Fortschritte der Physik / Progress of Physics: Band 14, Heft 4 1966 [Reprint 2021 ed.] 9783112500286, 9783112500279

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R PHYSIKALISCHEN GESELLSCHAFT IN D E R DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

BAND 14 • HEFT 4 • 1966

A K A D E M I E

-

V E R

LAG

•

B E R L I N

I N H A L T Seil.-

P . E . B E C K M A N N : Electromagnetic F o r m F a c t o r s of Nurleons

235

W . L A N G B E I N : Zur heuristischen B e g r ü n d u n g der Feldgleichungen freier Teilchen . . 259 CAO C H I , N G U Y E N V A N H I E U , B. S R E D N I A W A : Meson Resonances 2+, 1+, 0+ in t h e Higher S y m m e t r y T h e o r y

281

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Fortschritte der Physik 14, 235-257 (1966)

Electromagnetic Form Factors of Nucleons 1 ) PETES E.

BECKMANN

Institut für theoretische Physik der Universität

Mainz

Introduction If high energy electrons are scattered by protons at large momentum transfer the observed cross sections are considerably smaller then the ones expected for the scattering by an electromagnetic point source. A typical value is a decrease by a factor of 1/30 for an energy of 1 GeV of the incident electrons and for a scattering angle of 112° in the laboratory system. A similar phenomenon occurs at low energies if charged particles are scattered by an extended charge distribution, e.g. by an atom. This case is treated in many textbooks on quantum mechanics [J]. The influence of the distribution of the charge is represented by a form factor, which is a function of the momentum transfer only and which multiplies the amplitude for the scattering by an electric point source. If the charge distribution is rigid and fixed, the form factor is the Fourier transform of the spatial charge distribution. I n the scattering of charged particles by atoms it therefore is related to the distribution of the electrons within the atom. The behaviour of the proton at large momentum transfer can also be described by two form factors. In analogy to the nonrelativistic case they are interpreted as being due to a spatial distribution of the protons electric charge and magnetic moment. But, since the proton is not fixed and takes u p considerable recoil, the form factors are not simply the Fourier transforms of the spatial distributions. The spreading of the protons charge and magnetic moment is related to the fact t h a t it is a strongly interacting particle — a hadron — and as such is coupled to other hadrons e.g. to mesons. If one considers the electromagnetic interaction of the proton one therefore has to take into account the electromagnetic interaction of the other hadrons as well. Important contributions to the protons electromagnetic structure arise from an interaction of the electromagnetic field with pions, which through strong interactions are coupled to the proton. The role of the mesons as intermediaries of the protons interactions is frequently expressed b y saying t h a t a „meson cloud" surrounds the nucleon. A theoretical background of this picture, which should not be taken too literally, is provided by the dispersion relations for the form factors. There the mesons enter as important intermediate states. 1

) Based on lectures held at the CERN Easter School 1965 at Bad Kreuznach.

18

Zeitschrift „Fortschritte der Physik", Heft 4

236

P . E . Bbckmanst

The electromagnetic interaction of the neutron also decreases with increasing momentum transfer. Since free neutrons are not available as target for scattering experiments, one uses neutrons bound in deuterium. I n deducing the neutron data one has to apply theoretical corrections for the strong proton-neutron interaction. This introduces additional uncertainties, which make neutron data somewhat less reliable than proton data. The electromagnetic structure of the neutron is also determined by strong interactions. Since in strong interaction physics proton and neutron are considered to be the two states of the nucleón, their form factors are usually treated together. Recently new electron accelerators with energies in the several GeV region have considerably extended the range of momentum transfers accessible to experiment. I n addition, current experimental interest focusses on tests of the assumption t h a t the electromagnetic interaction between electrons and nucleons is sufficiently well described by the exchange of a single photon. I n this contex the polarization of the recoil proton is being measured and electron-proton scattering is compared with positron-proton scattering. Furthermore, nucleon-antinucleon annihilation into electron-positron and muon pairs is being studied in order to obtain experimental information on the behaviour of the form factors for timelike momentum transfers. I n the theoretical field the general kinematical structure of electron scattering is being studied with the aim of extending the relativistic treatment to the scattering by particles with higher spin. The influence of strong interactions in the determination of the neutron form factor through electron-deuteron scattering is being investigated. Further work is devoted to various approximations which within the frame of dispersion relations relate experimental results with other results of high energy physics, e.g. the various resonances. The consequences of higher symmetry schemes are investigated and parallel with the extension of experiments to larger values of the momentum transfer the question of asymptotic limits is being discussed. The following is intended to be introductory survey. I t supplemets other summaries as those by D r e l l and Z a c h a r i a s e n [2] and by W i l s o n and L e v i n g e r [3]. I n order not to obscure the general line we sometimes omit lengthy computations or details of the application of standard techniques. Instead, the reader a t these is referred to the original papers or more specialized summaries. The list of papers cited is not a complete bibliography. But it contains the more recent papers, which might serve as a starting point for a detailed study of special questions [4], 1. General notation We consider the process of electron-proton scattering according to e~ + p

h+ p

e- + p =

k' + p' ;

(1.1)

k, p, k' and p' denote the four-momenta of the particles such that 2 )

k2 = k2-k2 2

) We use units such t h a t h = c = 1.

