi' m.i.sos of the x is laryo, corr<'.-.£)on<5lng to a baryoninm with low bin ling energy. 1'hr calculation is at present being extended by the same yrojp to the case I « 2 , that could be much more realistic, since It should be less lnriuunc«3d by the annihilation channel.
Tilt; search for ressonat trig states is also difficult as far as both its theoretical and experimental aspects are concerned. The difficulties associated with the latter aspects are related to th« fact that the pp annihilation produces the destruction of thin targets. This requires the use of Intense p beams, that have only come into existence with the LEAR facility at CERN. Thick targets, on the other hand, introduce the possibility of multiple scattering, distorting the results.
Difficulties of this kind were responsible for the uncertain experimental situation of the resonating state with mass 1935 MeV. Between 1966 and 1979 several experiments have accused the existence of such a state, considered at that time to be a meson. The evidence of this "masonic" states S has been disputed by the first time in a phenomenological study in
1979 of the reaction pd -* pi's (ALB 79a). This analysis was based on the following diagrams
lo)
Fig. 10
The Inclur, Ion of diagrams of ty|><- c, representing th? sc.it. turinti of ono of i ho f i n a l plons by fh<? remaining nucleon ban Urcn 8u>j>j.'Htt'J by a previous «tuúy of tho distribution
102
of momentum of the final proton of the same roqvrinfnt (AIS 79b).
In the study of the differential cross section with respect to the missing mass, da/dH , the baste Ingredient was the cross section for NM annihilation, where the intensity of the contribution of two resonances (mi • 1879 NeV and mi = 1934 MeV ) aro modulated by means of the parameters x\
and Xi : -Wi'jnfr formulas and the background ijivon by 4 i / q .
Thi: a n a l y s i s of tho experimental r e s u l t s uslmj t h i s model h.is r;liown t h a t thr value of xj was nopipatlblc with zaro, mitklnq i t c l o a r th.it the rnscmanrc wlfh miss m2
was not n«cesBary. As far a s the resonance with m.ias ntj was concerned, no il<;t'initu coticliüiions could he riMi-'lird duo to expcirlmcnt t l unci^rtaf nl. it-». Th«? value of tin1 par.nnctor x j obtained in thin study contradict od the pri-vlous . l n . i l y s i s of Kiloqoropuiiioü ••infloytnq il-it.i from another ox|>i'rlm>'iit (KAl. 7ri).
T h . i K t i - K i i l t w . i r : f m l v e r y r c i 1 í . i M < • , s l m n - t l »1 i - i u i l i i l i l < • f i n t h i ' t c s u i i . i m v i » , i s . ' I i h . . h o i . I . i i > l l l n ' § > I i • > • . < ' ' . | ' i i - . ' n l I h i !
« l . i l . i . i n I l i e p r i ' s ' - i i f i \ r ; i . t l i l u < l I I t i < M I 1 1 y i s m i l p i • • • • • n l ,
101
since It Is at the center of the Mass spectrum.
zuu tho "musonic" S s t a t e . Recent experiment» havo shown the poiiHibll t t.y of rc;ionanc<-8 with misut-rt 2.022 ' 0.006 MeV and 2.026 • 0.005 MeV, nsifociatfri with croon s e c t i o n s of 1.2 i 0.4 i<>
and 1.1 > 0.4 lib r(!»(M>ct(v<<l.y (A/,O 6 2 ) .
The moat Important f a c i l i t y for experimental NÍJ [lhysics Is the Low Energy Antioroton Ring (I.Er.H) .it CtPiJ. I t
is already in o p e r a t i o n and Is expected to produce in thr; next ycr.iS a l o t of new and precissu information «bout i n t e r a c t i o n s with p laboratory energy between 5.3 MeV and 1.3 GeV, c o r -r<><-,pondintj to mom.Tita in th.' > inqc 0.1 GeV/c t o 2 ficV/c . The main c h a r a c t e r i s t i c of LEAH Is the a v a i l a b i l i t y of i n t e n s e antiproton beams, obtained by the d e c e l e r a t i o n and c o o l i n g of p a r t i c l e s produced in the Proton Syncrotron (GRE 8 3 ) .
An i n d i c a t i o n of the experimental p o s s i b i l i t i e s t o bo explored dt LE'R in t h e n e j r f u t u r e can be obtained from
the follow! nu H u t of proposfd experiments (the numbers w i t h i n
p . i r n i U h i " - i s r . r f f c r t o v x p e r i n i - i M j l p i o p o s n l < ; ) :
• i l p p i i f . ' . M t . ' i m u w i t h I ' M t ' i ^ y . i r o u n d 0 MeV: s t u d i r s o f p n j i ^ m m ( i ' S 1 7 1 , 1 7 4 , ! 7 5 ) . i n d b . i r / o n i u m Í P . i 1 8 2 , 1 8 3 ) )
l > ) f i [ j s . r i t t o i . i n < | w . i h f i v . ' i q Y J C O I I H I ! 1 0 0 M ! ' V : s r i k l i o w o i t o l j l , i l l f f . T r . i t í a l . u n i i ' o \ , u L / . a t t o n c r v . s - y . - I i n n s ( P K W ..', I V ) ) ;
c l p - r i ' i f ! I ' M I I i n i m >•*» i i > n s : B t i i f i l i - i ; o f c r o s s - s o r t t i i ^ ; ( P S 1 7 9 , l a l , I d l » , 1 3 7 ) . i m l , m l i j - ' t o t o n l e . i t . j m i ; ( l ' .r; 1 7 4 , 1 7 ' ) , 1 7 b ,
186) ;
i | ) , ) ( | u i p h , I ' . M I I I , n . i : i i l i i . l i i ' S o f I n - , i v y l i y p t T i m r l i M ( | ' ! > 1 7 / 1 , t i n , i . h - l i l { t i i i r l i i c t t i ' i i o f h y : » ' i o n - n i t l h y i ' i ' i m i U ' S 1 8 0 ) , J M . - I I X I r I i "-f r' •!« i - i l l ' ' ! t i " I " i n I ' . i f l u t ' ( I ' M 1 7 0 ) , . n i t i . t i i ' i i » I n u | ' i o f o k " l i i i i
11)5
106
h.ivo tin; formation of di<i':>'ii i .1, which .11 • %ritcs composed of two |>ciirs yq an represented in f u j . 12fl . T.'ujye e f f e c t s cor-r«S[>'jn<l to i n t e r m e d i a t « s i t u a t i o n s between the d e s c r i p t i o n s cf tli.: ; ; ' ! i n t e - r a e t l o n s In terras of mesons or q'jaiks, 'ihowinq th.it they a r e not mutually exclusive. I n s t e a d , they «ire conip] Piri':iit.'ry p i c t u r e s of «J single p r o c e s s .
