Electromagnetic structure of the hadrons

RIVISTA DEL NUOVO CI3][ENTO VOL. 2, N. 3 Lug l io -Se t t embre 1972

Electromagnetic Structure of the Hadrons.

B. BARTOLI~ F . FEI,ICETT[ a n d V. ~][LVESTI~IN[

Labora tor i 2~azional i del C N E X - F r a s c a t i

(ricevuto il 10 Luglio 1972)

241 Int,roduction. 243 PART I. - Spacclikc region. 243 1. Validity of the underlying hypotheses in the spacelike region. 243 l ' l . One-phot,on-exchan~c hypothesis. 248 1"2. Tests of the photon propagator and of the point.like [eptons. 251 2. Nucleon form factors. 262 3. Pion form factor. 265 4. The A(1238) transiti(m form factor GMA. 267 5. Deep inelastic electron-nucleon scattering. 272 PART II. - Timelike region. 273 1. Validity of the one-photon-exchange, pointlike lepton and 1/q 2 photon

propagt~t,or hypotheses in the timelike region. 273 l ' l . ()he-photon c, xchange. 277 1"2. ])(tint,like leptons and ]/qe photon propagator. 279 2. Proton form f~ctors. 281 3. Th('~ pion form factor. 284 4. The kaon form f~.etor. 286 5. Multihadr(m production in ,':%- interactions. 291 6. Comments and conclusions.

Introduction.

Nature has provided us with good probes of the electromagnetic s t ructure

(E.SLS.) of stable particles: as we know, these probes are the charged

leptons (~, e).

By means of a high-energy lepton l the s t ructure of a target particle T,

considered as a whole, can be studied by performing elastic-scattering exper-

iments of l on T. This line of invest igation has been pursued during the last

15 years, and has provided a large quan t i ty of information about tile E.M.S.

of tile proton, tile neu t ron and tile pion.

16 - Riv is ta del iVuovo Ciracuto. 241

242 B. BAI~TOLI, F. FELICETTI and v. SILVESTRINI

Complementary information can be obtained through the production of a TT pair by lepton-antilepton annihilation. Experiments in this category have become possible only in the last few years through the operation of e+e - storage rings at Orsay, Novosibirsk and Fraseati.

As a consequence of some rather general hypotheses, the amplitudes for both the elastic-scattering and the pair production processes can be written in terms of a small number of form factors, which summarize the information on the E.M.S. of the target hadron T.

The form factors are analytic functions of one single variable q~, the square of the four-momentum transferred from the projectile lepton to the target hadron. T h e form factors are explored for essentially negative values of their argument in the scattering reactions~ and for positive values in the pair production experiments (*).

If T were really elementary, the above categories of experiments would provide all the possible information on the E.M.S. of T. This is however not true, and we know that the impact can excite T, or produce particles on it~ or break it into its constituents. Additional information on the E.M.S. of T can be obtained by studying these processes--the inelastic-scattering processes. While in the elastic scattering the kinematics is completely known once the momentum of the final lepton (or of the recoil target particle) is measured, in inelastic processes the complete knowledge of the final-state configuration cannot be achieved by detecting a single particle. The study of inelastic processes based on a detailed study of the final state becomes therefore very difficult both experimentally and theoreticMly.

However. some relevant insight into the E.M.S. of T can be achieved also through that partial knowledge of the final state which is obtained by measuring only the momentum of the final-state lepton. These experiments--the so- called inclusive experiments--have been an important field of investigation during the last few years. The corresponding process in the timelike region is of course the production of many-hadron systems from e+e - inter-

actions. The following four categories of experiments are the object of this purely

experimental review article:

a) elastic-scattering experiments of leptons on nucleons and on virtual

pions~

b) inelastic-scattering experiments of leptons on nuclear targets (in-

clusive),

c) hadron-antihadron pair production from e+e - interactions,

d) production of multihadron systems in e+e - collisions.

(*) Throughout this paper, the following convention is used: q2= q, qt, = E2-- Ip~[ , so that positive q~ are timelike, and negative q2 are spacelike.

ELECTROMACTNETIC STRUCTURE OF THE HADRONS 243

In addition, we will review the experimental evidence for the validity of the hypotheses which allow us to express the results of the experiments in terms of structure functions of the hadrons involved in the reactions.

I. Spacelike region.

1. - Validity of the underlying hypotheses in the spacelike region.

The hypotheses which allow one to express the elastic- and inelastic-scattering amplitudes in terms of structure functions of the target hadrons are essentially the following [1-3]:

i) the scattering amplitude is well described by the first-order, one- photon-exchange (1-PE) Feynman diagrams;

ii) ch'~rged leptons behave as pointlike Dirac particles;

iii) the photon propagator is given by 1/q", the inverse of its four- momentum squared.

The validity of more general hypotheses--Lorentz inw~riance, parity and time-reversal symmetry, invariance under rotations in the isospin space and gauge invariance--will be assumed to hold in electromagnetic inter:~ctions without discussing their experiment:d foundation.

Let us instead review the experimental support to tile above hypotheses i), ii) and i i) in the spacelike region.

1"1. One-photon-exchange hypothesis. - Due to the smallness of the electromagnetic coupling constant ¢¢ -- c'-/ftc ~1/137 the first-order one-photon-exchange contribution A1y (proportional to ~) to the scattering amplitude in lepton scattering is expected to dominate over the second-order (proportional to a s) and higher-order contributions.

Despite the technical difficulties [4], the contribution of higher-order diagrams can be calculated in detail in the case of lepton-lepton scattering described by pure QED. This contribution, which depends in part on the features of the experimental apparatus, does not exceed in general a few percent; it can therefore be treated as a correction (radiative corrections), and the 1-PE contribution to the scattering amplitude can be accurately extracted from the experimental d~ta. In the case of lepton-h~dron scattering, the problem is treated in a similar way: at each order, the knowledge of the hadron structure which is needed for the calculation is derived from the results of the lower- order approximation.

244 B. BARTOLI, F. FELICETTI and v. SILVESTRINI

Some doubts, however, have been raised about the correctness of this procedure. In particular, it has been pointed out that the interference term 2A~ ReA~ (*) between the 1-PE term and the contribution A~v from the graph

~ t ~

when evaluated with the above procedure could result in large uncertainty, since possible resonances could enhance A~v [5, 6]. Since the magnitude of A~v has been determined by several experimental investigations, we can consider these as tests of the correct evaluation of A~v with the standard radiative correction techniques. These experiments are based on cross-section measurements, on the comparison of g--proton (g--p) with i+-proton (i+-p) elastic-scattering cross-sections and on polarization measurements.

1"1.1. C r o s s - s e c t i o n m e a s u r e m e n t s . R o s e n b l u t h p l o t s . Lorentz invariance, gauge invariance and the 1-PE hypothesis lead to the form of the electron-nucleon elastic-scattering cross-section known as the Rosenbluth formula [3]:

_ _ = [G2-- G A ± ~ ~ c t g 2 ] d ~ ~-

(1.1.1)

where

(1.1.2) n =

( 1 . 1 . 3 ) ~ ----- - -

e 2 ~2 1 I d a \ t 20 ~ ! sin2(O/2)(1 + (2EoIM)sin e (0/2)) = ~d~)Ns g 2 '

- - q 2 •

4M 2 '

0 is the electron scattering angle, E0 the energy of the incident electron and M the nucleon mass. The form faetors G, and G~ are real functions of q~ only, related to the Dirac and Pauli form factors F1 and F~ (see Sect. 2).

Plots of R = (1/A)da/dY2 vs. ctg~(O/2) at fixed q~ are known as Rosenbluth plots and are straight lines of slope (G~-?~G~)/(I÷~) and intercept 2~G~. Deviations of the experimental points from the Rosenbluth plot would imply the break-down of one of the hypotheses, the weakest of which is the 1-PE approximation.

Experimental tests of the Rosenbluth formula have been extensively made [7]. We report an example in Fig. 1.

(*) Because of gauge invariance and current Hermiticity Alv is real.

100

1250

5G

"d z

~ 2

1

o 500 i

I

I I

I 4

lOO 200 300 ctg2(G/2)

Fig. 1. - Typ ica l R o s e n b l u t h s t r a igh t - l i ne p lo t for e l ec t ron -p ro ton elast ic s c a t t e r i n g a t _ q2 = 0.39 (GeV/e) 2. T he q u a n t i t y R = [(d~/df))/(d(;/d~Q)Ns] c tg z (0/2) is p l o t t e d aga ins t c tg 2 (0/2): o JANSSEN8 et al. [7], a BEHREND et al. [7], • BARTEL et al. [7], • ALBRECHT et al. [7].

1000

*5

._.z 750

500

250

L

0 I0~00 20'00 3000 bOO0 ctgZ(8/2)

E L E C T R O M A G N E T I C S T R U C T U R E O F T H E H A D R O N S 245

Fig. 2. - Typ ica l R o s e n b l u t h s t r a igh t - l i ne p lo t for m u o n - p r o t o n e las t ic s c a t t e r i n g a t _ q2 ~_ 0.39 (GeV/e)L The q u a n t i t y R ~ [(d~/d~9)/(da/d~9)~s] c tg ~ (0/2) is p l o t t e d aga ins t c tg~(0/2) : o ~ - + p ~ - + p [ 8 ] , . F + + p - ~ + + p [ 8 ] .

2 ~ 6 g . B A R T O L I , F . F ] ~ L I C E T T I and V . S I L Y E S T R I N I

The l~osenbluth behaviour has also been tes ted in ~-p scattering [8], as shown in Fig. 2.

Agreement has always been found within the errors. I t is worth recalling, however, tha t only the real par t of a possible 2-PE amplitude A~T gives an aa contribution to the cross-section, and in addit ion even a nonvanishing l~e A~T does not necessarily destroy the Rosenbluth behaviour [6].

1"1.2. C o m p a r i s o n of ~+-p a n d t--p c r o s s - s e c t i o n s . While the squared moduli of both the Alv and the A~v amplitudes are obviously even in the charge of the incident lepton l, the interference te rm is odd:

dff ± (1.1.4) dD = A~v + [A~v[ 2 ~: 2A1~ Re A2~;

a + and a- are the cross-sections for e+(~ +) and e-Qx-) scat ter ing in the same kinematical si tuation.

The comparison of the e+Qz +) and e-(~-) scattering cross-sections has been performed by many authors [9-15].

8

_ 4 1 1 I I 1 1 1

~0-2 10 -1

I L I I I I l i t i i

~0 ° ]0 ~

- qZ[(GeVlc) 2 ]

Fig. 3. - Comparison of the e--proton and e+-proton cross-sections. The ratio of the real part of the 2-photon-exchange amplitude to the 1-photon-exchange amplitude R e A 2 v / A l v = ½((a*-- a-)/(a+ + a-)) is plotted vs. - - q2: . ref. [9], o ref. [10], • ref. [11], [] rcf. [12], x ref. [13], a ref. [14], , ref. [15].

The summary of all the experimental results available is given in Fig. 3

and 4, in which

R R e A2~ 1 a + - - a -

A l v 2 o ~- + a-

is plot ted v s . - - q % For electrons, /? appears to be smaller than ~ 1 % for 0 . 0 1 < - - q 2 < 1 (GeV/c)% and smaller than ,--2% up to - - q 2 ~ 5 (GeV/c) ~. In

ELECTROMAGNETIC STRUCTURE OF THE t IADRONS 247

201

o

-10

-20

-30

T ti o ' 2 - - - & - - oi~- 0'.8 1.o

_ q2 I(G e V/c) 2]

Fig. 4. - Comparison of the ~--proton and ~+-proton cross-sections. The ratio of the real part of the 2-photon-exchange amplitude to the 1-photon-exchange amplitude Re A 2 y / A I y = ½ ((a+-- a-)/(a+-4- a-)) is plotted vs . - - q2. Data at two different primary- rouen energies are shown: open circles 6 GeV, black circles l l GeV (ref. [8]).

the case of muons [8], on ly - -q2 ~< 1 (GeV/cp has b e e n explored wi th a lower accu racy .

1"1.3. P o l a r i z a t i o n m e a s u r e m e n t s . I n the 1 - P E a p p r o x i m a t i o n ,

gauge inva r i ance a nd the t t e r m i t i c i t y of the e.m. cu r r en t r e s t r i c t the elast ic

~-p sca t t e r ing cross-sect ion to have no spin dependence . I n o the r words, b o t h

0

b

o~

- 6

I - 8 0 ~ ~ L J

0.2 4 0,6 0.8 1.0

-q2[¢Gev/c/21 Fig. 5. - Values of the recoil-proton polarization or of the asymmetry in the scattering from polarized protons for elastic electron-proton scattering: [] ref. [17], ~ ref. [18], × ref. [19], o ref. [20], • ref. [21].

2 ~ B. BARTOLI, F. FELICETTI and V. S I L V E S T R I N I

the polarization P of the recoil proton and the asymmetry A in the scattering of leptons from a polarized target must vanish.

Due to the presence of an imaginary part of a two-photon amplitude, however, A and P could be different from zero. If time-invariance holds, then A--~ P are linear homogeneous functions of the imaginary part of the two-photon-exchange spin amplitudes [16].

Both A and P have been experimentally investigated in e-p scattering [17-21]. The results are summarized in Fig. 5.

lqo evidence for A or P different from zero appears in the explored q~ range (0.2 < -- q~< 1 (GeV/c)~). The experimental accuracy is typically (1--3) %.

1"2. Tests ol the photon propagator and o] the pointlike leptons. - We now consider the experimental tests of the two hypotheses, ii) of the pointlike leptons and iii) of the 1/q ~ photon propagator. These experiments go usually under the name of tests of the validity of QED (quantum electrodynamics).

Here, we review only those which are relevant in the spirit of testing the leptons as good probes of the e.m. structure of hadrons, i.e. we are interested in the experimental proof of the hypothesis that the vertex function and photon propagator in the graph

(1.2.1)

(f and t', the initial and final leptons, being both on their mass shell) can be written according to the rules of pure QED, i.e. 7~ and 1/q ~ [22] respectively.

The most general modifications of QED allowed in this case by gauge and Lorentz invarianee are

(1.2.2)

1 M(q 2) (1.2.3) -- - + - q~ q~

y~ -~ 71~Fl(q :) ÷ kg,~q~F2(q ~) ,

The restrictions on F1, F~ and M required by general principles [23] are not relevant in this context.

The g- -2 experiments [24] allow us to conclude that kF2(q 2) ~ 0 (*) with, from our point of view, absolute precision. Thus, the most general form of

(*) The measurement of a static quantity like g--2 gives information also on the values of the form factors at q2¢ 0. For details see, for instance, K. OKA~OTO: Nucl. Phys., 4 B, 226 (1967).

ELECTROMAGNETIC STRUCTURE OF THE HADRONS 2 4 9

(1.2.2) becomes

(1.2.~) y, ~ _Fl(q'~)y~ .

For spaeelike values of q2, F~(q2) and M(q ~) have been measured , in the case

of electrons, b y means of the e - c - (MoLLE~)[25] and e+e - (BttABttA)[26]

elastic scat ter ing. The F e y n m a n graphs which describe these processes to

first order are

e- e e e e e-

e e e e

BHABHA MOLLER

The ampl i tude corresponding to each graph is obviously propor t iona l to M(q2)F~(q:). Since q: is not the same for the two graphs cont r ibu t ing to each

process, the ra t io R = %J a ,h , in t e rms of which the expe r imen ta l da ta are

convenient ly expressed, is not propor t iona l to M~(q~)~'~(q~). However , in the

k inemat ica l regions explored b y the exper iments , one g raph (the one corre-

sponding to the lower value of the m o m e n t u m t rans fe r squared q~ which is

spacelike glso in the Bhubha scat ter ing) usual ly dominates the other so t ha t R

is app rox ima te l y propor t iona l to M~(q~)I'~(q~). I n Fig. 6, R (for both Moller

1.5

! 0.50 I l

0,5 1.0

-q~[(OeV/J]

iI ! ! l:S 2.0 2.5

Eig. 6. - Comparison of experimental data on e-e and e+e - elastic scattering with QED predictions as a function of the spacelike momentum transfer --q2. The results arc expressed in terms of the ratio aexr,/ath. Statistical errors arc shown as bars, while boxes represent the systematic uncertainty: s BoR(~I,~ et al. [30], • BARTOLI et al. [31], o BARBER et al. [27], x AUGUSTI• et al. [32], a BARBER et al. [28], t ALLES-BORELLI et al. [29].

