Revista Mexicana de Física 24 (1975) 97-104

SYSTEMATICS FOR ELASllC SCATTERlNG OF

ELEMENTARY PARllCLES

Leona Marshall Libby

R & D Associates, Santa Mónica, Califa 90402

and

Engineering Sehool, VCLA, Los Angeles. Calij. 90024

(Ree ibido: abril 1', 1974)

97

ABSTRAer" : A rclativistically invariant scaling law is proposcd which

yiclds elastic •.scattering curves for elementary particles very

similar in shape to rhe well •.known Morse potencial for atomie

intemetions. The maximum depch of the curves is three times- .larger for p"'p and 7T_P scattering than for K -Py p_p and n-p; atentative explanacion for this difference is put forward.

In chis festival we honor Professor Manuel Sandoval Vallarta. Hevery early had great foresight in recognizing the imporrancc of planetaryphysics and cosmic rays to an understanding of che physical world. Thesewere chen new •.born fieJds; we are now seeing an explosion of knowledge inthese fields.

" Supplied by the Editor.

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Fig. l. Examplcs oí [he diagram of porential cnetgy (oedinare) versus separariondi.sranc(' (abscissa) for [WO aromic spccies, taken ftcm Molecular Speclraand Molecular Structr~,(,. 11. ln/rared a"d Rama" Spectra 01 PolyatomicMo/ecules. by Gcrhard Hcrzberg, D. Van I'\ostrand Co., Inc .. Princcton~cw Jersey. Pp 99 & 199, (19.15).

SY5tematics lor f-'/a5tic _~catlf'ring. , , 99

One eonsequ{'nee of research in cosmic rays has been the discoveryof almost aH of the elementary particles of lifetime about lO-la seconds orlonger. Today, beams of these particles are obtained at cosmie-ray energiesfrom the large partiele aceelerators. The most abundant of particles in suchbeams are protons, fleU[fOnS, positive and negative pions, posidve and ne,ga-tive kaons, andprotons, etc. \l'e shoot particle bcams at a protun targct andobserve scauerin,g, riJar is, rhese particles are inreracting \\'itll prutons roresulr in proron-protun scattering, pion-proton scatterin,g, etc., evideflce forinterac don en('r~ies. lIo\\' can thcse intemetions be representcd?

About 40 }'('ars ago, P.~J. Morse found a theoreticaI representarionfor rhe inrcraerion energies of various atom pairs as a function uf interacriondistance bnwecn t1H.' atums, /H,) vs , (see Fig. 1). Fur elem<,'ntaryparticles,rhe many succ<.'SS('S of the quark model of an elementary particle has led tospecularion as ro wilat forces and potcntial energies are involved to hold rhequarks together. Furth<.'rnlOre, similar force s and porenriaI cncrgie,~ mustrhen be invoh'cd to account for particle- particle intcranion I .,

I,lre have found2 that the functioo (da/d In 1)1 is a sealing la\\' forelastic scattering uf eiem('nrary particles 00 prorons, and \\'iril sorne jusri-ficarion we havc related it ro a potenrial energy R{,) and ro an interparticledistance " according tu,

leF. - F.o(da/d In 1)' F.o - h'/MV , (1)

where t is the four-momentum transfer and M and Vare a mas s and an [nter-action volume characterizing the rwo colliding particles. This function isrelativistically invariant, as it should be, becausc we are studying particlesat cosmic-ray cnergies, many times their rest masses.

Why is ir appropriarc ro use eIastic scattering? Bccause the sum totalof all interactions bc,tween a pair of panicles evidences itseU in the elasticscattering.

The function (I) has be en obtained directly from Fermi's -GoldenRule". The reJation of t ro an interparticle distance is suggesred by dlCoptical model for clas[ic scattering in which "(,, z: p/h) and , entcr symmetri-calIy. Experiments are mOldeby varying " and soIvinR (or T

O' bur in principie

one could vary , and .••olvc for an appropriate "o' Thar is,k, -, •.

O o' (2 )

100 l.ihhv

FipIrt's 2-8 si",,, ,lit' scalll'fill~ dol!.• f", partide ener~, ,,1 ';-2'; (,('\.plottnl ,u.:ClHding ro funetions (1) <lml (2) in (!le form E(r) \'S r. llne '{"{""

eh,u lhe PO(,IHia! (,Ilcr,gy of ill(('f.leti(lll i:-. Zt"rtl fm Lu.cC' distanc('s, hectlllH:'

l.n,J.;t.' ami Ileg;ui\"(' al in(Cfllledi;lI{, •..•ep,H<ltioIlS, anO hecomes bq.!.l' ,lnd p(l ...•i-[¡V(' .11 .•..•mal] distances. The sillliLlrity of tlJ(" ..•liapes of fhe ClIfn .•..•kad.o,; lll(~

ro ;) ....1.; lhe qucsr;nll whcllief dH'r{, is a universal scalinf!, function for eleml'tltar}'.

panicle illll"r,lctinlls •.3 If fiJe feduced lIla,ss is used in Eq. (1), lIJen th('

rnaxilllUlll deplh of rhe curve hCCllfTll'S idell(icl! for K+-p. p- p. ;llld 11- p...•catlnin}!., \\hile th<lt for p-p and piOIl-pr(llOn is abour rhree [illle .... larf!.ef.

1 han' fUI firm undefstanding of (his differencc bu[ spcculate as (ollo\\'s.