N N
si)
( c )
Fig 12
107 RKPKKKHCES
(AI,B 79a) -G. Alberi, C. Alvnar, E. Castclli, P. Poropat, M.
t
Sessa, L.P. Rosa and 7.D. Thome , Phys. Lett. B8 3 (1979) 247.
(ALB 79b)-G. Albert, C. Alvear, L.P. Rosa nnd Z.D. Thome, Nuovo Cimento A53 (T979I 191.
(ANJ 80) - J. Anjos, M.F. Barroso, L.P. Rosa and Z.D. Thome, Proc. of the 5th. Eur. Symp. on NN Interaction, Bressanone, Italy (19S0I 41S.
JAZO 82) - F. Azzor et «1., Rutherford Lab. preprint RL-82-108.
(BER 79) - L. Bertocchi, Internal Report IC/78/61-ICTP-1979.
(BER 82) - R. Bertini et ai., Nucl. Phys. B209 <1982) 269, (BIZ 74) - R. Bizzani et al., Nuovo Cimento 22A (1974) 22$.
(DOS 78) - H.G. Dosch and B. Porh, Heidelberg Report MPI-H-1978-V6.
(GRE 83) - A.M. Green, Preprint Univ. Helsinki HU-TFT-83-17.
(HOL 81) - K. Holinde, Phys. Rep. 68 (1981) 122.
(HOT. 84) - K. Holinde, Nucl. Phys. A415 (1984) 477.
(KAL 75) - F.R. Kalogeropouloa and G.S. Tzanakos, Phys. Rev.
Lett. _34 (1975) 1047.
(MAU 79) - R. Vinh Nau in Mesons In Nuclei, eds. M. Rho and D.H. Wilkinson, North-Holland, 1979, page 152.
(MON 80) - I'. Montanct, G.C. Rossi ,ind G. Vcnezlano, Phys.
Rep. 63 (1980) 149.
(PAV 78) - P. Pavlopoulos et al., Phy». Lett. 720 (1978) 415.
(PAV 821 - P. ravlopoulos et al.. Preprint CKRN - EP/62-177.
(RIC 83) - R. Rlchter et al., Phys. Lett. 126B (1983) 284.
loa
CENTER OF MASS AND RECOIL C O R R E C T I O N S IN SAC MODELS
H.Betx Instituto de Física
Universidade Federal do Rio Grande do Sul 90000 Porco A l e g r e , R S , Brasil
Approximate methods of eliminating spurious center-of-mass notion and taking into account r«lativistic recoil in bag model* are reviewed. The t e u t o n hag model la used as a framework to define a relativistic centcr-of • u s position operator. A procedure to c o m t r u c t aoving aoliton baga is described. Application!
to the calculation of charge radii, oagnetic moments and form factors of the nucleon arc discussed, with particular attention paid to the MIT-bag u n i t .
109
I. lntroduct inn
The HIT bag nodal it a simple modil of hadronic structure. In it» usual formulation, it consists of • static spherical cavity in the interior of which existi a certain energy density, I'Sii.jlly denoted by B. Inside the cavity, the quarks move as free Dirac particles. We shall consider 'here only u and d quarks, whose mass inside the bag can be taken equal to z e r o . In chis case, Che stationary quark wave functions obey the equation
(1)
where C is ths quark energy. R is the bag radius and the origin of the coordinate system is taken at the hag center. Quark confinement is imposed by the following boundary condition at the bag surface:
(2)
T h i s c o n d i t i o n g a r a n t e e s t h e a b s e n c e of q u a r k c u r r a n t f l u x t h r o u g h the s u r f a c e .
P o s i t i v e - p a r i t y h a d r o n i e ( r o u n d s c a t e s a r a maim o f q u i r k * in rlir lnw.'iL e n e r g y e i g R n f t a t e , i.a. ' * | / 2 'n " " stand.inl spti t r o n c o p i c n o t a t i o n . In t h i s c « * a , the s o l u t i o n s of ( 1 ) t a k e t h e f o r m
-(Lu) X _
*') • N
(I (1)I )0
w h e r e •* a n d X d « n o t e j :* . 1 j í i - a ; r i n w t t o r . i t i t f ei P , v j 1 i s p i n o r , r * * p « c t i v « l y . T h e f u n c t i o n s j r , j ( a r e o r i l i i . u y B e s s e l f u n c t i o n s . T h e f n e r g y e i g e n v a l u e z i s d e t e r m i n e d b y l h e c o n d i t i o n ( 2 ) , v i i i c h r e d u c e s C o jt )( t R > - j)( ? : ii ) . . l o s e l o u i t t s o I t i t i e n i s c • . , / P u r i t . i i u - 2 . O A . T h e c o n s t a n t V . i s f i x t d b y t h e n o r m a l i z a t i o n .
T h e t o t a l t n e r i( y o f c h £ s y s t e m i s
,m ^f-i^-B <«>
w h e r e N i s t h e n u m b e r o í q u i r k s i n t h e b . i f t - T h - : * v d l u e o f R i s f i x c á h y t h e r e q u i r e m e n t t b i t t h e q u a r k p r e s s u r e o n t h e s u r f a c e b e S a l a n i ' e d b y t h * ; p r < 4 S i ; i " t : e x e r t e d b y t n e o u t s i i l e v a c u u m o n t h e b a g . I n t h e s t a t i c c a v i t y d e s c r i p t i o n . C h i s c: o n i í t i < , n i s e q u i v a l e n t t o t h e m i n i m i z a t i o n o f ^ * * ) - w i c i i r »%s p ' i * . í t o ft, t h e r e s u l t ^ » f w h i r l » i s
(6)
T h i - d u f i i t h - i i f l i r i n t i t l ** 1 j u s t J f M - r i i ' r i ! i s u « ; u . i I L y i o : i n í d w i t h t h e i n c l u s i o n , i n p r r t n r b . i t i o n t h e o r y , o f t h e p f f e c t s o f g l u u n e K c h j n g o i n s i d i - L i n » h . i n . i n . ) . i f r l . , - p i . i i i r . n u i t o n t s i J , ! .