9"~0 B. BARTOLI, F. FELICETTI and v. SILVESTRINI

and Bhabha scat ter ing) is shown as a funct ion of - - q~. In the Stanford I [27] and I I [28] exper iments the absolute value of the cross-section was not meas-

ured; therefore the uverage value of R has been normalized to 1. In the Fruscat i data [29-31] and in the Orsay point [32], in addi t ion to the st~tisticM error (bars), the systematic uncer ta in ty is also displayed (boxes) including the overall normalizat ion unce r t a in ty in each exper iment .

The above data f rom e--e- and e+-e - storage ring exper iments a.llow us

to conclude tha t the form factor of the electron is tes ted to be equal to i to

wi th in ~ (1+2)° /o (F~ wi th in ( 4 + 8)%) up to - - q~ g 1.5 (GeV/c)~; and wi thin

~ 5 ~ o (F~ within ~ ± 2 0 ~ o ) u p to about 2.5 (GeV/c)t M(q~), appearing in R only squared, is measured wi th about twice us large a re la t ive error.

In the case of muons, no scat ter ing measurement on lepton targets has been performed. However , the muon s t ruc ture in the spacelike region has been invest igated by comparing the muon-proton elastic cross-section d%/dY2 wi th the electron-proton elastic scat ter ing daJd/2 in the same kinematical si tuation.

In the rat io Z~.~ = da~/dao the pro ton s t ruc ture cont r ibut ion cancels out (and the same is t rue for a possible modification of the photon propagator i(q~)), so tha t Z~.~ turns out to be propor t ional to F~(q~)/_F2(q2). The resul t

of this comparison [33, 34] in the case of lepton-proton elastic scat ter ing is

shown in Fig. 7. The rat io F~.o = F~(q2)/t~(q 2) is displayed as u funct ion of

1.5

1D-

1.0

0.5

I fill 1

0.8

i

L 1 1.0 0.2 0.6 1.2

_q2 [(G e V/c)2]

Fig. 7. - Comparison of muon-proton and electron-proton elastic scattering. The ratio _~v.e=2'~(q~)/~'o(q2) is displayed as a function of _q2. F~(q 2) and ~(q2) are the vertex modification functions: o CAMILLERI et al. [34], . ELLSWORTH et al. [33].

_q2 (0.1 ~ - - q2 ~< 1 (GeV/c)2). ~v.o does not appear to have any appreciable q2 dependence wi thin the errors; its absolute value, however, appears to be ___0.96. This 4 % discrepancy, six s tandard deviations outside the stat ist ical

errors, corresponds to an ~ 8 ~o difference in the cross-sections. The au-

E L E C T R O M A G N E T I C S T R U C T U R E OF T H E H A D R O N S 251

thors [33, 34] do not exclude a possible systematic normalizat ion error of this

size in e i ther the muon or the electron exper iments . Similar tests have also been per formed by comparing ~-p with e-p inelastic-

scat ter ing exper iments . These inelastic processes allow us to compare the muon with the electron s t ruc ture even be t t e r than elastic-scattering exper-

iments. In fact ~, larger category of kinematical si tuations can be explored

in this case. In par t icular the large w:Aues of the inelastic cross-sections allow us to have a stat ist ically significant measurement of/7~.o up to - - q2 ~_ 3 (GeV/e) 2.

The resul t is [35, 36] tha t /7~.~ can be f i t ted with a s traight line N ( 1 - - q ~ . K ~)

with N = 0.946 ± 0.042 and K 2 = (0.021 -~ 0.021) (GeV/c)-L

Again the slope of/7~.o is compatible with zero, the absolute value (although smaller thtm one) being consistent wi th 1 within the errors.

2. - N u c l e o n f o r m f a c t o r s .

The most general form of the ma t r ix e lement of the e.m. current of a

nucleon be tween two states of four -momentum p and p ' (p2 =p ,2 _ M ~) is [1, 2]

(2.1) (p ' lJ t , lp> = (p']/71(q~)~, ÷ kat,~q~/7~(q:)lp> ,

! where q, = p , - - p , is the difference be tween the final and init ial nncleon four- momenta ; k is the anomalous magnetic moment # - - 1 . F~ and F2 (the so- called Dirac and Pauli form factors) are res t r ic ted by the requi rement of thc Hermi t i c i ty of the e.m. current to be real for q~< 0 (spacelike region).

As a consequence of the hypotheses discussed in the previous Section, the cross-section for lepton-nucleon scat ter ing turns out to be a ra ther simple expression of E (energy of the incident lepton), 0 (scattering angle),/7~ and/~2. By performing at least two cross-section measurements a t the same value of q2, but a t different angles and energies, F~ and ~ can in principle be evaluated, al though i t is quite a complicated numerical procedure [1]. This procedure becomes qui te simple if we solve in terms of two other functions, G~ and G~.

G E and G~, the so-called <( electric )~ and << magnet ic ~ form factors, are related to /71 and /7~ as follows [37]:

(2.2)

~GM ÷ G~ G~ = F 1 - ~kP2 , F1 --

G~ = F1 ÷ kF2 , F2 -- - - (1 ÷ T)k

As usual, f rom here on we will a t tach when needed a second index p or n to the form factors according to whether they refer to the proton or to the neutron.

9'52 B. BARTOLI~ F, FELICETTI BAld V. SILVESTRINI

The i sovec to r a nd isoscalar f o r m fac tors are also c o n v e n i e n t l y i n t roduced :

(2.3) = e.o - e . o ,

I n t e r m s of G~ a nd G~, t he e l a s t i c - sca t t e r ing cross-sect ion assumes t he

s imple f o r m (1.1.1), a n d G~ a n d G~ are s imp ly r e l a t ed t o the slope and i n t e r c e p t

of the R o s e n b l u t h plots . I t is w o r t h no t ic ing , however , t h a t as T = - - q ~ / 4 M ~

"~ lO

v o

v

v

10 - 2 t o I o

v

J

0.8 ~ ~ ~ ~ - x .

0,6

I 0.4 2

_q2(fm-2 )

/ / i / . / / / /

I I t I l l I

o 10o fm-2

f l I I

1 2 3 4 -- q~ [C6ev/c) ~ ]

Fig. 8. - Electric form factor of the proton G~p(q 2) plotted against --q2: . ref. [38], \ ref. [39], o ref. [40], • ref. [41], n ref. [42], a ref. [43], • ref. [46], v ref. [45], • ref. [44], x ref. [47], • ref. [48]. The Figure on the top (right) is an expanded _q2 scale at low q~ (errors in the data points =h 1%).

E L E C T R , O M A G N E T I C S T R U C T U R E O F T H E t t A I ) R O N S 253

grows larger than 1, the cross-section becomes less and less dependent on G~. As a consequence, a t large values of --q~ G~ is difficult to measure.

Elast ic-scat tering exper iments of electrons on nucleons have been performed for more than 15 years.

A summary of the pro ton results is p resen ted in Figs. 8-13 [52, 53].

a

10 -1

r

10 2

i i

~v

" t o t

' ~ 80' L__ J i ]20' J ' ' I"06

frn 2

• ~ L_ I I

0 3 z+ 5 6

L _. ~ L ± _ 40

I 1 2

Fig. 9 . - Magnetic form f~ctor of the proton G~(q 2) plotted against --q2 up to --q2<150fm-L Values of Gi,(q 2) for q2~100fm-2 ~re obtained with the usual ~ssumption tt~,GEp(q 2) = (~p(q~): . r~f. [38], \ ref. [39], ~ ref. [42J, • ref. [41], • ref. [49], v ref. [45], , rcf. [44], / ref. [47], ~ ref. [43], (> ref. [50].

In Fig. 8 G~(q 2) is shown. In Fig. 9, G~p(q ~) for _ q 2 < 150 fm -2 is pre-

sented. When _q2 is larger than ~ 5 0 fm -~, the errors in G~p become very

large, and no measurement is available for _q2 ~> 100 fm-h At higher values

o f - -q~ , the G~p contr ibut ion to the cross-section becomes in fact so small

tha t only G ~ can be de termined. This is usually done by assuming in this

high-(--q 2) region the same relat ion between G ~ and G~, which experimental ly approximate ly holds a t lower momen tum transfers, namely GE~G~p/# ~ (see the following). Actually, the cross-sections in the --q~ region above ~ 1 5 0 fm -~

2 5 ~ B. BAI~TOLI, F. FELICETTI &Ild v . BILVEBTRINI

depend so l i t t le on G~, t ha t the da ta points on G,~ presented in Fig. 10 would

r ema in the same wi th in the errors unless GE, exceeds GMp by more t han one

order of magni tude . The fac t t h a t GEp(q 2) ---- G~x~(q2)/#,---- 1 for --q~ -* 0 is

,,--,up

i0 o

%

0

v

o_

10-1

10 -2

10 -3

0 0

0

o:

¢to 0

0

' ~00 ' 2 0 0 ' O0 . . . . . . . 3 4O0 5OO 6OO fm -2

1'0 1'5 20 215

Fig. 10. - Magnetic form factor of the proton G~p(q ~) plotted against --q~ for all the measured --q~ range. Values of G~,(q ~) for _ q 2 ~ 100 fm -2 are obtained with the usual assumption tbG~(q 2) = Grip(q2) • v ref. [45], a ref. [42], o ref. [50].

required b y the i n t e rp re t a t i on of the fo rm factors as Four ie r t ransforms of the

charge and magnet ic m o m e n t dis t r ibut ions of the p ro ton [6]: this p ic ture is rigorous in the s ta t ic l imi t --q2 _~ 0. On the cont rary , a t large m o m e n t u m transfers the Four ie r - t ransform in t e rp re t a t i on is far f rom being rigorous [37],

and an a t t e m p t to expla in the rap id fa, ll-off of the fo rm factors for increasing --q~ w th a hard core of the nucleon is therefore a rb i t ra ry .


2.5

2.0

1.5

o .

1.0

0.5

l

L I _ _ 1 I , _ ~ i i ,

50 100 fm 2

I ~ i i I i i I

1 2 3 4 -q2 [(GeV/c)2]

Fig. 11. - Comparison between the electric and magnetic proton form factors. The ratio [#pG~p(q2)/GMp(q)] 2 is shown against the momentum transfer _ q 2 : o ref. [52], x ref. [47], v ref. [45], A ref. [49], / ref. [51], , ref. [48].

I t is eas i ly seen f r o m Fig . 8 and 9 that G~ ~ G~,/tt , . To what e x t e n t this

famous re lat ion ((~ sealing law ~) real ly holds is be t t er seen in Fig. 11, where

the q u a n t i t y (#~G~,/G~) 2 is shown. We see that for _ q 2 ~>30 fm -~ dev iat ions 9 ( 2 0 - - 3 0 ) % are exper imenta l l y found.

2 5 6 B. BARTOLI, F . FELICETTI & n d V. SILVESTI~.INI

13

12

~" 11

101 ]

09

II l

0 . [ t I I I I I I I I [ ~ I I ~ I I

40 80 120 160 fm -2

i i i I I l I 1 I I o ~ ; 4 ~ ~ 7

-~[<o~v/~) ~]

Fig. 12. - (~ Dipole fit ~> for the magnetic proton form factor. The ratio Gip(q2)/ttpG~(cl2 ) is plotted against --q~ for values of _ q 2 < 180 fm 2: A ref. [43], . ref. [38], \ ref. [39], • ref. [41], D ref. [42], • rcf. [49], <> rcf. [50], ~, ref. [44], x ref. [47], v ref. [45].

The scaling law for the p ro ton fo rm ~actors is to be considered an approx- ima te mnemonic rule. I t s theore t ica l foundat ion (quark model , SU~ [54]) is

ra ther weak. I n addi t ion, in the t imel ike region, i ts va l id i ty near threshold

would cause serious problems (see Sect. 2, P a r t I I ) .

Another useful mnemonic rule (with no theoret ical foundat ion) is the so-

called dipole fit, i.e. G , o / Z o _ ~ e ~ - - ( 1 - - ( q ~ / 0 . 7 1 (GeV/c)~)) -~. The compar i -

son be tween G,~/ttp and Gv is shown in Fig. 12, 13 where the quan t i ty G~p/ttpG ~ is presented . The dipole fit appears to hold wi thin ~ 30 %.

The s i tua t ion on the neu t ron form factors is much less clear. This is due to the fact t ha t f ree-neut ron t~rgets are not avai lable , so t h a t the informat ion

on e-n sca t ter ing is ex t r ac t ed by means of a qui te compl ica ted and model- dependent procedure f rom e-deuteron (e-d) elastic and inelastic scat ter ing.

Some informat ion a t v e r y low m o m e n t u m t ransfers is also ob ta ined b y scat-

ter ing low-energy ( thermal) neut rons on high-Z a toms. An excellent review

]ELECTROMAGNETIC STRUCTURE OF THE HADRONS 25~

]2

1]

o _

o _

0.~

m

0.80 ~ ~ _ _ ~ lO0 200 300 400 500 600

I fm-2 0 5 I~0 1r5 210 215

q~ ~G ev/c) ~]

Fig. 13. - (~ Dipole fit ,) for the magnetic proton form factor. The ratio Gx~(q2)/~pG~(q ~) is plotted against --q~ for all the measured --q2 range: [] ref. [42], v ref. [45], <> rcf. [50].

of the methods used to ex t r ac t G~ and M ~ f rom the exper imen ta l da ta can be found in ref. [6].

The difficulty in the expe r imen t s of sca t te r ing of neut rons on a toms (first

pe r fo rmed b y FEBMI and M ~ _ ~ S ~ L [55]) is to separa te the coherent scat ter ing

ampl i tude on the nucleus f rom the sca t te r ing ampl i tude of the neutrons on the a tomic electrons.

Different a tomic ta rge t s and analysis methods have been used in different

exper iments . The results , expressed in t e rms of dG~n/dq21q,=o are in fair ng reement wi th each other :

T A B L E I .

Author dG~n/dq 2 (GeV/e) -2

KROHN I [56] 0.459 ± 0.02

KROHN I I [56] 0.50 ~= 0.01

MELKONIAN [57] 0.575 =J= 0.019

HUGHES [58] 0.512 q- 0.019

17 - Riv i s ta del •uovo Cimento.

2 ~ B. BARTOLI, F. FELICETTI ~n(~ V. SILVESTRINI

dG, Jdq ~ is re la ted to the root-mean-square charge radius of the neu t ron [6]

(2.~) I dG~.

dq 2 [~,~o ' O

which can thus be considered known---including uncer ta int ies f rom the mod- e l s - t o wi th in ,~ (5--10) %.

At higher values of --q~ ( 0 . 3 < - - q ~ < 2 0 f m -~) G~ can be ex t rac ted from elastic e-d scat ter ing measurements , once the electric pro ton form factor

G~ and the deute ron s tuc ture are known. In fact the e-d elast ic-scat tering cross-section can be wr i t t en as [6]

(2.5) ( ) [ da da A(q 2) ÷ B(q 2) tg ~ . d--~= ~ ,~o~

A(q 2) and B(q 2) contain the charge, magnet ic and quadrupole form factors of the deuteron, in addi t ion to the isoscalar (the deuteron has T = 0) nucleon form factors. For e lectron scat ter ing angles 0 smaller t han 15 ° the t e rm B(q 2) tg ~ (0/2) contr ibutes less t han 0 .1% to the cross-section and can be neglected. Equa t ion (2.5) takes t hen the simple form

(2.6) da ( d a ) A(q2) ' d ~ -- d-~ ,~o~t

so tha t cross-section measurements allow one to de te rmine A(q2). Notice however tha t for --q~ ~ 1 5 fm -~, A(q 2) is as small as ~ 10 -4, so tha t the exper iments become v e r y difficult.

In turn

8 r2G~(q 2) + 2 ~ G~d (2.7) A(q2) -- G~d(q2) ÷ -9 1 + ~ 1 ÷ : 1 ÷ ~

where

(2.8)

G~(q 2) = 2GE~(q2)F~a(q2) ,

G~d(q ~) = 2G~(q2)Fqd(q 2) ,

G~(q ~) -~ 2 ~ G~z(q2)F~d(q:) .

/VEal , 2'~a and F~d , which describe the deute ron s t ructure , can be calculated f rom the nonrelat ivis t ie deu te ron wave funct ions for the S and D states. Several models are available for this purpose (ItA_~ADA and JO~NSTON [59],

MC GEE [60], F E s m ~ C ~ and LO~ON [61], etc.), l~elativistic corrections have

been evaluated by G~oss [62].