In I\.+-r, p-p. amI n-p inter.lnion •• 1lHlihilation Joes no! occur (no

....(ran¡.:.e particle is kno\\"n havin¡.:. double positive charg.e), but in [he case of

pioll.prowrl <lnd p-p inceraction annihilation may occur, prodllcifl~ dt.ltas

\\'ith r\\'o ch,H¡.:.es and zcro char~t., and piolls of toral zero charg.e respeni\'( ..ly.

Sinc(' rhe diffraction partern of eL1sric scaucrin~ results frum thc rtHality

of all in(('raceion.s including annihilation, (he differences in \\"("11 deprhs

b('(\\'('e11 [iJe (\.lo'O sets of curves lJIay represcll( the differencc be(wcell (he

pft' S{'IJ(:e or absencc of al1l1ih i lar j 011.

o O'k o EPo o o o o o

~ ~el!'"8: 6' ¡¡N 5:o,-Cl OO~E ~& ••

z 1.0 J;N..... ;r- 1.5~b

1P~"O2.0~

o

2.5

.5 1.0

tl/2 IN (GeV/e)

1.5 20

• ¡Fi,¡.;o 2. (ld(IT/dl) versus I for neutron-prOwll clastic scatr("rill~o

Sys'e".a'ics lo, elas,ic scallering ...

O

101

(d"/d Int)''' vs (' FOR {pELASTlC SCATTERING FROM 5 ~ 15GeV/C LABúRATORY MOMENTUM

~LEGm-• 148 GeV/C ,128 098-/.3 '68 '497

= 10~--L>~b

L> 15-O 2 4 6 8 10 12

tl/2 IN GeV/G1.4 16 18

Fig.3. (ldC!"/dl)' versus" for K+p elastic sC¡Htcring.

o • 1::.,1::. 1::. 1::. •• • ••••• la• • 6 Afr¡J'.N

"-

oit".o - 5 '"E o

z o o J."oo

#~~N o"-

o

~ o Differential eross sectionb -10 i .0.

"'lo for n+ p elastie seatteringL> fe "~J

- ., O "íP' O l' O S 8s 90' baryeentral angle.)~~ 6-17 GeV/c laboratory momentum~.•

-1.5

O .5 1.0

tl/2 IN (GeV/c)

1.5 2.0

Fig.4. (ldu/dO! versus 1" for pion (+)-proron elaslic scattcring.

102 Libby

o

zN,"O

b -1.0"O

Diffenntial erass seetianlor Il-p elastie seatteringo s Bs 90' baryeentral angle6-2~, GeV/c laboratory momentum

o

-15

.5I

1.0

t'/2 IN (GeV/cJ

,1.5 20

Fi~. '>. (td(J'/d,~tversus '~for pian (-)-proran clastic scattcring.

18

(; do-/dt¡''' vs ¡'" FOR PROTOI,-PROTOI, ElASTIC SCATTERING FOR615 GeV lABORATOR! ENERG!

-l[GElID-• 1463 GeV/C -1188 ,/111 "1984'196,191 +184 0167 '148-118,114 .66,68 ó85 0108,1094

,14 I 610 1/

t'/2 IN GeV/C8

,64

'" °r=r

"'" ó

•~=;::? -101. •= l ,,~== .,

- "-10l 0"t

'"

Fi~.(l. ('d(J'/d')~ \'(Or.<;u'.; "for proton~proton clascic scattcrilI~.

Systematics /0' elastic scatte,ing ... 103

1Id,,/dll'" vs t'" FOR PBAR- PELASTIC SCATTERING FROM 020 GeVLABORATORY ENERGY

-LEGEND-BARYCENTRAL SCATTERING ANGLE

• (60<5)' • (50<5)'o (40<5)' • (30<5!'

o""==~ -2=<..0~= -4==

O!

u -6~bu- -8

O 4 6 8 1.0 1.2

jl/1 IN GeV/G

o •

14 1.6

•

1.8

Fig. 7. (td(f'/dt)! versus t! for antiproron-proton elasric scattering, at30 dcgrecs to 60 degrees in rhe center oí mass .

1I d,,/dj)'12 vs t'" FOR PBAR-PELASTIC SCATTERING FROM 0- 20GeV LABORATORY ENERGY

-LEGEND-BARYGENTRAL SCATTERING ANGLE'• (70<5)' , 180<5)' • (90< 5)'o (100<5)' • 1110<5)'

•1

'" O<O

"'"~=<-0 -2~===~

-4-U~b -6U-

-8O

.,

2 4 6,

8 10 12¡'/2 IN GeV/C

14 16 18

Fig.8. (tdr:r/dt)'1 versus t2 for antiproton-proton elastic scatteringt

at 70 deg[ees to 110 deg[ees in the cente[ of mass.

104

\\'e ha ve enjoyed a (case of cxciring papers ae chis festival (o honor

Don .\fanuel and \\"(' wish (here ro be maoy happy, rewarding \'allarra Fesrivalsro come.

REFERENCESl. See for examplc H.P. FeYDman, Science 188 (1974) 601.2. L.~1. Libby, Prnc. Nat. Acad. Sci. 69(1972) 3524.3. 1 rhank Dr. Edw,Hd T('!ler fOf chis su~¡:estion.

RESUMEN'

Se propone una ley de escala rclarivi,'Hamcnrc invariante que propor-Ciona curvas de dispersión elástica entre panículas elementales dc forma

muy parecida al bien conocido potencial de '\lor5(, para interacciones atómi.caso La profundidad máxima es mayor por un factor de tres para p-p y 7T-p

que para K +- p, p"p y n - p; se ofrece una expl ¡cae ión renta t ¡va para e sta di •.fereneia.

:\ car~o del editor.