H i - a i < ! . • a p r u v i i ! m e i t . w ' , , ' r i i r i : . ( n c I » r y . l i - •, c r i p t i o n o f h r f . J r <i i - ; ) • • • . : I ( :< , f ti e h - M ' , P I u l r I ( ' • . • i t » i li i : i j I r. u 1 , i i i . n o f f o i ffl t .!'• ! . i r •. . i n J i •• I I | i i ! > , > . ' i i i •• . ( . a : ; i . c h a r g e r . i . J i i . i;» | m . i g n c i i r
,-. , , I . - I ; i ! , . I h . •, i . l l i i i . » i I I v . - ! i I i i :. . ! • . . • ! • « « i i l l - r i n ••• : ' . l i . n l I m i - i s
I l l
a s t a t i c c t i j r g i ' o r t i i i n i i i d i s t r i b u t i o n :
A l l t h e s e c a l c u l a t i o n s . i t t b . i á r d o n t h e a s 5 •; p t i o n t h a t t h i * h . t d n m c a n 1 >• t r e a t e d a s a h e a v y c m n y - q u a r k s y s t e m . I n t\\r c o n p u r u i " n o f m a s s e s u s i n g ( 4 ) , c h i s a s s u m p t i o n m a n i f e s t s i t s e l f b y t h e i n c l u s i o n o f a s p u r i o u s r . c o n t r i b u t i o n d u e t o C h e m o t i o n o f t h e c e n t e r o f m a s s ( 1 : , 111. ) i n t h e c a v i t y . T n t h e c a l c u l a t i o n <> í Í ; • r iu f a c t o r s , t h e r e c o i l o 1 t h e w h o l e h a d r o n u n d e r t l i e a c t i o n o f f l u -e x t -e r n a l c u r r -e n t i s n -e g l -e c t -e d . S u c h a p p r o x i r m H o n s H r -e » s i u l ! y v a 1 Í J f o r m a n v - b o < l y s y f . t i - ^ s , r n r r *•(• t i o n s b e i n g t y p í r a 1 1 y o f u r d c r 1 / N . F o r a t h r e e - q u a r k s y s t i r i r , w e m a y e x p e c t c o r r r c . t i i > n s o f r u u g l i l y J O S ; i t «* t t - i n s r h •• r e f 0 r «• n I T 1? s s a r y t o i m p r o v e c I n - t iv \i r.^-n r o f c m . m o t i o n b e f o r e i n c I u d i 11 R o t h e r e f f e c t s ,4 n d r o n p . i r i n i ; t h , p r e d i c t i o n S o f t h e m o d *? 1 w i t h i - K p r r i n n n t .
A l t h o u g h a L u r t?fi t / - i u v a r i .1 n t f o u n i i l . i i i o n u i t h e M i l b a g m o d e l i a p o s s i b l e , a t 1 t > . i < K a t t h e c l a s s i c ,i\ l e v e l , s u c h 1 t h u ; > r y i s r a t h e r i m p r a c t i c a l . F o r t h e p u r p o s e o f s t u d y i n g r e c o i l
• n d c m . c o r r e c t i u i > . 3a i t i s i r j r « c o n v r n i r n t t o s r . i r t t r . > m «1 ' . o n v n i t i ü n i l t e l â t i v i a ! i < q u .1 r t i t m E i »% 1 d t l i c v r y . T h i : - u l i L o n h u j ; T n o d « ' l o f F r i e d h e r g and I. r e p r o v i d e » « u c h • s t a r t i n g p » i n l . I n t h í s m o d i ? 1 , c m i f i it « m e n 1 i •? p r i n t u c i d I» y n m .% \;» r Í i • 1 'I w l i i h i n ( i r , M t « w i 1 h x \w ( p t i 1 V •• . n u i n m i - 1 1 D I M r I y w i t h t I >. »• 1 I . I h r M I I
U 2
b a g m o d . I r , m b e r e c o v e r e d b y ' . i n . a d e q u a t e c h o i r s o f p a r . i m < t i r s , I m ! t i i *1 s o 1 i t o n b . i g i s m o r e M »• x i h U - a n d a l l o w s .1 c o m p a r a t i v e
5 1 u * l y IJ t" v . n i o u s t y p t r s <» f c e n f i n » - m e n 1 . H i * r e w e s h J I 1 f n r u s o u r a t t e n t i o n o i t d e v e l o p m e n t s b a s e r f o n t h t s o 1 i t o n m o d e l . A * t h e r e
»• > j s 1 5 o t h e r a p p r »•• . » r h t ' s t o t l i f » p 1 o b 1 e m o f c - m . m o t i o n i n b a g m - i . U l s , t h e p r e s c n l a r L i c 1 e i s n o t a c o m p r e h e n a i v *1 r e v i e w , b u r
r a t h e r a d i s c u s s i o n o f s e l e c t e d w o r k s .
I it S e r 1 i o n I I , t h r s o l i t u n b a g m o J e 1 i s b r i e f l y r e v i e w e d .
I n S e c t i o n H I , I hv r o n u - p t o f r e l a t i v i s t i c t : . m . o p e r a t o r i s U M I I t o s t u d y c o r r e c t i o n : . » .j> b a r y o n m a s s e s a n d c h a r g e r a d i i . T h e
r c M i i » t r * i < r i MU t ' l q t i - t i k U J V P f u n c t i o n s i n 3 m o v i n g b a g i s p r e s e n t e d i n S n t i < M i I V . f h r S i ' w ; t v *» t u i i c i i o n1) .1 r < - u .s e . ! t o c a l c u l a t e f o r i r f j i ' t o t s .1 n 1) i 11 v f :• r i i ç a t K c o r r e c t i o n s t o n n r l . ' n n m . i ^ n * ? i i c m o m e v. t s .