ELECTROMAGNETIC STRUCTURE OF T~E HADRONS 2 5 9

0.10 ~-

/ - /

1 °5 ~ ~ ~ 5 10 15

--q2(fm-2 )

Fig. 1 4 . - The neutron electric form factor G~,~(q. 2) derived from elastic electron- deuteron scattering measurements. The Yeshbach-Lomon deuteron wave function was used. The dashed line is GE.=/&vGE~, which corresponds to the a s s u m p t i o n / ~ = 0; the dash-dotted curve is GE~= ( ~ / ( l q - 4r))Ge~; the solid curve is the best fit to the data points with a curve G r . = (/~,v/(1 4- bv))G~ (b, the free parameter, turns out to be 5.6): • ref. [66], x ref. [63], D ref. [65], a ref. [69], o ref. [67], . ref. [68].

The expe r ime n t M values of A(q 2) are c o m p a r e d wi th t he di f ferent models .

I t t u r n s ou t t h a t no mode l fits t he da t a i£ G~. is p u t equal to zero. This leads

to G~. ¢ 0, t he uc tuM va lue o£ GE. d e p e n d i n g h o w e v e r on the wave func t ion

of the d e u t e r o n used in the analysis .

A r ev i ew of the resu l t s for G~ o b t a i n e d wi th this m e t h o d can be fonnd

in ref. [63]. I n F ig . 14 the resul ts o b t a i n e d wi th the F e s h b a c h - L o m o n model

are shown [63]. Re la t iv i s t i c correc t ions ~re inc lnded [62] ; t he four-pole fit [64]

for the e lectr ic p r o t o n f o r m fac to r is used (to e x t r a c t G~ f r o m GE~ , G~D m u s t

0.03

0.02

= 001

0

-0.01

o

\

I

0.5 110 115 (Oev/c)~ I

l I

10 210 3'0 4,0 50 -q2(fm-Z )

Fig. 15. - G~n as determined from inelastic electron-deuteron scattering is plotted T GM~ against --q~: ~ ref. [72], [] ref. [73], 2 ~ (v2/(lq_4T)2)G~n,

. . . . . V2p.

c 10 -~

tin

////

////

zu~z

z

' ~

~ 4'

0 '

' ~

~0

' 10

--20

8

I frn

--2

I I

0 1

½

3 -q

[io v/

cl F

ig.

16.

- G

z~=

as

det

erm

ined

fr

om

in

elas

tic

elec

tro

n-d

eute

ron

sc

atte

rin

g.

o re

f.

[76]

, ~

ref.

[7

5],

• re

f.

[77]

, v

ref.

[7

8],

× re

f.

[79]

, ~

ref.

[6

7].

I I

[ I

L ]

L

120

160

] I

,,

Th

e sy

mb

ols

"/

////

/z

rep

rese

nt

up

per

li

mit

s:

• re

f. [7

4],

[-@

©

O~

]ELECTROMAGNETIC S T R U C T U R E OF THE H A D R O N S 2 6 1

be known, see (2.3)). The da t a points are f rom ref. [63, 65-69]. The dashed

curve is G r ~ = # = r G ~ [ 7 0 ] , which corresponds to the a s sumpt ion ~ = = 0 ;

the dash-dot ted curve is GE= =- (/~ ~/(1 + 4~))G~ [71], while the solid curve is the bes t fit to the da ta points wi th a curve G~= = (#=~/(1+ b~))G~, where

the p a r a m e t e r b turns out to be b = 5.6. At higher values of q2 (_if2 ~ 15 f m -~)

the neu t ron fo rm factors can be de t e rmined by means of inelast ic electron- deute ron sca t te r ing exper iments .

At sufficiently large m o m e n t u m t ransfers the nucleons can be t r ea t ed as

approx ima te ly free (impulse approximat ion) , bu t wi th a m o m e n t u m dis t r ibut ion

g iven by the deuteron ground-s ta te wave funct ion. (Again the knowledge

of the deuteron wave funct ion is needed [6].)

Inelast ic e-d sca t ter ing exper iments have been per formed, de tec t ing e i ther

only the sca t te red electron (noncoincidence exper iments ) , or the sca t te red

electron in coincidence wi th the p ro ton or the neu t ron (coincidence experi-

ments) . Small corrections are needed in this case due to the tai l of the deuteron wave funct ion and the f inal-state in te rac t ion . 18]

1.4

c 1.0

c

0.6

0.2

Fig. 17. - Test of the (( sealing law ~>: /~pG~=lttnG~p is plotted as a function of --q~. The symbols ~/////A represent upper limits: . ref. [74], ~ ref. [67], o ref. [76], v ref. [78], x ref. [79], z ref. [75], • ref. [77], v ref. [73].

_ _ . . . L ~ 1 - - ~ L 8 0 1. l,O 120

f m - 2

I i i [ i i i l i

0 1" 2 3 4

262 B. BARTOLI, F. FELICETTI ~nd v. SILVESTRINI

In Fig. 15, 16, G~ and G~ as determined in inelastic e-d scattering measurements are shown. In Fig. 15 G~ is shown, and compared with the same models as in Fig. 14. In Fig. 16 G~ is shown. The data of some old experiments, not shown directly in the Figure, have been used in more recent work to improve the quality of the analysis, and are therefore quoted under the reference of the most recent analysis. The points at --q2 > 50 fm -2 are upper limits.

In Fig. 17 the ratio t t G ~ / t t G ~ is shown. We see that also the <~ scaling law ~ [54] G~o/tt = G~/tt ~ approximately holds.

3 . - P i o n f o r m f a c t o r .

The most general form of the matrix element of the e.m. current of a spin-0 particle ~ is

( 3 . 1 ) ' ~ ~ ' •(q )(p~ + I J ; i p ) = p . ) .

As a consequence of (3.1) and of the usual hypotheses, the cross-section for the elastic scattering of electrons on ~ can be written as [6]

d--~ = (q2),

where ~(q~), the form factor of the target particle, is a function of q2 only and is real in the spaeelike region.

Elastic-scattering experimengs of electrons on free pions are however not possible, and elastic-scattering experiments of pions on electrons have not yet been performed. The information available on F,~(q ~) in the spacelike region has been obtained by scattering electrons on virtual pions, namely through eleetroproduction experiments of pions on nucleons near threshold. In this case, however, the connection between experimental data and F,~ is much more complicated than (3.2), and is in addition model dependent.

This is due to two facts:

a) in electroproduction processes (e.g. e - + p - + e - + ~ + + n ) the contribution from the graph

% f ~ f

(3.3) (( pion pole , /

:ELE(~TROMACrNETIC S T R U C T U R E O F T H E I t A D R O N 8 263

(which is relevant for the determinat ion o f / ~ ) is due to an off-mass-shell piou interact ing with the nucleon;

b) the contribution from the graph (3.3) must be separated from the contribution of other graphs, e.g.

e e r e e ' e e ~

(3.4) ~ /-~ /~ ,

a) b) c)

whose contributions are expected to be impor tant in the electroproduction process near threshold.

The e!eotroprodtlction cross-seetion--nnder the 1-PE ~ssumption---can be wri t ten as [80]

d3a da (3.5) y ~

dE' d~odg* -- ~ , h ~ . ;

E ' = e n e r g y of the scattered electro~ in the laboratory system,

d[2~ =sinOodOod%, element of solid angle of the scattered electron in the laboratory system,

d/2~* " * * * sm0~d0=d~o~, element of solid angle of the produced pion in the c.m. system of the hadrons in the finul state.

The factor F~h contains the electrodynamics of the process (electron-photon ver tex and photon propagator) and da/d~2* is the cross-section for pion photoproduction by vir tual (polarized) photons [80]:

(3.6) dr2* = A + sB + sC sin 2 0~* cos 2 F* + ~/2s(1 + ~) D sin 0* cos ~v*,

where s, the polarization of the vi r tual photon, is given by s =- (1~- (2]pl~/Iq21) • • tg ~ (0o/2))-1 with p the th ree-momentum of the photon in the laboratory system• The functions A, B, C and D contain the information on the structure of the hadrons involved in the process, i.e. t hey depend on the form factors at the vertices [ y ~ ] , [~SY'J¢], [yS'A] (A ~ 1238 ~ a 3 resonance). The first term A is the differential cross-section for nnpolarized transverse vir tual photons, the second te rm is the contribution from vir tual longitudinal photons, the third term from linearly polarized transverse photons, and the last te rm represents the interference between the transverse and the longitudinal amplittldes.

2 6 ~ B. BARTOLI, F. FELIC]~TTI and V. 8ILVESTRINI

I n order to e x t r a c t f r o m the expe r imen ta l da ta the p ion fo rm fac tor F,~(q2),

i .e . the [Tr~] v e r t e x funct ion, the cont r ibut ion f rom the 1-p ion-exchange pole

d iagram (3.3) mus t be isolated. For this purpose a deta i led phenomenological

theory of e lec t roproduct ion is needed. Many different approaches have been

t r ied b y several authors . Dispers ion re la t ions [81-86], isobaric models [87],

cur rent a lgebra techniques and the PCAC assumpt ion [88] are the ma in ingre-

dients in the calculations. A r a the r comple te l ist of references can be found

in ref. [80]. The expe r imen ta l in format ion comes ma in ly f rom the react ions

(3.7) e ÷ p - - > e + n ~-:: + ,

(3.8) e + p -+ e + p +7: °,

1.0

0.8

0.6!

u.Y

0,4 ~

l'\ \ "! T I

I

0 . 2 0 I I I I I I I I I I 0.2 0 .4 0,6 0.8 1,0 1.2

- - q 2 [ (GeV/c)2 ]

Fig. 18. - The pion form factor E=(q 2) as determined from electroproduetion experiments. In correspondence with each point we quote both the reference of the experiment (e.g., NIISTRETTA [91]) and of the model used for the analysis (e.g., DOMBEY [89]). The slopes expected at q2= 0 for three different values of the pion radius x / ~ are also shown: MISTRETTA [91]: O ADLER [81], a ZAGURY [82], [] DOMBEY [89]; AKER- L0F [93]: e ADLER [81]; SOFAIt¢ [94]: • DEVENISH [85]; BROWN [95]: X BERENDS [84]; DRIVER [90]: -- BERENDS [84]; AMALDI [92]: v G E a : O, • G.~a= - - GMn'~/(1 + 4w), DE TOLLIS [83]; - - - - - - 0.63 ( V ~ fm, - - . . . . 0.81 ~ / ~ fm, 1 . 0 0 ~ / ~ fro.

ELECTROMAGNETIC STRUCTURE OF THE It&DRONS 2 6 5

in which one of the hadrons in the final state is detected in coincidence with the scattered electron. Reaction (3.8) is particularly relevant to a detailed phenomenologieal understanding of the electroproduetion process since in this case the pole diagram (3.3) does not contribute; in particular the so-called (( transition ~) form factor G~A(q 2) to the first A(1238) resonance (namely the ~,A~A vertex function) can be evaluated. This makes easier the interpretation of the more complicated data from reaction (3.7) in terms of 2~,(q2).

In Fig. 18, a summary of the experimental information available on F~(q ~) is shown. In several eases the experiment,sl data have been analysed using different theoretical models: this gives an idea of the dependence of F~(q 2) on the method of analysis. An additional uncertainty connected with the models, which does not show up as a difference in d~ta points coming from different types of analysis, is the uncertainty in the foundation of hypotheses which are common to different models.

An important example of this kind has been pointed out by DOMBEY and I~EAI) [88, 96], and is related to our ignorance of the lower vertex of diagram (3.3). In current algebra techniques (PCAC), the pion field is related to the divergence of the axial weak current associated with the nucleon. This brings about the axial form factor of the nucleon G A. Usually G~ = G~ is assumed. If one however does not make this more or less arbitrary assumption (which is actually justified by ehiral symmetry), at each value of q2 a large band of possible values of /~=(q2) can fit the experimental data very well by properly choosing an associated value for G~.

In Fig. 18 three slopes expected at q2= 0 for different pion radii ~/(r ~} are also shown. I t appears that st this stage ~/(r 2) is still poorly determined by eleetroproduction measurements.

4. - The A(1238) transition form factor G~a.

When the invariant mass of the =-nucleon system in the pion electroproduction processes is near the A(1238) mass, the reaction is dominated by the production of the A-resonance.

This is especially true for reaction (3.8) to which diagram (3.3) does not contribute.

I f we treat the A-resonance as a particle, then the cross-section can be written as [97]

(4.1)

where

ix Io 4

. c o s 2 (0° /2)

a s s = 4 E ~ s i n , (0o /2 ) [1 q- 2(E/M) s i n ~ (0o/2) ] '

26~ ]~. BAI~TOLI, F. FELIC:ETTI and V. SILV:ESTRINI

E = energy of the incident electron and p = three-momentum of the virtual photon in the A rest frame.

G~A(q~), GEa(q2) and GoA(q ~) correspond to the 3/1, J~ and C~ (Coulomb 3+ octupole) which can excite the J = ½+ to J----~ transition. Both on theo-

retical [98] and experimental [99] grounds GE~ and Go~ are expected to be much smaller than G~A , and are usually neglected in the analysis of experimental data. In this case eq. (4.1) assumes a very simple form allowing one in principle %o measure quite easily IG~I 2. Notice in particular that, according to (4.1), IG~A[ ~ can be measured in experiments in which only the scattered electron is detected.

In practice, however, there are problems connected with the contribution of nonresonant background and with the large width of the A-resonance. A detailed phenomenological theory of electroproduction is again needed, although the situation is much simpler than for the determination of the pion form factor . F = .

The information from coincidence experiments is in practice needed for the multipole analysis of the electroproduction data, in which case G~A is simply connected to the q~ behaviour of the M1 multipole contribution [97].

10 0

10-~

i

0 J ' :30 ' -2

I fm I I

o I 2 -q%eV/J]

' G'0

Fig. 19. - The transition form factor G~A to the first nucleon isobar A(1238) as determined from electroproduction experiments. The point at q2= 0 comes from photoproduction experiments, see ref. [97]: o photoproduction, . BARTEL et al. [100],

IMRIE et al. [I01], a ASH et aL [97].

ELECTROMAGNETIC STRUCTURE OF T H E H A D R O N S

U

267

~o.9

<1

0.7

I!

0.% , 1'o ' 2'0 3o I fro-2

I I

0 0.5 1.0

- £[(o~v/c) ~]

40 ' 5'0 J

I I

1.5 2.0

60

Fig. 20. - The ratio GMa/3G D of the transition form factor GMa for the first nucleon isobar to three times the dipole fit is plotted as a function of --q2: o ref. [I01], v ref. [97], . ref. [100], o photoproduction, ref. [97].

The exper imenta l s i tua t ion for G~a is summarized in Fig. 19. The value at q~ = 0 de te rmined f rom photoproduct ion data. is 3.00=L0.01 [97]. With increasing - - q 2 G~ a drops more rapidly than the pro ton form factors, as shown in Fig. 20, where the rat io of G~I a to three t imes the dipole fit is presented.

5. - Deep inelastic e lectron-nucleon scattering.

An extensive systematic invest igat ion of electron-nucleon inelastic scat ter ing

has been carried out during the last few years at SLAC by a SLAC-MIT col-

laborat ion [102]. These data include measurements a t large values of ]q~l and

correspondingly large energy t ransfers to the nucleon {deep inelastic region)

and are of the inclusive type , i.e. only the scat tered electron was detected. Some coincidence measu remen t s - -wi th the detect ion of par t of final-state

hadronic p roduc t s - -have also been per formed [103], however these investigations are only now beginning and the results cannot ye t properly be included in a review paper.

2 ~ B. BARTOLI, F. FELIC]~TTI and V. SILVESTRINI

Also, since the inclusive- type data come essentially f rom one single experi-

me n t [104], we repor t here only a short summary of the results.

The cross-section for inclusive electron-nucleon inelastic scat ter ing can be wr i t t en as

d2a ~ W ~ - M 2 E ' 2

(5.1) d~2dE' -- 4~ 2 2Miq~ I E 1 - - ~ [~t + ~s] •

The symbols have the same meaning us in (3.5), (3.6); W, the missing mass

of the unobserved hadronic final s tate , is given by W 2 = 2M(E-- E') + M 2-q2. The cross-sections for t ransverse and longitudinal polarized v i r tua l photons,

a, and as, are functions of the invar iants q~ and W (or of q~ and v = / ~ - - E ' ) . As q~-->0, as-->0 and at(q2, v)-->%(v), where %(v) is the photoabsorbt ion

cross-section for real photons of energy v. Al ternat ively , d2er/d~dE ' can be wr i t t en in te rms of the (( s t ruc ture func-

tions ~) W~ and W~:

(5.2) d2a -- e4 c°s~ (012) [W2+2W1 tg ~ (0/2)]. dY2dE' 4E ~ sin 4 (0/2)

W1 and W2 are re la ted to a, and a, by

(5.3)

W1 = Ka~ ,

w 2 = K I¢1

W 2 - - M 2 K - -

8 ~ 2 Ms

and to the exper imenta l cross-section by

r d:a 1 /doX-1 /1 ]q~]v~+2tg:(O/2)-i t ( + ") + ) '

For small values of tg ~ (0/2), W2 depends much less cri t ically than W1 on R.

