IL NUOVO CIMENTO VOL. XIV, N. 1 1 ° Ottobre 1959

On ~-Meson Electron Scattering.

B. DE TOLLIS

l s t i tu to di F i s i ca dell' Universi t5 - R o m a

Istit,teto Naziouale di F i s ica Nucleare - Sezione di R o m a

(ricevuto il 12 Agosto 1959)

Assuming a possible coupling of ,~-meson with a spinless boson neutral field (~-meson) (,-s), we siiall a t tempt in this note an evaluation of the form factors in ~z+-meson electron scattering (knock-on processes)(6). We consider both cases of scalar and pseudosealar G with non-derivative couplings.

The interaction hamiltonians are:

in obvious notations. In perturbative approximation the lowest order diagrams contributing to the process are:

, . 0, ol % [ i g ~T III V El

The contribution from diagrara II is zero for both scalar and pseudoscalar ~. I t is sufficient to evaluate the first of the following five diagTams. By subtracting from the resulting expression its value at q2= 0, one automatically includes the self-energy contributions of the diagrams IV-VII.

(1) j . ~CHWINGER: Attn. Phys . , 2, 407 (1957). (2) W. S. CowLAx:O: Nucl . Phys . , 8, 397 (1958). (a) I . R . (]ATLAND: Nucl . Phys . , 9, 267 (1958-59). (~) I . SA.kVEDRA: N u e L Phys . , 11, 569 (1959). (6) S. N. GUPTA: Phys . Re c , 111, 1436, 1698 (1958). (,) Processes of th i s k ind h a v e a l r e a d y been obse rved wit l i cosmic r a y V.-nlesons, sec for examp le

references in Progress irt Elementary Particle ar~l Cosmic R a y Physics, 4, 107 (1958).

254 B. :DE TOLLIS

0)

The resul t ing finite ma t r ix e lement is:

M = i e ~ ( 2 ~ ) 4 ~4(P' + p ' - - P - - p)(g~,)J~up) ~ , F~(q~)y~+ ~-~F~(q~)~,v " q~ ue ,

where : p ~ ( p , ie) t he is e lectron 4 - m o m e n t u m before scat ter ing, ;

P ~ ( P , i E ) the is tz+-meson 4 -momen tum before sca t te r ing;

p ' and P ' are the 4 -momenta af ter sca t te r ing; q = P ' - - P = p - - p ' ; h = c = l ;

is the ~-anomalous magnet ic momen t (units e / 2 M ) ; a ~ = [VvY~ - - Y ~ ] / 2 i .

Moreover here and in the fol lowing:

M is t he ~+-meson mass,

m is the electron mass,

/~ is the , -meson mass,

= I ~ l M 2 ; ~ = m l M , ~ 0.5.10 2.

All energies and m o m e n t a are divided by M and are therefore pure numbers . F 1 and F~ are electric and magnet ic form factors and for the two cases (S and P) are given b y :

(2)

1 1

F~(q 2) = 1 Y "[ x2 - v x + v + q2x~y(l - y) ~

0 o

1 1

~s F~(q2) = d Y x 2 - - ~x + V + q~x~Y( 1 - - Y) ' o 0

(2 - ,j]j x ~ - - ~x + '

(3)

1 1

G ~ [" [" j q 2 y ~ 2y F~(q 2) = 1 - - I d x | d y x 3

4 , ~ J J x ~ - ~ x + v + q~x~y( 1 - y) 0 0

1 1

G 2 f F xay - - T J d x J dy , x + , + q,. y(l - "

o 9

+ x ~ - - V x + '

(G 2 = g2/4~) .

The cross-section (val id in any reference system (')), eva lua t ed f rom (1), is

(4) q, v/(p/))~ - ~ ~'(E + ~) - - IP + P l c o s 0'

(*) See for example JAUCH and ROHRLICH: Theory o] Photon and Electrons (Cambridge Mass., (1955)) p. 254.

ON [/,-MESON EJELCTI~ON SCATTERING 2 5 5

f l '= IP' }/E' ; ro = e~/4-~ m ~ 2 .8 .10 -~a cm ,

q2 X = F~[4(pP)(pP') - - q~] q- (F~ + xF2)~[q ~ - - 2). ~] + n~.F~[4(pP)(pP') - - q~] ~ .

R o u n d b racke t s d e n o t e 4-vec tor products• Quan t i t i e s w i t h apex are r e l a t ed to scat- t e r e d ~+. The eq. (4), in t h e l imit , reduces, of course, to t h e well k n o w n R o a e n b l u t h f o r m u l a (v).