C t n : i i i i u n c s a n i i r u 111 I u s t o n 5 a r <• p. 1 1 b • 1 v J i n S e r t i o n V .
. ihc s
j |: e- s o l i r o n n .(, ! i ' l • • ! b r 1 . - « I b « • T i ; i - , . i I v . •C '* * i -, . i p ) i » • n . M - . - p . j 1 ,- j i t 1 I m . ' . i i e 1 o t f j i . u k t i > r i f i n r m < r ? f .1 - H ' i i t i n " i i f : r j , ^ f
I 11 ; i i • -. 1 . . . 1 •• 1 , c . ' n r 1 n i • i t i f . - i t i -; n o r 1 |( |- . . - c >\ u u . - i , , . ] , n h o u r . t . i r y .. t u . ' ! í * ; k> n , N u t i «: . » < b i " v - i l l . v t i n - m i t o J ' r i » > ; ; , » ( . i f i <• t i t i o M
. i : • f 1 . " J * i ( h i t i ' u í t » l ,• 1 t . I 1 • • [ • ; I t r H 1 . , 1 Í < . - d 1 . ( d
I 13
j n U A , b . « , g a r e n u d e I p a r . i m e t t>r « . T h . - f o r r a ( I 0 1 i s d i , - t a t ,• J l>
t h i * r e q u i r e m e n t o i r 41 n o r n a 1 i z . i b i 1 i r y . F . n . i ü u - t t i :i A r i.» c S >:; • ;t s m - ! t h a t h "" > i . n j a i u l l> - O . t n l i ' r s t1 *: • > ti <J i c i J t t P r i u1 [><>r e » t i <t l D Í ' ) h i
t w o :ri i i i i [ L I . I , f .•> r . ' e U i i i i t
3 . i/!.2 flic
o » - - I- b + \ / h M l ) vac ijc I V i \
T h e c o n s t a n t p i s f i x e d s o t h a t t (*.T ) = f J ; o n e h a s 0 * 0 . s o t i n t
r v a c r
• i r= r* i s t h e .1S s o 1 i J r e m i n i n u T , . va<.
H J Í Í r r- n i c. s I .i f e Í , o t c c r t f i t i e d q u , i r k Ü c a n b e c o n s t r u r t f d i n t h e m c . t n - f i e l d a p p r o x i m a t i o n ( M F A ) t o t h e a h o v e t h e o r y , i . *•.
b y t r e a t i n ^ t h t s r ; i 1 a i í i e 1 \\ n s a t i ot*> -- i n t) e p •' n <1 e n t r - t n i m h t r t l i n t t i o n •" . . T h e q u a r k t i v I d i n . t y i > e o x p i r u l i- ci . i s
w h i ? r e { ^ f i s * t o r t p I *• t e o r t í i o n o r m a l s e t u f s p i r n t r f u n c t i o n . ' , s a t i s f y i n g
w i t h cv c h v q u a r k e n e r g y «< g r n v u l u r , A baryonic b»n s m n - i s ^iv. n hy
1M
w h e r e Cy v v i t a n a p p r o p r i a t e c u e f f i c i e n t . T h e m e a n f i e l d i s r a l e j l . i t f d b y r e p l a c i n g t h e s c a l a r q u a r k d e n s i t y b y i t s
«• x p e r t a t i » > n v a l u e , i . e .
V2 o • U'(u )
w i n - i i- t h e s u m i s o v e r o c c u p i e d p o s i t i v e - e n t- r g y s t a t e s o n l y , ( n t i e . i b s e n c e o f q u a r k s '_' - « . P a r a m e t e r s c a n b e t h m t n i n c h
t h a t , f u r .1 q u a r k « i e n s i t y s u f i *• i e n t I y l a r ^ e , t h e L o w e s t t M i e r > ; y f i j i i t i d u r a t i o n t u r r e s p o n i i - s t o "i n e . i r z e r u . I n t h i s c . i S * 'p t h e c o u p l e d e q u a t i o n s ( I 3 ) - ( I > ) a d m i t l o c a l i z e d s o l u t i o n s s u c h t h a t r f i e ' j i j . ' i r k m . i s s i s n e a r z t r o i t i t h e i n s i d e r e g i o n a n d g o ã t i n f i n i t y . I n t h e l i m i t o f v e r y l . i r ^ i - g o , c o n f i n t n ^ t i t i s o b t a i n e d .
I n t h i s m o d e l , t h e t i i a l e n e r g y o f a p » s i r i w - p a r i t y
! i . i d r > > ! i i<- t > r u n n d s u c r i s
E * N> • h "
r i s C h e l o w e s t f i f i c n v A l i n - u f
T i n - .*; n 1 u t Í o n s of e q u a t i o n * ( 1 3 ) . m i l ( I 5 ) tiri v*> b e t - t » s t t i . t i r . f , . n . i I y ( i Ktt) \y i n p u t i c u l . i r r i f e i h y F r i c d b c r g « n d L c c
t u t ! n n í H i i i . i i l y h y C o J t i l 1 . i m # i n » l W i 1 *• ( n . I t t u r n » o n I t o bv p . < s s i h i e I .» r l n n r . f j ' . n . i m f l « r . M M h l l i . i t I li»- c r . i t f . i l t,t,t I i . . m
O t .» ,» , ,-,> » •• 1-, , M I •. i n •( i t ' i ' i ' H i < i ( -i r b i i r . i r i I y « m i 1 1
|J ' v.n:
i 1 5
t l l U ' k l l t - K » , . 1 1 1 . 1 » l H - h t i l . l l t i l . H l ' i l 1 ( 1 . 1 1 O f ( | / ) ( t i n - M M l . l .