The exper imenta l findings, which are of ex t reme interes t and have s t imulated

a qaan t i t y of theoret ical speculations, can be summarized in the following points:

a) R, al though ra ther poorly measured (see Fig. 21), appears to be qui te small and is compatible with being constant over the full q2 and v range

explored. The average value is /~ = 0.18 4-0.10. This fact is of ten inter-

E L E C T R O M A G N E T I C S T R U C T U R E OF THE HADRONS 269

0.9

0.6

b ~

b ~

II

0.3

I 3

I

6

-q~ [(G~V/c#] I [ 1

0 ' 9 12

Fig. 21. - The ratio a,/a, (~, and at are defined as in (5.1)) plotted as a function of --q~, for different values of W. W is the invariant mass of the unobserved hadronic final state in GeV: o W = 1.9--3.1, ref. [104]; . 2.0, ref. [102]; ~ 2.5, ref. [102]; x 3.0, ref. [102]; n 3.3--3.5, ref. [102].

preted as an indication tha t the contr ibut ion to the total cross-section from

the scattering on vir tual scalar (or pseudoscalar) mesons is small. Actually in

the inf ini te-momentum limit when all the particles involved in the reaction

travel along the beam direction (a si tuat ion which is not necessarily satisfied

in the experiment) helici ty cannot be conserved if a longitudinal photon is

absorbed by a bosom

b) When W is well above the resonance region, d2~/df2dE ' has, at fixed W,

a ra ther weak dependence on q~. This is shown in Fig. 22, where

/Id,/

is p lot ted as a funct ion of - - q: for different values of W. With increasing W,

F becomes flatter. This is one of the facts which has suggested the idea of

the proton buil t up of pointl ike const i tuents [105] (partons) which, due to

the above observation a), should be most ly fermions.

270 B. BARTOLI~ F. F]ELICETTI a n d V. SILVESTRINI

10

10-

b- 10-:

lo-3

10 -].

\ ° ~ 0 . \ \

\ ' \

\ \

i i i I

2 3 4 5 6

-q 2~GeV/c)2J

Fig. 2 2 . - The ratio of the experimental cross-section to (~Mot~ in deep inelastic electron-proton scattering is plotted as a function of --q2 for different values of W. For comparison, the same ratio is also shown for elastic electron-proton scattering: • - - W = 2 G e V ; × - - - W = 3 G e V , a - - - - - - W : 3 . 5 G e V , - - . . . . elastic scattering.

c) vW~, de te rmined in the hypothes i s t h a t R is cons tan t and equal to 0.18 i 0.10 (W~ however depends r a the r weakly on /~, so t h a t this hypothes i s is not v e r y drastic), appears to depend on the ra t io ~ = 2M~,/q 2 r a the r t h a n on q2 and v separate ly . The exper imenta l s i tua t ion is p resen ted in Fig. 23,

where vW2 is shown as a funct ion of --q2 for var ious values of co. At (o----4, vW~ is clearly independen t of q2 wi th in the errors (see also Fig. 24). At different

values of ~o, the da ta suggest t h a t a p la teau is be ing reached a t higher values

of Iq2]. This behav iour was first suggested by BJOR~EN [106] (<( scaling law ~))

to occur in the so-called Bjorken l imi t Iq21-->co, v - ~ c ~ (~o constant) , and the

fact t h a t this l imi t appears to be reached a t r e la t ive ly small values of ]q2] and v

was considered surprising. Other var iables (e.g. ~o' = ~o + M2/q ~ = 1 + W2/q ~)

in t e rms of which the scaling behav iour is reached even earlier have also been

proposed [107].

F r o m an in tu i t ive po in t of view, the sealing law of the s t ruc tu re funct ions

can be unders tood as follows. Consider a funct ion of the type (one-dimensional l ight-cone singulari ty)

(5.5) / ( x ) ~ ( x - t ) ,

where x is the direct ion of the incoming v i r t ua l pho ton of m o m e n t u m q, ___

E L E C T R O M A G N E T I C S T R U C T U R E OF THE t t A D R O N S 271

0.4 h)

0.3

0.2

0

0.4

0.3

0.2

0

0.4 c~)

0.3

0.2 ~'~

I I I I

I I

0 I 2

e)

I I i

3 4 I - [(G eV/c)2]

2 3

Fig . 23. - T h e s t r u c t u r e f u n c t i o n vW2 is p l o t t e d as a f u n c t i o n o f - - q ~ for d i f f e r e n t

v a l u e s of w = 2Mv/q ~. / ~ = 0 . 1 8 w a s a s s u m e d , see ref . [102]: x = 10 °, . = 6°;

a) 4 < w < 6 , b ) 8 < ~ < 1 2 , c ) 1 2 < ( o < 1 8 , d ) 1 6 < ( 9 < 2 4 , e ) 2 4 < c o < 3 6 .

~_(~,~-M/~o, 0, 0, v). The Four ie r tr,~nsform of (5.5) is

Therefore u funct ion of the type (5.5) h~s ~ se~ling Four ier t ransform. BJOttKEN first pointed out th,~t the e.m. cur ren t commutators , whose Fourier

0.5

0.4

0,2

0,1

I

0 6 --q2 ~(GeVc) 2 ]

Fig . 24. - T h e s t r u c t u r e f u n c t i o n v W 2 is p l o t t e d as a f u n c t i o n of - - q 2 for 04= 4, see

r e f . [ 1 0 2 ] . R=a~/at=O.18 w a s a s s u m e d : + 6 °, × 10 °, D 18 ° , o 26 ° •

272 B. BARTOLI, P. FELICETTI and v. SILVESTRIhrI

transforms are closely connected with the structure functions, must be dominated, in the infinite-momentum (Bjorken) limit by singularities on the light-cone (x~,x ~' = 0). The experimental result on the scaling law was therefore surprising mainly because scaling is reached at relatively low energy.

The main trend of the present theoretical work on scaling [108] is essentially along two lines: a) techniques to evaluate the light-cone singularities of the e.m. current commutators and b) parton models, in which the proton structure assumes quite naturally the form of objects moving in the infinite-momentum limit with the velocity of light (the simplest form of the charge distribution of the proton being ~,@~(x~--t) with Q~ the parton charges).

Both these approaches have a nonnegligible predictive power: in the first case the transformation properties of the currents under different groups (Poincar~ group, SU2, SU3 etc.) can be used to correlate cross-sections and to predict sum rules; in the second case many predictions can be made once the partons are identified, e.g., with the quarks.

Experimentally, some data on the comparison [102] of deep inelastic scattering on protons and neutrons are also available.

Finally, the results of ref. [36] (see Sect. 1) allow one to conclude that deep-inelastic-scattering data of muons on proton are consistent with the electron data.

IL Timelike region.

The investigation of the e.m. structure of hadrons in the timelike region has only recently begun with the operation of e+e - storage rings and has been pursued up to now only in few and rather small laboratories, essentially at Orsay, l~ovosibirsk and Frascati.

The experimental information is very important for an understanding of the physics of elementary particles, but is still quite meagre. This is due not only to the fact that this kind of physics has a short history and a rather limited geography, but also to the fact that hadrons are produced in e+e - interactions with very small cross-sections (in the range of nanobarns--10 -Sa cm 2 - when the total energy is above 1 GeV). In addition, in storage rings beams of ~1011 particles are sent one against another, to be compared with the use of beams against targets in the conventional machines. The fact that the beam-beam impact occurs ~107 times per second does not compensate the difference in intensity. For this reason, very small beam dimensions are generally used in storage rings to obtain a high target density, which, while making the machine operation quite delicate and difficult, brings the counting rate just to the limit of experimental feasibility. Counting rates of a few events per day, or even per week, are not umtsual in typical experiments.

E L E C T R O M A G N E T I C S T R U C T U R E OF T H E H A D R O N S 273

1. - Validity of the one-photon-exchange, pointUke leptons and 1/q 2 photon propagator hypotheses in the timelike region.

1"1. One-photon exchange. - In the timelike region the smallness of the nsuul two-photon exchange contribution from the graph

e+

(1.1.1) t , ~ F

has not been experimentally tested, and we have therefore to trust the calculations based on the absence of strong enhancement mechanisms. I t is worth noticing that in the timelike region the number of exchanged virtual photons is directly related to the charge conjugation eigenvalue C of the final state. If the charge of the produced particles is not recognized (us was the case for all the experiments performed up to now with e+e - storage rings), then the interference between even and odd states of C c~mcels, so that the two-photon- exchange process can contribute only to ~n ~ order.

There is however an additional two-photon contribution which is expected to be quite important in e+e - interactions, namely

e - e-

(1.1.2) F

(1.1.3)

18 - Rivis ta del Nuovo Cimento.

274 B. BARTOLI, F. FELICETTI a n d v . SILVESTRINI

The addi t ional ~2 fac tor appear ing , e.g., in d i ag ram (1.1.3) is in fact expec ted

to be a t least pa r t i a l ly compensa ted b y the fact t ha t this process can involve

much lower m o m e n t u m t ransfers t han the usual one-photon-ann ih i la t ion graph.

The cont r ibut ion f rom the above d iagrams migh t give rise to an i m p o r t a n t

background in the e+e - exper iments . However , they deserve also some in te res t

b y themse lves since t h e y can p rov ide useful in fo rmat ion on the coupling of

hadrons wi th a two-photon s y s t em when F of d iagram (1.1.2) is of hadronic

na ture (F could be, for ins tance, an ~] or an ~]' part icle) . This will be par t i cu la r ly

t rue wi th h igher-energy storage rings, since the cross-section is expec ted to

10 2

10 ~

E u

I o

H o

10 °

10

~ a ) b)

c)

d)

f)

1.0 2.0 3 0 E (Gev)

Fig. 25. - The cross-sections for different e+e - induced processes via two-photon interactions as calculated by BRODSKY et al., ref. [109]. We see that as the energy E of the electron and positron beams is above ~ 1.5 GeV, the total cross-sections for two-photon interaction processes are expected to overtake the cross-sections for the usual e+e--->V+~ - and e+c--~=+7: - annihilation processes: a) e + e - ~ + ~ -, b) e e ~ -~ee~+~ - (equivalent photon), c) e+e--> =+=- (pointlike), d) ee-->ee=+= - (equivalent photon), e) ee-~ee= ° (exact), ]) ce-->ee~] (exact).

increase logar i thmical ly wi th increasing energy, while the usual annihi la t ion

cross-sections are expec ted to decrease wi th increasing energy ~ (as 1 /E 2 for product ion of point l ike par t ic le pairs , see Fig. 25). For tuna te ly , the processes (1.1.2)

can be separa ted since in the final s ta te the incident electron and posi t ron

survive. This separa t ion is of ten possible also wi thout detect ing the scat tered

electrons, since the angular and energy d is t r ibut ion for these two-photon

react ions is qui te peculiar . For ins tance in the react ion e+e--->e+e-e+e - two

E L E C T R O M A G N E T I C S T R U C T U R E OF THE tI.kDI%ONS 2 7 5

leptons are expec ted to be emi t t ed in a nar row cone along the b e a m directions,

while the other two, when emi t t ed a t large angles, should have a A¢ dis t r ibut ion s t rongly peaked a round A¢ = 0 [109] (A¢ is the angle be tween the planes

defined b y the b e a m axis and the two emi t t ed part icles) .

Exper imenta l ly , the A¢ d is t r ibut ion for the react ion e+e--->e+e-e+e- has

been inves t iga ted a t Novos ib i rsk [110]. The resul ts are shown in Fig. 26, and

3oi

20 i

~1o ©

T J

! ' ? 1

-60 -40 20 0 20 40 6 5~(degrees) Fig. 26. - Distribution of events from reaction e+e--+ e+e-e+e - as a function of the angle A¢ defined as the angle between the two planes which contain each of the two large-angle emitted electrons and the beams axis. The data were collected at three different values of the total energy (2E = 1020, 1180, 1340 MeV). The iull line is the theoretical calculation of BAIER and YADIN [109]. The dashed line corresponds to a uniform A¢ distribution weighted with the detection efficiency of the apparatus.

are compared wi th the theore t ica l calculat ion of BAIE~ and FADIN [109].

A m e a s u r e m e n t of this reac t ion has also been pe r fo rmed in Frasca t i wi th

Adone, where also a few candidates f rom the reac t ion e+e--->e+e-F+ ~- have

been observed. I n this expe r imen t counters have been placed near the machine

vacuum chamber in order to de tec t the electrons (and/or positrons) emi t t ed near the beam directions, using ~s spec t rometers the magnets of the machine itself. Some addi t ional in format ion on the k inemat ics of the react ion can thus

be obtained. I n Fig. 27 the d is t r ibut ion of the 29 observed events as a func-

t ion of fl, the centre-of-mass ve loc i ty of the leptons emi t t ed a t large angles,

is shown. The sign of fl is defined as nega t ive when fl has the same or ienta t ion

as the single de tec ted electron along the b e a m direct ion (no event in which

bo th the small-angle leptons were de tec ted was observed) . A comparison wi th

theory is also g iven in Fig. 27. Al though the yield of events wi th fl < 0

depends cr i t ical ly on the lower expe r imen ta l cut in the energy of the detec ted part icles, i t appears t h a t there is a large cont r ibut ion of events with a kine-

mat ica l fea ture not foreseen b y the s tandard theoret ica l calculations [109],

2 7 6 B. BARTOLI~ F. ~ELICETTI a n d v , SILVESTRINI

a)

10

0

.... i fl<0

t I f

i \ /

! \ / \

#

~"~ detected S ~

,8>0

/ t I ,J]'\\ P

t 15

(h

~ 5

0 -20

~)

-10 0 10 20

Fig. 2 7 . - The results of the Frasca t i ~ yy-group >> on the react ion e+e--+e+e-e+e - (ref. [111]): a) Dis t r ibut ion of the events as a funct ion of fl, the e.m. ve loc i ty of the pair of leptons emi t t ed at a large angle. The convent ion used for the sign of fl is specified in the upper pa r t of the drawing. The resul ts are compared wi th the theoret ical calculat ions: PARISI [112], - - - - - - BRODSKY et al. [109]. b) Dis t r ibu- t ion of the events as a funct ion of A¢, compared wi th the calculat ion of BAIr, R and I~AD IN [109].

ELECTROMAGNETIC STRUCTURE OF THE HADRONS 277

whose approximations are based on the hypothesis of the dominance of the kinemat ical configuration

(1.1.4:)

e-

- I

e- forward

e+ wide angle

e- wide angle

e+ forward

PAI~ISI [112] has evalua ted the contr ibut ion from the kinematical configuration

(1.] .5)

e~ e_- forward

~ ! forward

wide angle

e÷ wide angle

which appears to account for most of the observed events .

Pre l iminary data f rom a second exper iment [113], in which a lower-energy cut is set on the observed wide-angle electrons, show a contr ibut ion of events f rom the configuration (1.1.4) bu t also a nonnegligible contr ibut ion f rom configuration (1.1.5). Be t te r theoret ical calculations would therefore be welcome. Also the possibility of some higher-energy storage rings to be opera ted both

with e+e - and with e -e - appears as ~ convenient faci l i ty to s tudy, in the e -e - mode of operation, the two-photon interact ions wi thout contaminat ion f rom

the annihi lat ion channels.

In any c~se, there is good exper imenta l evidence tha t the data repor ted in the next Sections as hadronie events f rom e+e - interact ions receive, if ,~t all, only a small contaminat ion from the two-photon in terac t ion graph (1.1.2) (see Sect. 5).

