On μ-meson electron scattering

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

    On ~-Meson Electron Scattering.

    B. DE TOLLIS

    lst ituto di F is ica dell' Universit5 - Roma

    Istit,teto Naziouale di F is ica Nucleare - Sezione di Roma

    (ricevuto il 12 Agosto 1959)

    Assuming a possible coupling of ,~-meson with a spinless boson neutral field (~-meson) (,-s), we siiall attempt 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 ~. It 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: NueL Phys., 11, 569 (1959). (6) S. N. GUPTA: Phys. Re c , 111, 1436, 1698 (1958). (,) Processes of this k ind have a l ready been observed witl i cosmic ray V.-nlesons, sec for example

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

  • 254 B. :DE TOLLIS

    0)

    The resulting finite matr ix element is:

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

    where: p ~ (p , ie) the is electron 4-momentum before scattering, ;

    P ~ (P , iE ) the is tz+-meson 4-momentum before scattering;

    p' and P ' are the 4-momenta after scattering; q=P' - -P=p- -p ' ; h=c=l ;

    is the ~-anomalous magnetic moment (units e/2M) ; a~= [VvY~ - -Y~] /2 i .

    Moreover here and in the following:

    M is the ~+-meson mass,

    m is the electron mass,

    /~ is the , -meson mass,

    = I~ lM 2 ; ~ = mlM ,~ 0.5.10 2.

    All energies and momenta are divided by M and are therefore pure numbers. F 1 and F~ are electric and magnetic form factors and for the two cases (S and P) are given by:

    (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 q2y~ 2y F~(q 2) = 1 - - I dx |dy x 3

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

    1 1

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

    o 9

    + x ~ - - Vx + '

    (G 2 = g2/4~) .

    The cross-section (valid in any reference system (')), evaluated from (1), is

    (4) q, v/(p/))~ - ~ ~'(E + ~) - - IP + P lcos 0'

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

  • ON [/,-MESON EJELCTI~ON SCATTERING 255

    fl '= 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~] ~.

    Round brackets denote 4-vector products Quant i t ies w i th apex are re lated to scat- tered ~+. The eq. (4), in the l imit, reduces, of course, to the well known Roaenb luth formula (v).

    Considerint~', now. that q,~,~ (max for a determined energy of the inc ident It-+) given by

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

    1 + ).2 + 2), ~L

    is only of the order of un i ty when the energy of the inc ident ~+ (in the laboratory system) is as h igh as 15 GeV, it is convenient to reta in te rms proport ional to q2 in the expansion in powers of q2 of the electric form factors, and to subst i tute the magnet ic form factors by one. That is:

    (7.~2

    f ~8 , 41~: , 12n

    s ;s s G2

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

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

    2n

    1 1

    gS(~) /dx 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 ~)~

    Wi th these approx imat ions, keeping terms in (;2 and neglect ing ).2 w i th respect to one, the cross-section, in the center of mass system, 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 in2_ :

    2 (pP) =- -LPk 2 - E~ ; (pP') = - - IPL 2 cos 0 - -E~ ;

    E2=]P]2+l ; ~=[P I2+X 2.

    (7) M N. I:[OSENBLUTII: Phys . l~ev. , 79, 615 (195(t).

  • 256 B. DE TOLLIS

    with all quantit ies referred to the C.M. system (i is the correction due to the form factors and it is wr i t ten:

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

    or, with no approximations:

    (6') (is,~' = 2(Ff,P __ 1) + ~