e t i i ' i g y ) i s t i e g l i i i i h l * • . » m (> . i • •• >l t . . I I n - - . I ' I O I U I t t h - v " l < n i n ' t i i I n t h i » a r t i c l e , w i > a l l i I 1 « I ' N s i i l r i . m l y i h i n l i m i r , w It i •• K c u r r t ' s p i H . d s l o l l u - M i l •••>!<• I . D i s i t i k s i n n » ••> o i n . ' i l y | > . - s n I c u t » ; i n i i m i i l l t a n h r l o n n . l i l l t l i r I i I r i . i t n i c ' ' '
I I I . C c m e r - o t - m a s » j n r K ' i ' t \\'n* l.\' ''"•' « • h j f f . u r i i > t i u » n i J t h e I > I I « T I ; J . ;
I l i e s o l i t k i n u i n l c l u . t ü n ü • • .1 b y l l i ( l i i . . | , t n l . t o e v a l u a t e c . m . c o r r e c t i u i m t u i m c l i ' u i i c l i n r n u r m l i i n n d e n t r ^ i v a . T h v b a s i c i d u j i * C o i l f t i i i c 3 11> I a t i v i » t i t c . m . p u s i t i o n n p i - r m i i r a n l l o i j o u l t o t h o c . m . p n v i t i ' i n n p u r t l l o r o f t h ü m m r r l n l i v l n l I t t h e o r y . T h e a u t h o r * o f R i f . ( 6 ) c l m u i t t h e d e f i n i t i o n :
t' I á i I ll^il) - - I <lx x \Ut) ' i / ( v ) * ; . ( i ) ( I N )
w h e r e H i * t h e K y n i m v 1 1 i z i ' i l v r r s i n n « f i t ) , i . i > . w i t h - i ^ ' n . V ' J ' r e p l a c e d b y - 4 Q > ' á . 5 i > - ( V * * ) , i i i ( / J . f t i > i m c f n l t u r e m a r k t t i . i t
Í i > r a I I I > ' i l I i i f l l u b I i i | n - i : i l o r K i l n f i i i r t l m»
Ki • M1" ( I ' M
w t i i ' i r M1' i l t h # i x i i n l K V I I I ' I . i I i / • • • ! * n y , i t \ n t - m u f n i i i l i n n I f i i s n r , I I " '
r a l a l l u n i t «imply
''•il l . || | | | . - | | . | | . | < I M I I I I I ' I . . ) K . I I I I I I I ' ) , l I I . I l l . . | i , , | , , . , 1 , I , , | .
116
117
not.it i o n inj u s i n g ( 1 3 ) ,
w h e r e tilt* f i r s t e x p r e s s i o n in v a l i d w h e n K a r t s o n a a t.»t e lu t h e r i g h t a n d t h e s e c o n d o n e w h e n it a c t s , to t h e l e f t . F o r t h e p r o t o n c h a r g e r a d i u s , o n e g e t s t h e r e f o r e
>
w h e r e < r > . is t h e » c a t í c - a p p r o x i m a c i o n v a l u e , a s g i v e n by ( 7 ) . T h e f i r s t t e r m o f ( 2 3 ) is s i m i l a r to the n o n - r e L a t i v i s t i c r e s u l t , w i t h m a s s e s r e p l a c e d b y e n p r g i e s . T h e s e c o n d t e r m is a rol.itivisi ir e f f e c t d u e to t h e qu.irk s p i n , s i n c e it a r i s e s f r o m t h e s e c o n d t e r m of ( I B ) . F o r a D i r a c p o i n t p a r t i c l e , t h e r e s u l t n n . i l o ^ o u u t o ( 2 3 ) is
T h i s t e r m is n o t i n c l u d e d in tliv e x p u r imenl.i 1 d o It rmin.i t i o n ul t h e c h a r g e r a d i u s . S u b t r a c t i n g ( 2 4 ) f r u n ( 2 3 ) , w e o b t a i n t h e r e c a i l - c o r r e c t « d p r o t o n ch.irge r a d i u s ;
I n t h e M I T - b a g l i m i t , f o r m u l a * ( 7 ) a n d ( 2 5 ) . t o g e t h e r w i t h ( ) ) a n d ( 5 ) , g i v e
< r V *) - .S3 R2 Ub.*)
< r2> ^ - . 3 9 RZ iib.b)
T h r r e c o i l c o r r e c t i o n i s c l r n r l y q u i t e s i g n i f i c . i n t .
L e t u s n o w c o n s i d e r t h e n u c l e o n m a s s . S i n c e f o r a l o c a l i z e d s t a t e < P > * 0 , t h e e n e r g y o f s u c h a s t a t e c o n t a i n s a c o n t r i b u t i o n c o r r e s p o n d i n g t o c . m . m o t i o n . A s i m p l e p r e s c r i p t i o n f o r t h e c o r r e s p o n d i n g m a s s i s g i v e n b y t h e r e l a t i v i s t i c r e l a t i o n
r 2 -i~\
M » < H > - < ?i- \
lf2
(27)
In the HIT mode 1 . the Dirac. equation (1) gives, for quark i.
( 2 8 )
S i n e ' t l i e b a g i L s e 1 1 c a r r i e s m i n i o n i e n t n i n , o n e g e t s
U s i n e , ( 2 7 ) , ( 5 ) a n d ( ? ' l ) , o u r o b t a i n * M2 - 2 . 1 0 [<;.• V f m ]2/ K2. 2 p 1 2 2 W i t h o u t < • . ! » . f o r i r c l i n n s , I h r r i ' s i i l l I K R(, • 2 , SH | C r V f m j / R . W i t h t h e h r l p o f ( 2 6 ) , o n e m a y r i i m i n a t c t h « bit)', r n d i u » f r o m I h r s p r c l . i t Í L i n n i n o h t.'t i n r r l a i i H U M l i i i u i r i i m . i ü s t " . n u l i h . u(r r . t i l i i :
F2 - '•'"•. [ c . V l,i,|-' ( M l )
1 19
i n t h f s t a t i c . i p p r o x i n . i t I o n a n d
' . H7 r "I
-M . - - -- i ; . V 1m ( 3 1 >
w L t li c . IT . c o r r e c t i o n s . I f t h e b .n\ p i c s s n r r B i :. c h<< s « - t i s u c h i s t o r e p r o d u c e t h e e x p e r i r n f n t . i l c h n r ^ e r ; i d i u s r • - , f ) 9 f m ,
..mi-2 2 . i 7
o b t a i n s E » 1 . 9 9 G e V i n t h e s t i n c a p p r o x i m a t i o n . i n j M* - J . í 9 C \ V "
w i t h c o r r e c t i o n s . S i n c e w e a n : i g n o r i n g g i u o u i c h y p r t t i nt -s p l i t t i n g -s , t h e c o r r e -s p o n d i n g e x p e r i m e n t a l v a l u e i -s t n e -s q u i r e o f t h e c e n t r o i d m a s s o f t h e n u c l e o n - A s y s t e m , i . e . M " • 1 . 3 7 G e V . C e n t e r - o f - m a s s c o r r e c t i o n s a r e s e e n t o i m p r o v e s i g n i f i c a n t l y t h t a & t e e i n t ' n t b e t w e e n t h e o r y a n d *.* x ji r i i m * n I . O n *• m a y h u p o t l i a c t h t1
r e n a t i i i n g d i s c r e p a n c y c a n b * » i L T o n n t t i d f o r b y g l u o n t ' c i n d p i o n i c p e r t u r b a t i v c f o r r e c t i o n s .