1"2. Point l ike leptons and 1/q 2 photon propagator. - Exper imen ta l tests of these hypotheses can be obta ined by measuring the ~+~- or e+e - pair produc- t ion cross-section in e+e - interact ions. However , as we already ment ioned in

in Sect. 1 (Par t I), the cross-section for the react ion e+e--+e+e - is dominated by the scat ter ing ra ther than by the annihi lat ion graph. Tests of the pointl ike

electrons in the t imelike region will be possible only by measuring the cross-

section for the process e+e--~e+e - with recognit ion of the charge of the pro-

duced electron pair, in which case the separat ion of the annihilat ion contr ibut ion

will be possible. Tile possibil i ty tha t the electron has a complex form factor Fo,

even if ]F~I: = 1, can also be tes ted in this react ion [114]. At present , only the annihi lat ion process e+e--+~+~ - tl,~S been experi-

mental ly invest igated. A comparison of the electron and muon form factors

2 7 ~ B. BARTOLI~ F. FELICETTI a n d v . SILVESTRINI

in the t imelike region is t hen available only at q~_~ 0.5 (GeV/c) ~ through the

measurement of the p-+ ~+F- and p-+e+e - decay rates: the resul t is [115]

F(p -~ e+e -) F(p -4 ~+~-)

-0.97 =0.17

The exper imenta l s i tuat ion on the react ion e+e---> [z+ F- is p resented in Fig. 28. The results [116-118] are expressed in terms of the rat io

_ _ , . _ , , . _ , ,

ffQED

1.6

1.2

CL ×

b • " 0.8

0.4

q i I I

I 2 3 4

Fig. 2 8 . - Results on the reaction e+e---> t~+~ -, expressed in terms of tho ratio R----6,x,/6QED. The machine luminosity is monitored in this case with the wide- angle e+e - elastic scattering: o Novosibirsk ref. [116], a Yraseati (~ ) ref. [117], • Frascati (BCF) ref. [118].

The agreement is good within the large exper imenta l errors ( typical ly (15--20) %) up to m o m e n t u m transfers as high as

s ~-- E+ + E_ = q2 = 4.4 (GeV/c):.

This kind of data is of ten parametr ized in te rms of a cut-off pa ramete r A ~

(R ---- (1--q~/A~) ~) by assigning the form 1--q~/A 2 to e i ther /~. or /~,, or to

the photon propagator modification M. This parametr iza t ion is arbitrary~ and

sometimes gives rise to serious theoret ical difficulties (when assigned, for

instance, to M or to a lepton propagator modification [23]). However i t is

usually justified with the need of comparing different exper iments . This a t t i tude is misleading in our opinion, since i t invi tes one to consider a rough exper iment at high energy equivalent to a good-precision low-energy exper iment : in fact in the cut-off phi losophy deviat ions f rom QED are expected to increase with increasing energy. This might v e r y well be wrong. Actually, a break-down

of QED is expected due to vacuum polarizat ion effects originated by hadrons

E L E C T R O M A G N E T I C S T R U C T U R E OF T H E HADROIqS 279

coupled to the v i r tua l photon: this kind of break-down is not expected to be

more impor tan t at higher energy. This is demonst ra ted exper imenta l ly in

Fig. 29 where we show the results of ano ther exper iment on the react ion e+e--~z+ F- [119] pe r fo rmed at Orsay at a lower energy than the experiments quoted in Fig. 28. The energy region explored is around the (~ mass, and a

vacuum polarization effect shows up, al though at the l imit of the exper imenta l

errors.

~15

+::::t Y~ b

- io G . . . . .

2E-M®(MeV) Yig. 29. - Results of the 0rsay group showing the vacuum polarization effect in the reaction e+e+-~ ~z+tz-. Open circles refer to events collected at five fixed energies E. Blaok circles refer to events collected with a continuous cycling of the beam's energy in a range of total c.m. energy 2E of =k 6 ~eV around the ~)-meson m~ss. The full line corresponds to no polarization effect; the dashed line represents a best fit to the data with vacuum polarization effect included: B - - F ~ o + o - F ~ . ~ - / F ¢ : - - - - - - B = =2.6"10 -4, 7e=7.1; - - B- -0 , Z2--15.

The re levant point is the comparison between exper iment and theory;

and the pe r t inen t paramete r , a t whatever energy, is the precision of the

exper iment ra the r than the cut-off pa ramete r A.

2. - P r o t o n f o r m factors .

A first measurement of the cross-section for the react ion

(2.1) e+e ---~ p~

has been recen t ly per formed at Frascat i by the Naples group [120]. 214-5 events f rom the react ion (2.1) have been observed at q~ =4 .4 (GeV/cp. The separation

of background f rom react ion (2.1) is achieved b y means of energy E and d E / d x

measurements in thick scintillation counters, time-of-flight determinat ion and

coll ineari ty of the observed tracks as measured in optical spark chambers.

The sample of events is t hen qui te clean in spite of the v e ry low counting ra te

( , -~levent / (2--4) days). 14 of the events show the an t ipro ton annihilat ion star, as expected on the basis of the known detect ion efficiency of the apparatus.

To determine from the observed number of events the to ta l cross-section, extrapolat ion over the full solid angle of the count ing ra te is needed. (The

2 ~ 0 B. BARTOLI, F. FELICF, TTI and v . SILVESTRINI

expe r imen ta l appa ra tu s covers ~ 0 . 6 of the to ta l 4z solid angle, a l though

the de tec t ion efficiency is equul to 1 only ~round 90°.) For this purpose , the

angular d is t r ibut ion of the events m u s t be known.

The eqa iva len t of the l~osenbluth formu],~ in the t imel ike region is []21]

da ~ ~ ( cos20) 1 ) - ~ a ~fl I(~,]'(1 ÷ le~l' s in to

dY2 _ q2.

fl, 0 = ve loc i ty and angle of emission of the proton.

10 -31

10:32

E

~ i0-33

b

D2o] : E122J

D23] LLLLL/

1~ 34 c)

10-35 i I

2.0 2.2 2.4 2.6 2.8 3.0 2E(GeV)

l _ _ 1 I I i I 4 5 6 7 8 9

q2 [(Ge V/c)2]

Pig. 30. - The result of the Naples group (ref. [120]) on the reaction e+e-~pp . The black circle . represent the cross-section as determined by using all the detected events; the cross + represents the cross-section as determined by using only the events in which the antiproton annihilation star is detected. Previously available upper limits ~/////~ from the reaction p~->e+e - are also shown (ref. [122, 123]). The full lines represent theoretical predictions for some particular values of the form factors: a) / v = 1, /v2=1.79; b) G ~ = G ~ = I; c) G ~ = G ~ = I/(I ÷q~/0.71) 2.


The angular d is t r ibut ion depends on ]G~I'/[GEI2 and some hypothesis is

needed, unless the exper iment covers a wide enough 0 region so tha t [G~] ~-

and [G~[ ~ can be separ'~tely de te rmined; this is not the c~se for the Naples experiment . However this exper iment has been performed near threshold where one would expect the angular dis t r ibut ion to be isotropic, i.e. IG, t'2= IG, I'.

In fact, unless G E = G M ,~t threshold (z =--q2/4M2= --1) , fx~ and F2~ (see Subsect. 2"2, Pa r t I) become infinite, producing an unwanted divergence in the e.m. current of the proton.

Wi th the hypothesis of an isotropic product ion dis tr ibut ion the authors

obtain a value of a,+._op~ = (4.6 q- 1)-10 -34 cm 2 (G+.--~o5 -- (6.2 4-1.8) .10 - ~ cm 2 using only the events wi th the de tec ted annihi lat ion star). In Fig. 30 these values are compared with previously available upper limits f rom the reaction

+p--> e + + e - [122,123] and with calculations for a few part icular choices for

G~ and G~. In the above hypothesis IGEI = IG~I, the measured value of the cross-

section corresponds to IGEI = IGeI = 0.19 4- 0.03. This exper iment is obviously only a first approach to a new field of inves-

t igation. New storage ring exper iments permi t t ing the de terminat ion of ]G~I and ]Gut separately (and also the form factors of unstable baryons) are planned

both at Frasca t i and Stanford.

3. - T h e p i o n f o r m fac tor .

The pion form factor is simply related in the t imelike region to the cross-

section for the react ion

(3.1) e+e---> ~+~:-

by the relations [121]

(3.2)

do',:+,~-

d cosO

O'~;+rc-

_ ~ ~ z = ( q ~ [ 2 s i n 20,

A measurement of ~(e+e--+=+~ -) therefore permits one to measure IF=(q~)I ~.

Notice tha t a ~+r~-sys tem with 1 = 1 (as i t mus t be in the one-photon-

exchange hypothesis) must have T = 1 so tha t F,:(q 2) is re la ted to the coupling of isovector photons with h'~drons. Actually, F= coincides exact ly with F v below threshold for 4~ product ion and in pract ice below q2 = 1 (GeV/c) ~. Phase-

space factors ~re expected to depress strongly the e+e--~4~ channels below q~--1 (GeV/o)L as confirmed by the first exper imenta l measurements (see Sect. 5).

9 . ~ B. BARTOLI, F. F~,LICETTI o,nd v . SILVESTRINI

l~easurements of a(e+e---*~-~ -) ut q~ < 1 (GeV/c): have been qui te exten-

s ively pe r fo rmed during the last ~ 5 years a t Novos ib i r sk [124] and Orsay []25].

The phenomenology is domina t ed b y the product ion of the vec tor meson p, the foreseen in ter ference t e r m wi th the ¢~ cont r ibut ion (via the e lec t romagnet ic

decay mode ¢o -*~+r:-) hav ing also been observed. The resul ts are summar ized

in Fig. 31, showing [F.12 as a funct ion of q2.

50

40

30

20

10

01 0,3 014 0.5

' '\

Fig. 3 1 . - [F~(q~)[ 2 as determined by the Novosibirsk and Orsay groups through measurements of the reaction e+e--+ =+r:- at q ~ 1 (GeV/c) 2. The full line is the Breit- Wigner best fit to the Novosibirsk points. The dash-dotted line is the best fit to the Orsay points using a Gounaris-Sakurai formula and taking into account the co contribution via the decay channel ¢o-->r:+~-. • ~ovosibirsk 1, ref. [124]; . . . . . o Orsay 1, ref. [125]; • Novosibirsk 2, ref. [124]; • Orsay 2, ref. [125].

The full line is the Bre i t -Wigner bes t fit to the Novosib i rsk points ; the

dashed line is the bes t fit to the Orsay points including the ¢o --> ~+~- contri-

but ion. On the top of the O peak the cross-section is ~ 1.5 ~b. I n t e rms of

p pa rame te r s , the resul ts can be summar ized as follows.

Orsay Novosibirsk

m~(MeV) 775.4± 7.3 754 ± 9

Fp (MeV) 149 ± 23 105 ± 20

/~p~°+o-//~p~ot~, (4.0 ± 0.5). 10-~ (5 ± 1 ) . i 0 -5

/'p__~+.- (keV) 6.1 ± 0.7 5.2 ± 0.5

ELECTROMAGNETIC STRUCTURE OF THE HADRONS 2 ~

The difference in the pa r ame te r s obta ined ut Novosibirsk appears to be

essential ly due to the fact t ha t the co cont r ibn t ion is not t aken into account.

Actually, the b e s t fit to the Orsay points (which provides the addi t ional infor-

mat ion (F,~,~/F,~tota~) ½ =: 0.2-4-0.057 the phase angle be tween the co and p ampl i tudes being ¢ ~ = (87.05 -+- 15.04) °) appears to fit perfect ly also the ~ovo-

s ibirsk points . The fact t ha t the cross-section for 7:+~- product ion can be accounted for

by the p and co contr ibut ions alone (actually the p, co ~nd ~ account for all

the hadronic produc t ion below 1 GeV) was a s t rong suppor t in favour of the

~ vector dominance ~) hypothes is [126]. The da ta above 1 GeV, however, give

10

o lO

10 -1

\\\t \ \

' t \ \ \

\ \ \ e ta,~k

\ \

\

10-2 1.0 1.5 2 0 I 2 E(G e V)

i r L L

1 2 3 4 5 q2 [(G eV/c)2]

Fig. 3 2 . - ]F~(q2)] 2 as determined by measurements of the reaction e+e--+=+=- at q2 >~ 1 (GcV/c)% The expected contribution from the p tail is shown. Corrections for a possible contamination of kaons in the sample of pions are not applied: x Orsay ref. [144]; • Novosibirsk, ref. [129]; o Fraseati (BCF), ref. [142]; n Frascati (F~), ref. [143].

284 B. BARTOLI, F. FELICETTI and v. SILVESTRINI

evidence in favour of a nonnegligible cont r ibut ion f rom other mechanisms (*).

Above 1 GeV, the resul ts are p resen ted in Fig. 32. We see th'~t for

1 < q2< 4 (GeV/e) ~, lF~t ~ is larger than the expec ted cont r ibu t ion of the p tail. In this energy region, the corresponding cross-sections gre of g few nanobarn

([F,~I~ = 1 correspond to a a(e+e--+~+7~ -) cm2]

\ 20.10-33

w o u l d cross-section

A separa t ion of the ~ 's f rom the K ' s has not been ach ieved in the Fra-

seati points , so t ha t the in t e rp re t a t ion of the data of Fig. 32 as ]F=12 requires

the hypothes is t ha t the cont r ibu t ion to the count ing ra te f rom the channel

e+e - - > K + K - is negligible.

4. - The kaon form factor.

The kaon form factor is re la ted to the cross-section for the process

(4.1) e+e - ~ K + K -

by a re la t ion of exac t ly the same form as eq. (3.2). However , also isosca 'l~r photons can couple to a K + K - pa i r ( ra ther t han only isovector photons as in

~+~- product ion) . Around the ~ mass, ]FKI2 is domina ted by the process e+e- - ->~- ->K+K - .

The product ion of C-mesons in e+e - in teract ions has been invest ig~ted ~t Orsay [127] and Novos ib i rsk [128]. The resul ts can be summar ized in the fol-

lowing Table.

(*) However, the simple p, ¢o and ~ vector dominance model was already in some trouble due to the behaviour of the isoveetor form factors ~ r of the nucleons as a function of the four-momentum squared t ~ - - q 2 :

In fact, using the relation

1 Fv~ ~ .

oc ( I m Fv(S ) Fr(t) J t - - s ds

and since 1/(t--s)= 1/t+ (1/t)(s/(t--s)), we have

I f 1 ~sIm.F~,(s)ds 2'v(t)~ Im 2%(s) ds + ~-j t~ s s "

Er(t ) ~ 1/t 2 requires f I m Fv(S ) ds = 0. This however cannot be satisfied if only the p contributes to fly(s) since the p contribution has a large imaginary part with definite sign.

~L]~CTROMAGN]~TIC STRUCTURE OF THE HAI)RONS

TABLE II.

285

Orsay Novos ib i r sk

O-K+K_ (q2 = M~) 2.41 ± 0 .13) '10 -3° cm 2 (2.13 -~ 0 .17) .10 -3° cm 2

aKo~o ( q 2 = M ~ ) 1.47 - k 0 . 2 1 ) . 1 0 - a ° e m 2 (1.01 ± 0 . 1 5 ) . 1 0 - 3 ° e m ~

an+n-xo (q2 _ M2O) 1.01 ± 0.21). 10 -s° cm 2 (0.81 -k 0.21). 10 -as cm ~

aaUmod~ (q2 = M~) 4.99 ~ 0 .40) '10 -s° cm ~ (3.96 -t- 0.35)" 10 -3° cm ~

F~ 4.09 :J_ 0.29) MeV (4.67 ± 0.42) MeV

/~¢__>e+e-//~._>all 3.52 ± 0.28) .10 -~ (2.81 ± 0 .25) .10 -4

T'~_~K+K-/F~_~ . 0.483 i 0.043 0.540 ± 0.034

F~_~KO~O/F¢__>~, u 0.295 ± 0.040 0.257 =k 0.030

T'+_~n+r:-r:o/F,i,~, 0.202 ± 0.035 0.203 ± 0.042

F6_~+ ~- (1.44 ± 0.12) keV (1.31 ~ 0.12) keV

Above the (~ mass, only four events h:~ve been observed ~t 5Tovosibirsk ut three different values of q2.

10 I

10 °

10 -I

t 1 0 - 2

10 Ill

I 1.0 112

' ' . . . . . . I'4 12 13 2E(GeV)

l'.h 1:6 IT8 210

Fig. 33. - IFK(q2)[ 2 as determined by the Novosibirsk group (4 events in total). The meaning of the curves is explained in the text.