= m2

p2 = p% - p2 = M%\

(1.2)

Electromagnetic Form Factors of Nucleons

237

M and m are the masses of proton and electron respectively. With the process we can associate a diagram as that of figure 1. The scattering is described by a scattering amplitude T such that the cross section can be expressed as:

4 (2nf(2 7tfd^p'd^k'd(p —2p0 2k0 2p'0 2k'0

+

(2*)*

k-p'-k')

1 1

1

'

^

v

l®i — 2 is the relative velocity of the incoming particles. T is connected with the ^-Matrix through [5] 8

p'k'

~

2i

1

pk) = d(p + k - p' - k')T.

(1.4)

It is more convenient to use variables P, Q, s, t defined as

P=(p+k) = {p'+ k'Y, P2 = 8 Q = (P — P') = (k'-k); Q2 = t. 2

(1.5)

/

\

H/

7

rig. l.

k

\ *

\

Frequently q = — t is also used in the literature. For electron-proton scattering these variables have the following meaning : P : Q: e - p: s: t:

total energy-momentum four-vector four-momentum transfer square of the total energy in the center of mass system (CMS) invariant momentum transfer.

The process of electron-proton scattering through the substitution law is connected with proton-antiproton annihilation into an electron-positron pair. (Remember that an incoming particle with charge e and momentum p corresponds to an outgoing antiparticle with charge -e and momentum -p). For p — p annihilation P, Q, s, t have a different meaning and assume different values: e — p scattering p — p annihilation s: square of CMS energy inv. momentum transfer t: inv. momentum transfer square of CMS energy physical region: t si 0 I > 4M 2 The connection between e — p scattering and p — p annihilation has an immediate consequence: The squared matrixelements | T |2, averaged over spin orientations, entering into the cross section for unpolarized beam and target and without analysing polarization are given by the same function, of course for different values of the variables s and t. If p — p annihilation proceeds through a finite number of angular momentum eigenstates with I < L, the spin average | T\2 is of the form = A(t) + ^ ( i ) cos + A2(t) cos2 #H b Au cos 2i &t. (1.6) The dependence ons is fully contained in cos &t, where &t is the angle between p and e+ in the CMS of p — p annihilation: 2 2 2 (a — M — ra ) + t =

18*

(t—nw

—ft

2

(

L

7

)

238

P. E. BECKMANN

For e — p scattering this implies a particular dependence on cot 2 (0/2), where 6 is the scattering angle of the electron in the laboratory system: cos

»< = i 1 + TT-rcot2

(IF5

T

=

ii = A

cot 2 ( | " ) } / a ^ i W + ••• +

W f = ¿ 0 ( 0 + {l + cot2

+ |i +

(L8)

(1 9)

'

An interaction proceeding via angular momentum I in p — p annihilation corresponds to the exchange of spin I in e — p scattering. Odd powers of cos &t appear only if there is interference between contributions of different parity. 2. Structure of the scattering amplitude and of the cross section The electromagnetic interaction between electrons and protons can be described by the exchange of photons, i.e. of quanta of the electromagnetic field. We decompose the scattering amplitude into terms corresponding to different numbers of photons being exchanged (cf. fig. 2).

Fig. 2.

Such a decomposition arises if one uses perturbation theory for the electromagnetic interaction [6]. We shall restrict our discussion mainly to the one photon exchange contribution, which seems to describe most experimental data very well. This agreement might be correlated with the fact t h a t the multiple photon exchange terms contain higher power of the fine-structure constant a, which is small (¡s» 1/137). But we shall discuss explicitely methods to test the validity of the one photon exchange approximation. The one photon contribution, according to the Feynman rules [7], turns out to be 2Ï ' ' '

'

(2tt) 3

(2 nYQ*(

l )

^

n )

{

2 „y3

{jp) is the matrix element of the electromagnetic current between the states of the particle by which the electron is scattered: 's'\jli(0)\ps).

(2.2)

Here, s and s' are the spin quantum numbers of the proton. The electric charge e appearing as coupling constant in eq. (2.1) is normalized such t h a t e 2 /4n =