Considerint~', now. t h a t q,~,~ (max for a d e t e r m i n e d energy of t he i nc iden t It-+) g iven by

2 qm,= 4 IPc.~. I ~ 41PLI 2 ).~

1 + ).2 + 2), ~L

is only of t h e order of u n i t y w h e n t he energy of t h e i nc iden t ~+ (in t he l a b o r a t o r y sys tem) is as h igh as 15 GeV, i t is c o n v e n i e n t to r e t a i n t e r m s p r o p o r t i o n a l to q2 in t h e expans ion in powers of q2 of t h e electr ic f o rm factors , a n d to s u b s t i t u t e the magne t i c fo rm fac to r s b y one. T h a t is :

(7.~2

f ~8 , 4 1 ~ : ,

12n

s ;s s G2

F f -+ 1 - - 12z ]e(v)q2

(~_2 P P P x F ~ - + ~ - - gP(17) ,

2n

1 1

gS(~) / d x x2(2 x) / d . . . . . . . ; g~(~) = x x2 . . . . . V x ÷ V

o o

X 3

1 1

, (.,,~ ; : / . V) 2 ; /~(~) = g ~ ( ~ ) + o 0

X 5

(x 2 - - Vx ÷ ~)~

W i t h these a p p r o x i m a t i o n s , keep ing t e r m s in (;2 and neg lec t ing ).2 w i t h respec t to one, t he cross-section, in the cen te r of mass sys tem, becomes :

(5) d%..,i. = r, ~, q4(E + ~-)2

{2(pP)e + 2 (pP ' ) 2 q2}(1 + 6 ) d ~ ' •

O~V :

0 q2 = 4!pL2 s i n 2 _ :

2 ( p P ) = - - L P k 2 - E~ ; (pP ' ) = - - I P L 2 cos 0 - - E ~ ;

E 2 = ] P ] 2 + l ; ~ = [ P I 2 + X 2.

(7) M• N. I:[OSENBLUTII: P h y s . l ~ e v . , 79, 615 (195(t).

256 B. DE TOLLIS

wi th all quant i t ies referred to the C.M. system• (i is the correction due to the form factors and i t is wr i t t en :

G2 [ 3(qe--2~t~) ] (6) (Is,e= ~nq2 _ js.~'(~l) ± gS,e(~) 2(PP)~ ÷ 2(PP') 2 - q~

or, wi th no approximat ions :

(6') (is,~' = 2(Ff,P __ 1) + ~<s,P Fs'P2 q~(q2--2~2)

2(pP)2 + (2pP,)~ __ q2

Coming back to eq. (6) we shall t r y to eva lua te the magni tude of & W e t ake 0c.~.= n (where (i is max) and we choose, for example , E~ N 15 GeV. Then, as we have seen before, q~(Oc.M. = ~) ~q~m~x is of the order of uni ty . I t is easy to see t h a t t he cont r ibut ion of the second t e r m in (6) is negligible because (see also (6'))

can at most be ~ 0.001 (according to the recent exper iments (8)) and the o ther fac tor is ~ 3. W e have now to eva lua te t he cont r ibut ion of the first term• In order to do tha t , we shall have in mind t h a t G and V are correla ted since u ~ 0.001. The l imits of G are approx imat ive ly 0•005 when ~ = 0, and 2 when V = 103, and t h e y do no t differ g rea t ly for scalar and pseudoscalar a. F r o m this and by an examina t ion of the funct ions ]s,z (which are monotonic decreasing vs. ~) i t can be immed ia t e ly seen t h a t an appreciable contr ibut ion to t he cross-section (for ex. ~ 5 0 % ) is obta ined only when ~ is ve ry small, and only for scalar a, because ]P is 0.5 when ~ = 0, while ]s goes to oo when ~ - ~ 0 as 2 In (1/V~). However , also in this case, a cont r ibut ion of 50% can only be obta ined for In (1/~/~) ,-~100, t h a t is for unreasonably small va lues of vj.

W e can conclude t h a t for energies (in the electron rest system) of the incident t~+-meson up to N I 5 GeV (which will p robab ly become avai lable at the C E R N proto-synehrot ron) , the re should be no essential correct ions to t he usual expression for t he cross-section due to a possible s t ruc ture of the ~-meson like the one t h a t has been considered here. This conclusion cannot be ex tended to higer energies for which i t is necessary to consider t he general fo rm factors as wr i t ten in (2), (3), keeping in mind t h a t in (5), ~ mus t be expressed through (6')•

One should finally no te tha t , avoiding par t icu lar hypotheses on the ~-meson s t ruc ture , an appreciable cont r ibut ion to t he cross-section can only be given by the electric fg rm f a c t o r / ~ , because of the present s t rong l imi ta t ion on the magnet ic

form factor .

$ $ $

T h a n k s are due to the Prof. R. GATTO for hav ing suggested the p rob lem and

for he lpfu l discussions.

(s) R . L . GARW]N, D. P. HUT(~ItINSON, S. PENMAN a nd G. SHAPIRO: P~IS• Bey. Lett., 2, 213 (1959).