These results [129], in terms of ]F~] 2, ure presented in Fig. 33. The curve B-W represents the tail of the ~) Breit-Wigner; the other two curves are [-FKI2

286 B. BAI~TOLI, F . F E L I C E T T I and V. S I L V E S T R I N I

as expected in the vector dominance model calculated as follows [129]:

2 2 g ~ m~ g~KK m~ (4.2) FK(q~) __ g~KK m~ ÷ _ _ ÷ _ q~ 2 __ q2 q~ g~ m~ g~ m~ g~ m~- - "

2 2] I f we pu t m ~ m ~ ,

F ~ ( q D - _q~ ÷ + _q~ m~ \ gp g~ / g¢ m~

and use ivy(0) --~ 1, A = g~x/gp ÷ g~:/g,~ can be evaluated in te rms of g~x/g~," Then the curves ÷ and - - in Fig. 33 represent respect ive ly what is expected

if A has the same phase as g~KJg~,, or a 180 ° difference in phase. We see tha t the ex t remely poor statistics are still insufficient to decide

whether [/~K[ ~ is accounted for, in this energy region, by the simple p, co and

vector dominance model.

5. - Multihadron production in e+e- interactions.

Up to values of q2~1 .1 (GeV/c) = multihad~'on product ion is essentially l imited to the product ion of 7:+rz-n ° via the isoscalar vector mesons co and }. The contr ibut ion f rom the channel e+e--->~)-~TV~-z: ° is a l ready summarized in Table I I . The co product ion was also invest igated at Orsay [130]. The results can be summarized as follows:

a(e+e--~7:+~-~ °) (at the co mass) --~ (1.76-V0.13) ~ b ,

F~ ~ 12.2 MeV (from the world average) ,

F~__)o+ _ = (1.00 i 0.18) k e V .

Above q ~ 1 . 1 (GeV/c) ~, mul t ihadron product ion has been invest igated a t Orsay, Novosibirsk and especially a t Fraseati .

The first exper imenta l values for mult ipar t ic le product ion cross-sections

were p resen ted at the K iev Conference b y the Frascat i (~boson ~)[131] and

(~ ~7: ~)[132] groups; the values of the cross-section (9 30"10 -a3 cm ~) were at

least one order of magni tude larger than expected on the basis of an extra-

polat ion f rom the lower energy range data. On the basis of pulse-height analysis, shower recognit ion and invest igat ion

of the in terac t ion proper t ies in the spark chamber plates, i t was soon possible to conclude tha t the produced particles are hadrons (7: or K) with a contaminat ion f rom e and Ez which is a t most (5--10)~o [133-136].

In addit ion, i t was possible to conclude tha t the product ion occurs essentially

ELECTRO~WAGNETIC STRUCTI/RE OF TI-IE HADRONS ~7

via the annihilation channel, with at most a small contr ibut ion (a few percent)

f rom graphs of the type (1.1.2). I n fact none of the observed multihadron

events were detected in coincidence with a small-angle electron.

I n addition, the distr ibution of the events as a funct ion of the noncop]anari ty

angle A¢ appears to be flat [137] (Fig. 34) with no appreciable contribution

I0

-6 b)

0 i i i i i I I I I I I I

10

i

5

a)

- 60

i

L I

I T I

13 ° 13 °

, I, , ,; , , ,

' -~'o -2'o o 2'0 ~o do A dp (degrees)

Fig. 34. - a) Distribution of noneoplanar (multihadron production) events as a function of A¢, the noneoplanarity angle between pairs of observed charged tr~cks. The distribution is corrected for the detection efficiency, b) A¢ distribution for the reaction e+e-~e+e-e+e - as expected according to BAI~R and ~ADIN, see ref. [109].

2 8 8 B . B A ~ T O L I ~ F . F E L I C E T T I a n d v , S I L V E S T R I N I

160

1 2 0

E ¢3

80

b

40

\ \

i \

\ \

\

I i I I i I

1.0

? 0 I I 1 I I I T I I I I I I P I I

1 .5 2 . 0 2 . 5 3 .0 2 E ( G e V )

Fig. 35. - The to ta l cross-section ~tot(e+e---> more than two hadrons) as a funct ion of the to ta l c.m. energy 2 E : E+-F E - = ~ /~ . F o r reference, the dashed l ine shows the calcula ted to ta l cross-section for product ion of F+~- pairs : , Orsay, ref. [144]; x Novosibirsk, ref. [145]; [] F rasea t i (boson), ref. [137]; • F rasea t i (fz~), ref. [143]; o Frasea t i (yy), ref. [134].

50

40

3O

o

b

10 I

0 ]~ ~ I J I I i I I. 1 I I I I 1.0 1.5 2 0 2.5 3.0.

2E(GeV)

Fig. 36. - The cross-section to produce four charged pions a(e+e--+ =+r:-r:+7:-) as a funct ion of the e.m. energy 2E. In this and in the fol lowing Figures the symbols used to indicate the exper imenta l points f rom different exper iments are the same as in Fig. 35: , Orsay, ref. [144]; o F rasca t i (boson), ref. [137]; • F rasca t i (Fr:), ref. [138].

ELECTROMAGNETIC ST]R, UCTURE OF THE HADRONS 2 8 9

40

~ ' 3 0 E L)

,.q I O .~20 b

10

1J0 ~ _ ~ i 1 1.5 2E(GeV)

Fig . 37. - T h e c ro s s - s ec t i o n ~(e+c--+~+7:-~°= °

210 2.5

as a f u n c t i o n of t h e t o t a l c .m. e n e r g y 2E.

I 0

30 a)

20

10

0

20[ b)

I0

0 J 6O

c)

50

4O

30

2O

I0

i

1.0

I

t , r , t , , t

r ! I I I I _ 1 I I

J - - J I

1.5 2E(GeV)

r i

2.0 2.5

Fig . 38. - a) T h e c r o s s - s e c t i o n a(e+e-->r~+7:-r:+~:-r:°u °) as a f u n c t i o n of 2E, b) t h e c r o s s - s e c t i o n a (e+e - - ~ r:+=-r:+T:-~ °) as a f u n c t i o n of 2 E , c) a(e+c - - + 7:+7:-=+T:-r: °) + + a ( c + e - - + =+r~-T:+7:-Tr°T: °) as a f u n c t i o n of 2E.

19 - Riv i s ta del Nuovo Cimenio.

2 9 0 B. BARTOLI, r . FF, LICETTI a n d v . SILVt~STRINI

from the peaked dis t r ibut ion characterist ic of the events of the type

e+e - __> e+e-e+e -, e+e - __> e+e-~+~ -, e+e - -> e+e- ::+~ -.

Under the assumption tha t the charged detected hadrons f rom the reaction

(5.1) e+e - -*more than two hadrons

are pions, and tha t angular distributions are determined by pure phase space,

cross-sections have been evaluated for different product ion channels. The main

conclusions would remain unal tered within the errors if the angular and energy

distributions are determined by quas i - two-body intermediate states (e.g. e+e--->

--> A.+~:---~ ~+~:-~+7: -) [134, 136, 137].

The results are shown in Fig. 35-40. The total cross-section appears to be

60

50

~- 40 E U

~g 30 b

20

10

I i i .I i I L L I ~ I I h

0 1 0 1.5 2.0 2.5 2E(GeV)

Yig. 39. - The cross-section a(e+e---> at least four charged pions) as a function of 2E. The points indicated by the symbol I'1 are taken from ref. [139].

ve ry large, larger than the cross-section for product ion of a pair of pointlike

Iermions. After a steep increase between --~ 1.2 and 1.4 GeV, it shows a

slow fMl-off consistent with a 1/q ~ dependence. Different channels show dif-

ferent energy behaviom's: for instance, while the channel e+e--->~c+r~-~+~ -

shows a broad bump at a mass of N ] . 6 GeV (a new vector boson?), the

e+e--->::+~-~c+rc-~:+~: - shows a slow increase between 1.5 and 2.4 GeV.

Some a t t empts have been made to in te rpre t the above data in terms of

conventional vector dominance [140], or in terms of an extended vector

dominance including the contr ibut ion of a higher-mass vector meson p' [141].


~'-" 10

c~

I o ~ 5 b

J

1.5 210 ' 2 E (GeV)

I I I I I

0 1.0 2,5

Fig. 40. - The cross-section a(e+e--> 7:+r:-r:+r:-T:+7:-) as a function of 2E.

The most general at t i tude is however to consider these large cross-sections

as a different manifestation of the same phenomenon which is observed in

deep inelastic electron-proton scattering.

6. - C o m m e n t s a n d c o n c l u s i o n s .

The scattering of charged leptons on hadron targets, as well as the production of hadronic systems in e+e - interactions, has been an active field of investigation.

Due to the validity of three experimentally confirmed hypotheses--one- photon-exchange approximation, pointlike leprous and 1/q ~ photon propa-

g a t o r - t h e experimental results have a straightforward phenomenlogical interpretation, in terms of a small number of form factors or struct~tre functions closely connected with the electromagnetic structure of the concerned hadrons and of the e.m. current generated by hadrons. The experimental information is quite abundant for nucleons in the spacelike region. On the contrary, the experimental knowledge of the e.m. structure of unstable particles, as well as the data in the timelike region above q: ~ 1 (GeV/c) ~, is still scarce or lacking.

In spite of the experimentt, ol difficulties--connected with the extremely small cross-sections and, in the timelike region, with the need of using the technique of beam-beam interact ion--a general effort to overcome our present lack of experimental knowledge is to be expected--and stimulated--in the near future. Actually, it appears improbable to us that a real understanding

of the interactions of the elementary particles will be achieved until their

e.m. structure will be known.

From the theoretical point of view, we are still far from being able to predict

the behaviour of form factors and structure functions, of connecting them with one another, of understanding the timeHke region in terms of the data in the

spacelike region; that is to say, as yet we do not have any theory. The experimental information on any particle, in any q~ region, is therefore extremely useful and will complement the already available phenomenological knowledge

292 B. BAItTOLI~ F. F:ELICETTI a n d v , SILVESTRINI

One of the mos t exci t ing and e legant ideas in e lementa ry-par t i c le physics

was first suggested by the behav iou r of e .m. fo rm factors : the idea t h a t a

gauge field is genera ted by each conserved q u a n t u m number . The exper imen ta l

da ta on hadron produc t ion f rom e+e - collisions below 1 GeV have nicely con- f i rmed this idea, suggest ing t h a t Na t u r e was app ly ing i t in i ts s implest f o r m - -

the vec tor dominance model. ~ e w data a t higher energy~ as well as resul ts

in the spacelike region, show tha t this s imple model is not adequate , bu t do not des t roy the va l id i ty of the idea. As a compensa t ion for this complicat ion,

the new data have suggested ano the r s imple i d e a - - t h e pa r ton models. Also

this idea is cer ta in ly too simplified, and we a l ready know tha t the s implest

versions are in t rouble : in the p a r t o n - q u a r k identif icat ion, for example , we

known t h a t the s imple 3-quark s t ruc tu re of the nucleons is not adequa te ; a sea

of v i r tua l quarks is added to the valence quarks, etc.

Bu t we a l ready knew t h a t i t is unl ike ly t ha t the e x t r e m e l y compl ica ted

phenomenology of e lementary-par t ic le in teract ions can be comple te ly under-

s tood in the f r ame of a s imple model . However , the close connect ion be tween the expe r imen ta l numbers and

fundamen ta l q u a n t i t i e s - - c u r r e n t commuta to r s , spectra l funct ions, Wigner te rms , e tc . - - t e l l s us t h a t these difficult expe r imen t s will cer ta in ly give im-

po r t an t resul ts since t hey will p rov ide the founda t ion on which to build all

fu tu re theories .

Note added in proo]s.

After this paper was submitted to Rivista del Nuovo Cimento for publication, new data on e+e - interactions have been presented at the X V I International Con/erence on

20

~15 E o

I 10 ~.10 ~i0 410 ~.0 ~.0

Fig. 41. - Experimental results on e+e-->e+o - elastic scattering obtained at Fraseati by the BCF group. The results are expressed in term of a yield per unit luminosity as a function of S = 4E 2. The full line is a best fit to the experimental data of the form yield = A/S ~.

ELECTROMAGNETIC STRUCTURE OF THE HADRONS ~9~

2.0

1.5

1.o

0.5

i I I i

2 3 4 5 - q2 [(GeV/c)2 ]

Fig. 42. - The pion form factor F~(q 2) as de te rmined by the measurement of the reac- t ion e+e--+ ::+=- at q2~>l (GeV/e) 2. Corrections for a possible con tamina t ion of kaons are not applied. The full l ine represents the expected cont r ibut ion of the O tail (Gounaris- Sakurai) . × Novosibirsk, ref. [129]; = Frasca t i (F=), ref. [146]; • F rasca t i (BCF), ref. [146].

High-Energy Physics held at Ba tav ia . We th ink it convenient to add in proofs these new data.

In Fig. 41 arc shown the rmw resul ts on e+e--+ e+c - elastic scat ter ing obtained at l~rascati by the BCF group. The authors prefer to present thei r da ta in te rms of yield per uni t lunainosity Y as a funct ion of S = 4E ~. The full l ine represents a fit

10

10

10-

10 1.0

I 1.0

I- 1.1 1.2 1.3 1.4 11.5

2E(GeV)

'll.~ ) 1./+ 'L6 11~ 2!0 2.2 -q~ E~G +v/~) ~ ]

Fig. 43. - The kaon form factor [FK(q2)[ 2 as de te rmined at • Novosibirsk, ref. [129] and i F rasca t i (~r~), ref. [146]. The meaning of the curves is explained in the tex t , P a r t I I , Sect. 4.

2 9 ~ B. BARTOLI, F. FELICETTI a n d v . S I L V E S T R I N I

of the form Y : A / S n, where n turns out to be n = 0.985 =t: 0.040. Both the absolute value A and the slope n are in agreement with QED predictions within (4--5)%.

In Fig. 42, 43 the present situation on the pion and kaon form factors z~:; and /~K is shown. These Figures update Fig. 32, 33.

Figure 44 shows a summary of the results for the total cross-section ~tot for multihadron production in e+e - interactions, presented in terms of the ratio (rtoJ(ra÷~-~

6 ¸

R

5

4

3

o

/

2

i L i

2 2E(GeV)

I i

2E(GeV)

Fig. 44. Fig. 45.

Fig. 44. - Summary of the results on mul¢ihadron production in c+e - interactions. The results are expressed in terms of the ratio R = atot(e+e--+more than two hadrons)/a(e+e--+tz+~ -) as a function of the total c.m. energy 2 E = E + W E - = V'~. • 0rsay, ref. [144]; • average of the results of the lVrascati (boson), (~=), (Y¥) groups, ref. [137, 143, 134]; o Frascati (BCF), ref. [146]; n CEA, ref. [146].

Fig. 45. - Average values of the multiplicities in multihadron production from e+e - interactions as a function of the total e.m. energy 2E. Black symbols represent total multiplicity, while open symbols represent charged multiplicity. <> Orsay, ref. [144]; • Frascati (~=), ref.[146]; u, • Frascati (TY), ref.[146]; A CEA, ref.[146];

. . . . (N}to~,~; . . . . . (N}oh~rgod.

The open circles (BCF group results) have been analysed under the assumption that the multiplicity distribution in multihadron production is the same as in proton- antiproton annihilation at rest, with phase-space corrections as the energy increases. The black circles are averages of the Frascati groups [137, 143, 134]. The point at 2 E = 4 (GeV) comes from the CEA by-pass experiment.

Finally, in Fig. 45 the average multiplicities in multihadron production are presented.

R E F E R E N C E S

[1] See, for instance, S. D. D~ELL and F. ZXCHAmAS~.N: Electromagnetic Structure o] 2Vueleons (Oxford, 1961).

[2] A . M . BINCER: Phys. Bey., 118, 855 (1960). [3] /~I. ROSV.NBLUTH: Phys. Rev., 79, 615 (1950).

ELECTROMAGNETIC STRUCTURE OF THE HAI)RONS 2~5

[4] P. TsAI: Phys. l~ev., 122, 1898 (1961); N. MEISTE~ and D. R. YENNIE: Phys. Bey., 130, 1210 (1963).

[5] U. GtiN~HEI~ and R. ROI)EN~ERG: NUOVO Cimento, 2 A , 25 (1971); S. D. DI~ELL and IV[. A. RUDERMAN: Phys. Rev., 106, 561 (1957); S. D. DRELL and S. Fu- BINI: Phys. Rev., 113, 741 (1959); N. R. WERTHAMER and M. A. RUI)ERMAN: Phys. Rev., 123, 1005 (1961); D. lYLAMM and W. KUMMER: NUOVO Cimento, 28, 33 (1963); S. D. DRELL and J. D. SULLIVAN: Phys. t~ev. Lett., 19, 516 (1965); G. K. GREENHUT: Phys. Rev., 184, 1860 (1969).