I V . M o v j _ n ^ J > a R S j n ^ f o r m f a o t n r
A s a l r e a d y m o t i I i o » t , - í l , t í i c r . m . y o s i t i * m o p r r ; j r n r i n t r o d u c e d i n S e c . I l l i s s i m p l y r •• 1 ;i I <• •! I n t h e b . m s t n p u i a t o r K , t h e g e n e r a t o r o f p u r e l . o t c n t / . I r . i n s f u r m n t i o n s . F o l l o w i n g R « f . ( 9 ) , w » » h a l l n o w a p p l y t i n ; h o o s t c o t u . p t t o t h e M K A o f t i n
• l o l i t o n m p d t l . T h i i w i l l a 1 1 1 w a d c i p c r a n J h r i i . u l u r s t u d y o f r e c o i l r f f s c t » i n h a d r o n i c f o r m l a r ' . u r » .
U n d e r A n n c t i v » L u i r n i / . r t . i n s f u r m . i t i • > ! ) , c l i . i r M I e i i i t ' l l b y f l u ; v i l o i i t y V , a . C 1 1 r « r r l u r • • .> n •• i •> t m i ;i »
UCv) - g11'™-1* . u-at;iPh v ( H )
In t h e M K A , q u a n t u m f l u c t u a t i o n s i n t h e s t . i l a r f i e l d a r c i g n o r e d a n d C h e L o r e n t 2 - t r a n s f o r m a t i o n l a w s f o r I h i s f i e l d a r c t h o s e o f c l a s s i c a l m e c h a n i c s . O n l y t h e q u a r k s a r e C r e a t e d q u a n c i c a l l y a n d t h e b o o s t o p e r a t o r b e c o m e s a o n e - b o d y o p e r a t o r a c t i n g o n t h e q u a r k v a r i a b l e s . U s i n g ( 1 8 ) , ( 2 0 ) a n d t h e n o t a t i o n o f ( 2 2 ) , w e u r i l i ,
K .v
w i t h
-r
H . < * ) -- \ . J • (Ç ' . ( I ) ft ()b) v i v
T h i ' $ » ( > s c r i p r v i m i i c i t e s t h a t i n t h r M K A , t h e i n o r f ; y d i ' i i f i t y , i n d I l i c r i ' l u r i1 I h i ' b o o s t i i p t ' i . i t . n d i - p t u d m i I h i ' v r l n c i t y o f l h e
l i n i t D n i t . U c o n w l i i r h t l n - y a c t . T l i i i ; r r s u l i s f r u m t h e l o r i ' i i K t - o o i i . 1 1 t i o n o f l l n n u ' . i n f i r 1 . 1 i n t i n ' d i r i ' i t i o n o f m o t i o n , i . e .
( U i )
- i * i i n i i i t t i i l I * >• i n i . i 1 I o i f i l l " I i , M I •• t •' i i n 1 1 i o n o f I l i r w . i v r f u n i l Í I H I , | , ( I ' M k I " I I I ' I ' . l l ' , t i l . I , I I I ( 1 . 1 ( U . I , ' Í V I
-I - i.l C . K . < • . « « ) -I W ) VI V
T h i s e q u a t i o n c m hi' s a l v e d w i i h t h t » h i ' l p o f ( I I ) ; t h e r r s u l t i s
* . ( x ) - S(v) j. (cosh a , • ix) e (IB>
V O x
vhcre
I t i s p o s s i b l e Co s h o w t h a t t h e e n e r g y a n d t h e m o r j f t i i urn L i t l i u l . n c v l w i t h ( 3 5 ) and ( 3 8 ) for tho q u a r k s , s u p p 1 emrn t L-II by the contrilmt ions itom Che c l a s s i c a l ' - f i e l d , form .) f o u l - v e c t o r u n d e r U r c n l t t r a i l s f o r m a t i o n s .
N u c l e o n form f a c t o r s a r e r e l a t e d t o c u r r e n t m a t r i x e l e m e n t s of t h e t y p e
(40)
w h e r e 11> > i s f o i i c - i n n n f n t i i i e i g i i i s t » t c o f t h v n u c K n n a n d i | - \>' I t i » c o n v e n i e n t t o u s e m o m e n t a r i l y a b u x n o r m a 1 i z n l i o n :
P it
w h a r * E [ ? * • W*J . I n t e g r a r i n g ( 4 0 ) over t h e v i . l u m c u f t i n -b o x , o n » g e t * .
(41')
w h i r «i I M il u ii ii 1 •• K .1 H l n l r i i í H I I I I I I I I I I urn / i - r o . m i l II ( v ) i t I I n '
trancfor.-n.it ion introduced a b o v e . Tbc static approxiaat ion is obtained (tom <&2> by i) ignoring t l i c boost t r«nsf or «at ion, i.e.
H<v) ' I, and ii) identifying | H > with a localized bag itit<
31 icsi - |M • 18 :-. In order to investigate relativistic recoil
r f I iir i s , ii si'cn.5 reasonable in maintain this la*t identification, hut tn take inlii account the honst transformation. One should taplusiie that chi» prucedutu unly garantee* that the states d e s c r i b i n g ooving nucleonf corrr.i|>nnd to the appropriate momfntum in expectation v a l u e ; they are nut momentum eigenstatct and are not orthogonal for different m o m e n t a . These defects could be removed using a projection method , but this has not been carried ou t.