[6] M. GOU~I)IN: Di]]usion des electrons de haute energie (Paris, 1966). [7] See, for instance: T. J),NSSENS, R. HOFSTAI)TER, ]~. B. HUGHES and M. R. YEA-

RIAN : Phys. Rev., 142, 922 (1966) ; K. BERKELMAN : in Nuclear Structure, edited by R. HOFSTAI)TElZ and L. SCIIIFF (Stanford, 1964); R. WILSON: in Particle Interactions at High Energies, edited by T. W. PREIST and L. L. J. VICK: (Edinburg and London, 1967); H. J. BEHREND, F. W. BRASSE, J. ENGLER, H. HULTSCHIG, S. GALSTER, (~. tIARTWm, H. SCHOPPER and E. GANSSAUGE: NUOVO Cimento, 48A, 140 (1967); W. BARTEL, B. DUDELZAK, H. KREHBIEL, 5. ~/[. ]~CELROY, U. ]~/[EYER-BERKIIOUT, R. o v. ]V[ORRISON, H. N~UYEN-Nooc, W. SCHMII)T and G. WEBER: Phys. Rev. Lett., 17, 608 (1966); W. ALERECUT, H . J . BEHRENI), 1 ~. W. BRASSE, W. I~LAUGER, H. HULTSCHIG and K. G. STEFFEN: Phys. Rev. Lett., 17, 1192 (1966).

[8] L. CAMILLERI, J. H. CHRISTENSON, M. KRAMER, L. M. LEI)ERMAN, Y. NAGA- s i l i c a and T. YA~IANOVCHI: Phys. Re~. Lett., 23, 149 (1969).

[9] 5. MAR, B. C. BARISH, 5. PINE, D. H. COWARI), H. DESTAEBLER, J. LITT, A. MINTEN, R. E. TAYLOR and M. BREII)ENBACH: Phys. Rev. T~ett., 21, 482 (1968).

[10] D. YOUNT and J. PINE: Phys. Bey., 128, 1842 (1962). [11] A. BR0WMAN, F. LITY and C. SCHAERF: Phys. Rev., 139 B, 1079 (1965). [12] W. BARTEL, B. DUI)ELZAK, It. KREItBIEL, J. ~ . MCELROY, R. J. MORRISON,

W. ScHMII)T, V. WALTHER and G. WEBER: Phys. Lett., 25B, 242 (1967). [13] B. BOUQUET, D. BENAKSAS, B. GROSSET]~TE, B. JEAN-~A-RIE, G. PARROUR,

5. P. P o u x and R. TCHAPOUTIAN: Phys. Lett., 26 B, 178 (1968). [14] R . L . ANI)ERSON, B. BORGIA, G. L. C~SSIOAY, Y. W. DE WIRE, A. S. ITO and

E. C. LOll: Phys. Rev. Lett., 17, 407 (1966); Phys. Rev., 166, 1336 (1968). [15] G. CASSIDAY, J. DE WIRE, H. FISCHER, A. ITO, E. Lo~ and J. RUTHERFOORD:

Phys. Rev. Lett., 19, 1191 (1967). [16] Although A and P are zero if Im A2v is zero, they cannot be expressed in terms

of Im A2v, since A~¥ of eq. (1.1.4) was already summed and averaged over the final and initial spin states. For explicit calculations of polarizations see, for instance, M. L. GOLDBERGER and K. M. WATSON: Collision Theory (New York, 1965).

[17] C-. V. DI GIORGIO, E. GANSSAUGE, R. GOMEZ, G. GORINI, S. PENNER, S. SER- BASSI, M. L. VINCELLI, E. AMALDI and G. STOPPINI: NUOVO Cimento, 39, 474 (1965).

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S. ROCK, G. SHAPIRO, H. WEISBERG, R. L. A. COTTRELL, 5. LITT, L. W. Mo and R. E. T~XYLOtt: Phys. Rev. Lett., 24, 753 (1970).

~ B. BARTOLI, F. FELICETTI and v. SILV]~STRINI

[22] See, for instance: 5. D. BJORKEN and S. D. DRELL: Relativistic Quantum Mechanics (New York, 1964).

[23] See, for instance, ref. [1, 2, 6], and N. KROLL: NUOVO Cimento, 45 A, 65 (1966). [24] J. C. WESLEY and A. RICH: Phys. Rev. Lett., 24, 1320 (1970); J . BAILEY,

W. BARTL, G. VON BOCHMAN, R. C. A. BROWN, F. J. M. FARLEY, H. J()STLEIN, E. PICASSO and R. W. WILLIAMS: Phys. Lett., 28B, 287 (1968}.

[25] C. MOLLER: Ann. der Phys., 14, 531 (1932). [26] H. J'. BHABHA: Proc. Roy. Soc., A 154, 195 (1935). [27] W.C. BARBER, B. GITTELMAN, G. K. O'NEILL and B. RICHTER: Phys. Rev. Lett.,

16, 1127 (1966). [28] W.C. BARBER, C~. K. O'NEILL, B. GITTELMAN and B. RICHTER: SLAC Report ,

CLNS-139 (February 1971), to be published. [29] V. ALLES-BoRELLI, M. BERNARDINI, D. BOLLINI, P. L. BRUNINI, ]~. FIORENTINO,

T. ~ASSAM, L. MONARI, F. PALMONARI and A. ZICHICHI: Nnovo Cimento, 7 A, 345 (1972).

[30] B. BORGIA, F. CERADINI, M. CONVERSI, L. PAOLUZI, W. SCANDALE, G. BAR- BIELLINI, IV[. GRILLI, P. SPILLANTINI, R. VISENTIN and A. MULACHIE: Phys. Lett., 35 B, 340 (1971).

[31] B. BARTOLI, F. FELICETTI, H. OGREN, V. SILVESTRINI, G. MARINI, A. NIGRO, N. SPINELLI and F. VANOLI: Phys. Lett., 36B, 593 (1971); B. BARTOLI, B. COLVZZI, F. FELICETTI, G. GOGGI, G. MARINI, F. MASSA, D. SCANNICCHIO, V. SILVESTRINI and 1~. VANOLI: NUOVO Cimento, 70A, 615 (1970).

[32] J. E. AUGUSTIN, J. BUON, B. DELCOURT, J. JEANJEAN, D. LALANNE, H. NGUYEN NGOC, J. PEREZ-Y-,]'ORBA, P. PETROFF, F. RICHARD, F. RUMPF and D. TREILLE : Phys. Lett., 31 B, 673 (1970).

[33] R . W . ELLSWORTH, A. C. MELISSINOS, J. H. TINLOT, H. yON BRIESEN jr., T. YA- MANOUCHI, L. M. LEDERMAN, M. J. TANNENBAU)~I, R. L. COOL and A. MASCHKE : Phys. Rev., 165, 1449 (1968).

[34] L. CAMILLERI, J. H. CRISTENSON, M. KRAMER, L. ~¢[. LEDERMAN, r . NAGASHIMA and T. YAMANOUCHI: Phys. Rev. ZeSt., 23, 153 (1969).

[35] B . D . DIETERLE, T. BRAUNSTEIN, J. COX, F. ~¢[ARTIN, W. T. TONER, M. L. PERL, T. F. ZIPF, W. L. LAKIN and H. C. BRYANT: Phys. Rev. Lett., 23, 1187 (1969).

[36] W . T . TONER, T. J. BRAUNSTEIN, W. L. LAKIN, F. MARTIN, M. L. PERL, T. F. ZIPF, H. C. BRYANT and B. D. DIETERLE: Phys. Lett., 36B, 251 (1971).

[37] See, for instance, R. WILSON: in Particle Interactions at High Energies, edited by T. W. PREIST and L. L. Y. VIce: (Edinburg and London, 1966).

[38] W. BARTEL, B. DUDELZAK, H. KREHBIEL, J. M. MCELRO'f, U. ]~][EYER-BERKHOUT, R. J. MORRISON, H. NGUYEN-NGoc, W. SCHMIDT and G. WEBER: Phys. Rev. Lett., 17, 608 (1966).

[39] W. BARTEL, B. DUDELZAK, H. KREHBIEL, J. M. McELRoY, U. MEYER-BERKHOUT, R. J'. )/[ORRISON, H. NGUYEN-NGoc, W. SCI-IIMDT and G. WEBER: Phys. Lett., 25B, 236 {1967).

[40] T. ,}'ANSSENS, a . HOFSTADTER, E. B. HUGHES and M. R. YEARIAN: Phys. Rev., 142, 922 (1966).

[41] H . J . BEHREND, F. W. BRASSE, J. ENGLER, U. HULTSCHIG, S. GALSTER, G. HARTWm, H. SCHOPI"ER and E. GANSSAUGE : Proceedings o] the X I I I Inter- national Con/erence on High-Energy Physics (Berkeley, 1966): Nuovo Cimento, 48A, 140 (1967).

[42] W. ALBRECHT, H. J. BEHREND, F. W. BRASSE, W. FLAUGER, H. HIYLTSHIG and K. G. STE~F~N: Phys. Rev. Lett., 17, 1192 (1966).


[43] K. BERKELMAN, 1~[. ]~ELDMAN, ]~. M. LITTAUER, G. RO~YSSE and R. R. WILSON: Phys. Rev., 130, 2061 (1963).

[44] K . W . CHEN, J. ]~. DUNNING jr., A. A. CONE, N. Y. RAMSEY, J. K. WALKER and R. WILSON: Phys. Rev., 141, 1267 (1966).

[45] L. E. PRICE, J. R. DUNNING jr., M. GOITEIN, K. HANSON, T. KIRK and R. WILSON: Phys. Rev. D, 4, 45 (1971).

[46] D. 5. DRICKEY and L. N. HAND: Phys. Rev. Lett., 9, 521 (1962). [47] B. DUDELZAK, G. SAUVAGE and P. LEHMAN: NUOVO Cimento, 28, 18 (1963). [48] CHR. BEGGER, E. GERSING, G. KNOP, B. LANGENBECK, K. RITH and F. SCHU-

MACHER: Phys. Zett., 28B, 276 (1968). [49] W. BARTEL, F.-•. BUSSER, W.-R. DIx, R. ~ELST, D. HARMS, H. KREHBIEL,

P. E. KUGLMAN, J. MCELROY and G. WEBER: Phys. Lett., 33 B, 245 (1970). [50] D. H. COWARD, H. DESTAEBLER, R. A. EARLY, J. LITT, A. ]V[INTEN, L. W. ~/[o,

W . K . H . PANOFSKY, ]~. E. TAYLOR, M. ]:~REIDENBACH, J. I. ]FRIEDMAN, H. W. KENDALL, P. N. KIRK, B. C. BARIStI, J. MAR and J. PINE: Phys. Rev. Lett., 20, 292 (1968).

[51] J. LITT, G. BUSCIIHORN, D. It . COWARD, H. DESTAEBLER, L. W. Mo, R. E. TAYLOR, B. C. BARISH, S. C. LOKEN, J. PINE, J. I. ]FRIEDMAN, G. C. HARTMAN and H. W. KENDALL: Phys. Lett., 31 B, 40 (1970).

[52] The data points from some old experiments are not shown in the Figures. However, most of the corresponding cross-section measurements have been used to improve the Rosenbluth plots of later experiments. The references of these experiments are: P. LEHMAN, ]R. TAYLOR and R. WILSON: Phys. Rev., 126, 1183 (1962); K. W. CHEN, A. A. CONE, 5. ]~. DUNNING jr., S. F. G. FRANK, N. IV. RAMSEY, J. K. WALKER and R. WILSON: Phys. Rev. Lett., l l , 561 (1963); B. DUDELZAK, A. ISAKOV, P. LEHMAN and R. TCH~OUTIAN: Proceedings o] the XII International Con]erence on High-Energy Physics, Vo]. 1 (Dubna, 1964), p. 916; W. ALBRECHT, H. J. BEHREND, 1 ~. W. ]~RASSE, W. FLAUGER and H. HULTSCHIG: Proceedings o/ the X I I I International Con- /erence on High-Energy Physics (Berkeley, 1966); E. E. CHAMBERS and R. HOFSTADTER: Phys. Rev., 103, 1454 (1956); Y. BUMILLER, M. CROISSAUX, E. DALLY and R. HOFSTADTER: Phys. Rev., 124, 1623 (1961); D. YOUNT and G. PINE: Phys. Rev., 128, 1842 (1962); D. N. OLSON, H. F. SCHOPI~ER and R. R. WILSON: Phys. Rev. Lett., 6, 286 (1961).

[53] M. GOITEIN, ]:{. J. BUDNITZ, L. CARROLL, 5. ]:{. CHEN, J. R. DUNNING jr., K. HANSON, D. C. I~RIE, C. MISTRETTA and R. WILSON: Phys. Rev. D, l, 2449 (1970) give results on electron scattering cross-section and comparison with the dipole fit. In terms of form factors, their data are included under ref. [45].

[54] See, for instance: A. L. LICHT ~nd A. PAGNAM]~NTA: Phys. Rev. D, 4, 2810 (1971) and references quoted there; T. NARITA: Progr. Theor. Phys., 42, 1336 (1969).

[55] E. FERMI and L. ~ARSHALL: Phys. Rev., 72, 1139 (1947). [56] V . E . KROHN and G. R. RINGO: Phys. Lett., 18, 297 (1965); Phys. Rev., 148,

1303 (1966). [57] E. MELKONIAN, B. M. RUSTAD and W. W. HAVENS: Phys. Rev., 114, 1571 (1959). [58] D. HUGHES, J. A. HARVEY, M. D. GOLDBERGER and M. J. STAFNE: Phys. Rev.,

90, 497 (1953); L. L. FOLDY: ]~ev. ~od. Phys., 30, 473 (1958). [59] T. HAMADA and J. D. JOHNSTON: N~cl. Phys., 34, 382 (1962). [60] I. MCGEE: Phys. Rev., 151, 772 (1966). [61] H. }~ESHBACH and E. LOMON: Rev. Mod. Phys., 36, 611 (1967). [62] F. GROSS: Phys. l%v., 142, 1025 (1966); 152, 1517 (E) (1966).

29~ B. BARTOLI, F. FELICETTI and v. SILVESTRINI

[63] S. GALSTER, H. KLEIN, J. MORITZ, K. H. SCttMIDT, D. WEGENER and J. BLECK- WENN: DESY preprint 71/7 etc.

[64] CII. BERGER, V. BURKHART, C-. KNOF, B. LANGENBECK and K. RITH: Physikalisches Ins t i tu t , Universit~t Bonn, preprint 1-075 (July 1969).

[65] D. J'. DRICKEY and L. N. HAND: Phys. Roy. Lett., 9, 521 (1962). [66] B. GROSSET~TE, D. DRICKEY and P. LEHMAN: Phys. Rev., 141, 1425 (1966). [67] D. BENAKSAS, D. DRICKEY and D. FR~REJACqUE: Phys. Rev., 148, 1327 (1966);

Phys. Rev. Lett., 13, 353 (1964). [68] C. D. BUCHANAN and M. R. YEARIAN: Phys. Bey. Lett., 15, 303 (1965). [69] 5. I. FRIEDMAN, H. W. KENDALL and P. A. GRAM: Phys. Rev., 120, 992 (1960). [70] A. 0. BARUT, D. CORRIGAN and H. KLEINERT: Phys. Rev. Lett., 20, 167 (1968). [71] R. BUDNITZ, 5. AFPEL, L. CARROLL, J. CHEN, 5. R. DUNNING jr., )/I. GOITEIN,

K. HANSON, D. [MRIE, C. )/[ISTRETTA, 5. K. WALKER and R. WILSON: Phys. Rev. Lett., 19, 809 (1967).

[72] K . M . HANSON, J. R. DUNNING jr., M. GOITEIN, T. KIRK, L. E. PRICE and R. WILSON: presented at the 1971 International Symposium on Electron and Photon Interactions (Ithaca, N.Y., 1971).

[73] W. BARTEL, T. W. BUSSER, W. R. DIX, R. FELST, D. HAR~8, H. KREHBIEL, P. E. KUItLMAN, J'. MCELROY, J. ~¢[AYER and G. WEBER: presented at the 1971 International Symposium on Electron and Photon Interactions (I thaca, N.Y., 1971).