Chousing to ealcul.itf in the Broil fr.ime, in uhich p1 . -5 • q/2, we write <AZ) in the Cora
tlir iiuiviii»-, h.-ig nc.it e
i s « o u s t m i ' I I'd w i t h t h t > 1 H . 1 r V y . i v i h i n c t i n n s ( J H ) . I n r l i r r . i s t1
o f ( t i . - «• l> i I i iim.i)'.iii' t i r i i i r r . n f , I I n1 •• 11 >• I r n ( <:. ) a n i l mty.nelie ( i ;M) I K I ' D I f . i r I H I I» r a n b e <• M r . i - < <•<! I r t<m I'» I ) u s i n y .
\^^
w h « ! i e s , s ' . 1 1 »' s p i n i m f i r *• s . n u ! \ i s .1 n i n l i . i i i Í V i t i ! • s p i n o t -L o t u s c -LI ti :• i d i ' r - « p i í n t> r i tr f l v t t u * <_ í i . i r £ • : í > i > l i u s . B y
* l f f i o i t i i » t » ,
*,.,..[
i - --b uU s i n g C ^ 4 ) , ( 3 3 ) j n d ( ^ 9 ) , i i n i ; i r . j y r e w i i t e ( 4 3 ) i n t h < - f o r r a
|- f>
P u t t i n g
( . ' . : - • >
c x p . i n t l i n g t u O ( < 1 ) a n d u s i n g ( A r> . n ) n n d ( 4 6 ) p i t n c Kt : l-S
3
I j e . ( x . - K) | B > (•'•'•>
i »I ' ' °
w h i c h i s i d e n c i r . i l t o ( ^ 1 ) C ' - ' i ) .
A l r h n u )•, h b u i i . s t i n g I l i e I • . « > • l e a d s t o t h e s . n : u - i - r. p i .• i s i t» ••
í o r l h e c h a r g e r . i d i u » « i s t W n » e o f t h t > r . m . n p r r s r u r , I h f f r u n r r flppruftch i s m u r t r n i i n t . i l . i t i j | > r i > c 1 n r i - f t . 1 n o w n - i n l t f u r I l i > ' | > n > t " i i m . i g m t i r « u n n r n l . K • u r n C D , ( ' • ' • ) , ( ' . H ) , ( l ' i ) . i m l ( I I ) , , > n r , • . • ! • .
124
íÈ~ *i * *f l
Eo
* 0 ( q 3 )Comparing t h i s t o ( 4 J . b ) , out obtain* the Magnetic moment
2E- V " Ç» / "
• « • ¥ í o *)•'•>
-where u is th« remit obtained in the static approximation (S) and a'j is a correction originatint from the second ttrn of (34) and corresponding to tha rotation of the quark spin in the lotjiti transformation (Wigner rotation). In the HIT model, the
«•v« function* (3) yield the formulae
„<•> . *£. H* | d> »3 io(ex) ],(£») (52.a)
° o
tnd th* numerical values
M( B ) - .20!» I t àv • .040» I (33.•)
MW-.rtt« (fl.fe)
0*« •••» «•»»» • » • kfncM»ttc«l correction O(f-) ana ch« sptm
125
r o t a t i o n e f f m l e t both of o r d e r / 0 - ! M . Howrvrr. • • • £ • t h e y arc of o p p o s i t e s i g n , t h e t o t a l c o r r e c t i o n i t only of about 5 1 . Ue should a e n t i o n t h a t s i m i l a r r e s u l t * have been o b t a i n e d by other authors . Although t h e r e i s unanimity on the spin-rotation c o r r e c t i o n , the p r e s e n c e of a k i n e m a t i c a l c o r r e c t i o n for the
• a g n e t i c moment i s not r e c o g n i z e d by a l l
With the h e l p of t h e quark wave f u n c t i o n s ( 3 ) and formula ( A 3 ) , f o r n f a c t o r s can be computed nmnuri c a l ly as functions of q . F i g u r e s 1-3 show r e s u l t s of such c a l c u l a t i o n s for t h e e l e c t r o m a g n e t i c f o r a f a c t o r s as u e l l as for t h e a x i a l - v e c t o r i o i n f a c t o r G,, which i s g i v e n by
<f .' |5 « J*(o> I- I s> - <± I T I 1 » ^ Ü
X;. Í - 5 X, C
A(,
2> (54)
where T i* an isospin Pauli m a t r i x and
(55)
In these c a l c u l a t i o n s , the b a g radius is chosen such as to reproduce the cxperimpnt.il p r o t o n charge radius (for the calculation with recoil c o r r e c t i o n s ) , i.e. K * 1.3 tm. O n e tees that (he agreement with data is much improved by chc inclusion of recoil effect!, although C£ remains too iron 11 for q J, 10 (•' . Of course, comparison with e x p e r i a e n t b e c o m e s acad.mic for l»r|« 4 since the bag mod** i« not e x p e c t e d to be realistic in that r e g i o n .
12S
127
Ack««»ledga»««tt
Tha author ia grateful to tha organiser* of the V Encontre Nacional de Física de Energia* Intermediária» for giving hi* tha opportunity to present the above salarial. Ha alao expresses his thanks to R.Coldfla». C.Krein, Th.A.J.Marls and L.Wilets tui many useful discussions. Financial support from CNPq and FINEP is gratefully acknowledged.
I2tt
129
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F a c t o r » , V o l . 8 3 o f S p r i n g e r T r a c t s i n M o d r r n P h y s i c s ( S p r i n g e r , B e r l i n 1 9 7 9 ) :
J . I H i c H u l l e n a n d H . D . S c i d r o n , P h y s . R e v . D 2 0 ( 1 9 7 9 ) 1 0 8 1 ; F . D . G a u ! t a n d N . D . S c a d r o n , N t u l . F h y s . B 1 5 7 ( 1 <) 19) 5 1 7 .
130
figure cjtjilion»
F i g . 1 - C o m p a r i s o n o f t l i t r M I T b a g m o d e l e l e c t r i c 1 .> i m l . i r t u i < : u i t l i e x p e r i m e n t . S o l i d l i n e : f j l c u l . L t i o n i n < I m l i n p r e c o i l e f f e c t s ; d a s h e d l i n e : s t a t i c i p ; n . • x i n a l u ' n . T h e e x p e r i m e n t a l v a l u e s a r e f r o m R c f . ( I S ) .
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F i g . 3 - T h e s a m e a s F i g . I f o r t h e a x i a l - v e c t o r f o r m f a c t o r 0 ' , . T h e d a t a i s f r o m R e f . ( 1 6 ) . T h e d i a g o n a I - 1 i n e b a m l
i n d i c a t e s t h e u n c e r t a i n t y i n t h e e x p e r i m e n t a l a n a l y s e s .
131
132
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fíj.3
134
INTERMEDIATE ENERC.Y Sl'CI.EON'-Nt'CLEUS X I REACTION CROSS SECTION IN THE tURAC
PHOtCMENilt.OGV*
B.V. Carlson am) M.P. lsidro Filho
Divisão de Física Teórica, Instituto de Rstudos Avançados Centro Técnico Aeroespacial
12200, São José dcs Campos, SP, Brasil
and
M.S. Hi'Ssein
I n s t i t u t o de F í s i c a , Universidade de Sao Paulo 20516, São P a u l o , SP, B r a s i l
t Supported i n p a r t by the CNPq
* Bai.«d on • til* pri-nuntcd by M.S.H. »t the V Fncuntru n.u:i<jna\ de F í s i c a d< Erergfu» I n t e r m e d i á r i a s , Gr.im.ido, RS - Q'l a 09 dr maio He 1984.
I1.iv/1 «1)4
I3S
Abstract
The total reaction cross section for intermediate energy n u c l e o n - n u c l e u s scattering systems is calculated within the Dirac-eikonil formalism. Comparison with data indicatetthat the recently proposed impuIseappro»imation Dirac optical potencial for n u c l e o n -nucleus scattering, is not absorptive enough.
1J6
I - Introduction
In the last f«v years the Dirac equation used with a mature of phenoaenological scalar and weccor interaction. h*§ been shown to provide a greatly Improved starting point for understanding intermediate energy proton-nucleus scattering* . Calculation b » e d on an impulse-approximation optical potential gave excellent agreement with data on elastic scattering
differential cross-section, spin polarization and spin rotation for systems such as 500 Mev ^f #|-f &Q %rA * ^ .
Parallel to the above development» several attempts to derive the relativistic nucleon-nucleus optical potential have been made. These range from extending the faailiar impulse approximation "£7*"-type ütrivacion to include explicitly scalar and vector components . to more ambitious plans starting from a relativistic many-body field theory of interacting nuclron*
and mesons. One such theory, which is extensively cited, is that of Walecka . Though originally constructed to describe nuclear matter as a(4) tyiiem of interacting nucleons and isoscalar scalar and vector meson*, it is also adequate for the description of spin saturati^,! sosoilar (closed-shell) nuclei such a* 0 and Cct- The inclusion of the isoveecor B and P nesnn*
in the theory was subsequently performed by SÊTA • In oojt of the
applications of the theory, special emphasis was placed on deriving the real part of the nuclenn-nudeus optical potential (the tingle p.irt icle potential).
In a recent work Horowitx , calculated the rel.itivistie iii«*in*ry potential to lowest order in nuclear «utter for the e». h.in, e of tf* , IV •"•<
1T -mesons. Of inuric the rtfl«ti"tst ic Jtj putential rrfrred to earlier dorf tuppjy a well-defined íiuglnary pottnliãl. whirh is dirrrtly rcl»cpd to ch«
scalar <n<l the time cumponrtit of th< vector mwlr.ir >U-n<iífy. It would hi' impo.-rant Co check these potential* in a direct way.
137
An important obs«tvable quantity that «directly relateA to the imaginary part of the optical potential i* the total reaction cross section 0*. Though obtainable froi« an optical oadel analysis of the elastic
scattering data, it is, nevertheless, of value Co calculate 0j_ directly.
Such a calculation would supply a further test of the adequacy of the theoretical imaginary potential and help analyzing its reactive content.
The propose of the present paper is c develop a theory of vi within a Dirac description of the elastic scattering of nucleons of nuclei. We usa the eikonal approximation in our discussion of the nucleon-nucleut elastic scattering amplicudc. Such a Oirac-eikonal approximation hat recently been put forward by Amado et.al and Friar and Wallace .
The paper is organized as follows. In Section II we present a denivation of Q|^ from the Dirac equation thac describes the scattering of nucleons from nuclei affected by complex scalar and vector interactions. Ue then use the eikonal model in Section II to express tfV in terns of an impact parameter integral involving relativistic nuclear transmission eoefficienta.
In Section IV we present the results of our calculation of (7g for >-f C x
and p f PI» in the energy range J © < E> (1000 Me? , and sake a
comparison with the data, as veil as with the non-relatlvittic calculation of Digiacono, De Vrict and Peng . finally, in Section V, we present several concluding remark*.
1 1 " The Total Reaction Cross Section Obtained from The Dirac Equation
The Oirac equation that describe the elastic scattering of anurIcon, trtatcd as a Oirnc particle, fro* a spin-saturated nucleus, 1* usually written in the form, using a time-independent description,
138
(1)
where it is aesuoad that the average, couple*, nucleon-nuclcot potential is
• sua of a scalar component. Y y , a nj the fourth (time) component of a vector potential, Ya> . The aatricea CC and B arc Dirac'i, and f ia the four-component vector wave-function,
lac us write \g and V9 aa
(2)
Equation (1) can be rewritten ai
(3)
obtained froa the usual relation» I J" í ^ »< JT» 5 /3
We now perfora the usual oanipulations of multiplying Eq.(3) froa tht left by V|< = Y }> a n4 constructing its con jugate with Che subsequent aulciplication froa the left by y , to obtain finally
(4)
O>
t h e u s u a l W r o n l h i j n A r g n t n e n l n n w iu|i|)li>> , n w u h ' »i >• ' i m i i n u i r y r | i K i J>I
(6)
with
» A t y
(7)the hadronic current.
Interating Eq.(6) over a large volime and using Gauss's theorem, giver us
(8)
where the integral is over a surface smiounding the por^ntial. in a regian where the potential has coapletrly vanished, .ind describ'-s the net inv.ird