[74] D. BRAESS and G. KRAMER: Zeits. Phys., 189, 242 {1966); D. BRA~SS, D. HAS- S]~LMANN and G. KRAMER: Zeits. Phys., 198, 527 (1967). Analysis of da ta from ref. [75].

[75] E . B . HUGHES, T. A. GRIFFY, N. R. YEARIAN and R. HOFSTADTER: Phys. Rev., 139, B 458 (1968).

[76] R. 5. BUDNITZ, 3". APPEL, L. CARROLL, 5. CHEN, J. R. DUNNING jr., lVI. GOITEIN, K. HANSON, D. [MRIE, C. MISTRETTA, J'. K. WALKER and R. WILSON: Phy8. Rev:, 173, 1357 (1968).

[77] J. R. DUNNING jr., K. W. CHEN, A. A. CONE, G. HARTWlG, N. F. RAMSE¥, J. K. WALKER and R. WILSON: Phys. Rev., 141, 1286 (1966).

[78] W. BARTEL, F. W. BOSSER, W. R. DIX, R. FELST, D. HARMS, H. KREHmEL, P. E. KUHLMAN, J. McELRoY, W. SCHMIDT, V. WALTHER and G. WEBER: Phys. Lett., 30 B, 285 (1969).

[79] C. W, AKERLOF, K. BERKEL~AN, G. ROUSE and ~ . TIGNER: Phys. t~ev., 135, B 810 (1964).

[80] E. AMALDI: Frascat i Internal Report , LNF-71/41 (1971) is an excellent review of the experiments and methods in elcctroproduction of pions.

[81] S. L. ADLER: Ann. o/ Phys., 50, 189 (1968). [82] N. ZAGVRY: Phys. Rev., 145, 1112 (1966). [83] B. DE TOLLIS and F. NICOL5: ~uovo Cimento, 48 A, 281 (1967). [84] F. A. BERENDS: Phys. Rev. D, 1, 2590 (1970). [85] R . C . E . DEVENISH and D. H. LY'rH: Universi ty of Lancaster preprint , April 1971,

to be published in Phys. Rev. (1972). [86] Cx. VoN CJEHLEN: Nuel. Phys., 9 B, 17 (1969); 20 B, 102 (1970); G. VON GEItLEN

and M. G. SCH~IDT: -~ucl. Phys., 20B, 173 (1970); G. VoN GEHLEN: Bonn Universi ty, P I 2-80 (1970).

[87] See, for instance: PH. SALIN: N'~ovo Cimento, 32, 521 (1964); ~ . C-OURDIN and PH. SALIN: 2(UOVO Cimento, 27, 193, 309 (1963); J. P. LOUBATON: Nuovo Cimento, 39, 59l (1965); P. KESSLER: Coll~gc de France, Report PAM No. 6505 (1965) ; 5. D. WALECKA: Phys. Rev., 162, 1462 (1967) ; J . D. BJORKEN

ELECTROMAGNETIC STRUCTUIgE OF THE HADRONS 299

and J. D. WALECKA: Ann. of Phys., 38, 35 (1966); P. L. PRITCHETT and J. D. WALECKA: Phys. Rev., 168, 1638 (1968).

[88] See, for instance: S. 1,'lrmN~ and G. IJURLAN : A)t?I. of Phys., 48, 322 (1968) ; A. M. CyLEESON, M. G. GUNDZ1K and J. G. KURIYAN: Phys. Rec., 173, 1708 (1968); G. FURLAN, N. PAVER and C. Vm~ZEGNASSI: NUOVO Cimento, 62 A, 519 (1969); 70A, 247 (1970); Y. N~t.~r~u and 5I. YOSIIIMURA: Phys. Rev. Lett., 24, 25 (1970).

[89] N. DOMBEY and B. J. READ: paI)er contributed to the 1971 Internal Symposium on Electron and Photon Interactions at High Energy (Ithaca, N.Y., 197I).

[90] C. DRIVER, K. tIEINLOTH, K. Ill)lINE, G. HOEMANN, P. KAROW, D. SCHI~IIDT, G. SPECHT and J. RATgaE: Phys. Lett., 35B, 77, 81 (1971).

[91] C. MISTRETTA, J. A. APPLE, R. J. BUDNITZ, L. CARROLL, J. CHEN, J. R. DUN- NIXG jr., 1g. GOITEIN, K. HANSON, D. C. IMRIE and II. V~'ILSON: Phys. Rev., 184, 1487 (1969).

[92] E. AMALDI, B. BORGIA, P. PISTILLI, M. BALLA, a . V. DI GIORGIO, n . GIAZOTTO, S. SERBASSI and G. STOI'rlNI: 2¥UOVO Cimento, 65A, 377 (1970).

[93] C. W. AKERLOF, W. W. ASH, K. BERKELMAN, A. C. LICtlTI,'NST~'IN, A. RAMA- NAUSKAS and R. H. SIEMANN: Phys. Rev., 163, 1482 (1967).

[94] A. SOFAIR, J. ALLISON, B. DICKINSON, E. EVANGELIDES, M. IBBOTSON, R. LAW- SON, R. S. 3IEABURN, H. E..~{ONTGOMERY, W. J. SIIUTTLEWORTH, A. B. CLEGG, l ~. FOSTER, Cx. ]]ITGIIES, P. KI:MMER and R. SIDDLE: Daresbury preprint DNPL/P 97 (December 1971).

[95] C. N. BROWN, C. R. CANIZARES, ~V. E. COOP]rR, A. ~[. EISN]~R, G. J. FELDMAN, C. A. LICHTENSTEIN, L. LifT, W. LOCRERETZ, V. B. MONTANA and F. M. PIPKIN: Phys. Rev. Lett., 26, 987, 991 (197l).

[96] R. -WILSON: Rapporteur talk in tit(" Proceedings o/ the 1971 International Sym- posium on Electron and Photon Interactions at High Energies (Ithaca, N.Y., 1971), p. 98.

[97] W.W. Astt, K. BERKELeiAN, C. A. LICHTENSTFIN, A. RAMANAUSKAS and R. tl. SIEM~tNN: Phys. Lett., 24B. 165 (1967).

[98] It. I IARARI and [[. J. LIPKIN: Phys. Rev., 140, B 1617 (1965); C. BECCIII and G. MORPURC,): Phys. Lett., 17, 352 (1965).

[99] M. GOURDIN and P. SALIN: ~UOVO Cimento, 27, 193 (1963); D. J. DI~ICKEY and R. F. MOZLE¥: Phys. P~ev. Lett., 8, 2.(tl (1962).

[100] W. BARTEL, B. DUDELZAK, H. KREHBIEL. ,}'..~[CELROY, U. ~'EYER-BERKHOUT, ~V. SCHMIDT. V. WALTER and G. WEBFR: Phys. Lett., 28B, 148 (1968).

[101] C. MISTRETTA, 1). [MRIE, J. ~k. AI'eEL, R. BVDNITZ, L. CARROLL, J. Cm.'n, ,)'. DUNNING jr., M. GoI'rEIN, K. HANSON. A. JATKE and R. WILSON: Phys. Rev. Lett., 20, 1070 (1968); D. I~rR~E, C. MISTRETTA and R. WILSON: Phys. Rev. Lett., 20, 1074 (1968). BREIDENBACIt, J. I. I~RIEDI~IAN, tI. ~V. KENDALL, E. D. BLOOM, D. H. CO- WARD, It. DESTAEBLER, J. DREES, L. W. 3[o and R. E. TAYLOR: Phys. l~ev. Lett., 23, 930, 935 (1969); SLAC-PUB 815 (to be published in Phys. Rev.); SLAC-PUB 796 (presented at the X V International Con]erence on High.Energy Physics, Kiev, September 1970); J. I. FRIEDMAN: invited talk presented at the Topical Seminar on l~lectromagnetie interactions, I C TP Trieste, 21-26 June 1971.

[103] D. E. ANDREWS, K. BERKFLMAN, D. Cv. CASSI.:L, D. L. HARTILL, 5. HARTMANN, R. KERCHNEIt, E. LAZARUS, R. M. LITTAIrER, R. L. LOV|,;LES8, R. ROIILI,'8, D. H. WIHTJ," and A. J. SADOFF: Phys. Rev. Lett., 26, 864 (1971). addition to the d%ta of ref. [102], only a few more data are available from a

[102] M.

[104] In

~00 B. BARTOLI~ ]~. EELICETTI and V. SILVESTRINI

DESY experiment: W. ALBRECHT, F. W. ]~RASSE, H. DORNER, W. FLAUGER, K. H. FRANK, J. GAYLER, H. HULTSCHIG, V. KORBEL, J. MAY and E. GANS- SAUGE: DESY Report 69/46, November 1969.

[105] R. FEYNMAN: Phys. Rev. Left., 23, 1415 (1969). [106] J. D. BJORKEN: Phys. Rev., 179, 1547 (1969); R. JACKIW, R. VAN ROYEN and

G. B. WEST: Phys. Rev. D, 2, 2473 (1970); R. A. BRANDT: Phys. Rev. D, l , 2808 (1970); H. LEUTWYLER and J. STERN: Nucl. Phys., 20 B, 77 (1970); B. L. IOFFE: Phys. Lett., 30 B, 123 (1969); Y. FRISHMAN: Phys. Bey. Lett., 25, 966 (1970).

[107] E. BLOOM and F. GILMAN: Phys. Rev. Lett., 25, 1140 (1970). [108] See, for instance: S. D. DRELL: rappor teur talk in the Proceedings o] the Amsterdam

International Con/erenee on Elementary Particles (Amsterdam, 1971), p. 307. See also G. P2EPARATA: Riv. NUOVO Cimento, 1, 423 (1971).

[109] L. LANDAU and E. LIFSHITZ: Sov. Phys., 6, 244 (1934); V. E. BALAKIN, V. M. BUDNEV and I. F. GINZBURG: J E T P Lett., 11, 388, 559 (1970); S. 5. BRODSKY, T. KINOSHITA and H. TERAZAWA: Phys. Rev. Lett., 25, 972 (1970); C.L.N.S.-152 (1971); V. N. BAIER and V. S. FADIN: 2urn. [~ksp. Teor. l~iz., 13, 293 (1971); Lett. Nuovo Cimento, l , 481 (1971); Novosibirsk prepr int (1971); N. ARTEAGA-RoMERO, A. JACCARINI, P. KESSLER and I. PARISI: PAM 71-Q2 (1971).

[110] V. E. BALAKIN, G. I. BUDKER, E. V'. PAKHTUSOVA, V. A. SIDOROV, A. •. SKRINSKY, G. M. TUMAIKIN and A. G. KHABAKHP&SHEV: Phys. Lett., 34 B, 661 (1971).

[111] C. BACCI, R. BALDINI-CELIO, G. CAPON, C. ~/[ENCUCCINI, G. P. ~V~URTAS, G. PENSO, A. R~AL]~, G. SALVlNI, hi. SI"INETTI and B. STELLA: Lett. Nuovo Cimento, 4, 709 (1972).

[112] G. PARISI: pr ivate communication. [113] G. BARBI]~LLINI: pr ivate communication. [114] D. C. CHENG: pr ivate communication. [115] V. SILVESTRINI: A review o] high energy work in quantum elcetrodynamics, invited

talk at the I I I Rencontre de Moriond sur les Interactions Electromagnetiqnes, 1968; L N F 68/17. Average of the da ta from the following experiments: J. K. DE PAGTER, J. I. FRIEDMAN, G. GLASS, R. C. CHASE, M. GETTNER, E. VON GOELER, R. WEINSTEIN and A. M. BOYARSKI: Phys. Rev. Lett., 16, 35 (1966); B. D. HYAMS, W. Koch-I, D. PELLET, D. POTTER, L. VON LINDERN, E. LORENTZ, G. LUTJENS, V. STIERLIN and P. WEILI-IAMMER: Phys. Lett., 24B, 634 (1967); A. WEHMAN, E. ENGELS jr., C. M. HOFFMAN, P. G. INNO- CENTI, R. WILSON, W. A. BLAMPIED, D. J. DI~ICKEY, L. N. HAND and G. D. STAIRS: Phys. Rev. Lett., 18, 929 (1967); P. L. ROTttWELL, A. BOYARSKI, R. C. CHASE, J. DE PAGTER, J. I. FRIEDMAN, M. GETTNE~, G. GLASS, E. VON GOELER and R. WEINST]~IN: International Symposium on Electron and Photon Interactions at High Energies (Stanford, 1967); J. G. ASItBURY, U. BECKER, W. K. BERTRAM, P. JOOS, M. ROHDE, A. J. S. SMITH, C. L. JORDAN and S. C. C. TING: Phys. Rev. Lett., 19, 869 (1967); S. S. HERTZBACI~, R. W. KRAEMER, L. MADANSKY, R. A. ZDANIS and R. STRAND: Phys. Rev., 1S5, 1461 (1967); 1~[. N. KHACHATURYAN, M. A. AZIMOV, A. M. BALDIN, A. S. BELOUSOV, I. V. CHUVILO, R. ~IRKOWS1KI, J. HLADKY, M. S. KHVA- STUNOV, J. ~ANCA, A. W. ~ATHYUSHIN, V. H. MATI~YUSHIN, G. A. OSOSKOV, L. N. SHTARKOV and L. I. ZHU~AVLEVA: Phys. Lett., 24B, 349 (1967); V. L. AUSLANDER, G. I. BUDKER, N. PESTOV, V. A. SIDOROV, A. N. SKRINSKY and A. G. KHABAKH~ASHEV; Phys. Lett., 25B, 433 (1967); J. E. AUGUSTIN,


J. C. BIZOT, J. BUON, J'. HAISSINSKI, ]). LALANNE, P. C. MArnE, 5. PlgREZ-Y- J'ORBA, F. RUMPF, E. SILVA and S. TAVERNIER: 0rsay Report LAL-1t81 (1967); Phys. Rev. Lett., 20, 126 (1968).

[116] V. E. BALAKIN, G. [. BUDKER, L. M. KURDADZE, A. P. ONUCHIN, E. V. PAKHTUSOVA, S. I. SER~EDNYAKOV, V. A. SIDOROV, A. N. SKRINSKY and A. G. KHABAKHPASHEV: Novosibirsk preprint 53-71.

[117] B. BORGIA, ~. CERADINI, M. CONVERSI, L. PAOLUZI, R. SANTONICO, ~-. :BAR- BIELLINI, )/[. GRILLI, P. SPILLANTINI, R. VISENTIN and F. GRIANTI: Lett. Nuovo Cimento, 3, 115 (1972).

[118] V. ALLES-BORELLI, M. BERN-ARDINI, D. BOLLINI, P. L. BRUNINI, ]~. ~IORENTINO, W. ~¢[ASSAM, L. MONARI, F. PALMONARI and A. ZICHICHI: Zett. Nnovo Cimento, 2, 376 (1971).

[ l l9] Results of the 0rsay group: 5. E. AUGUSTIN, A. COURAU, B. DUDELZAK, 1~. FULDA, G. GROSDIDIER, ,1". HAISSINSKI, J'. I~. MASNOU, R. RISKALLA, F. RUMPF and E. SILVA: presented at the 1971 International Symposium on Electron and Photon Interactions at High Energies (Ithaca, N.Y., 1971).

[120] G. DI GIUGNO, J. W. HU~IPHREY, E. SASSI, (~. TROISE, U. TR0YA, S. VITALE and M. CASTELLANO: Lett. :Vuovo Cimento, 2, 873 (1971). Latest results presented at the In/ormaI Meeting on e+e - Interactions, May 2-3, I~rascati, 1972.

[121] N. CABIBBO and R. GATT0: Phys. Rec., 124, 1577 (1961). [122] M. CO~VERSI, T. MASSA~I, TH. MULLER ~nd A. ZICHICHI: Nuovo Cimento, 40 A,

690 (1965). [123] D. L. ~tARTILL, B. C. BAt~ISH, D. G. FONG, R. GOMEZ, d. PINE and A. V. TOL-

LESTRUP: Phys. Rev., 184, 1485 (1969). [124] V. L. AUSLANDER, G. I. BUDKER, JU. N. PESTOV, V. A. SIDOROV, A. N. SKRINSKY

and A. G. KIIABAKHPASHEV: Phys. Lett., 25 B, 433 (1967); V. L. AUSLANDER, G. I. BUDKER, E. V. PAKHTVSOVA, J'U. N. PESTOV, V. A. SIDOROV, A. N. SKRINSKY and A. G. KHABAKHPASItEV: S0V. Journ. Nucl. Phys., 9, 144 (1969).

[125] J. E. AUGUSTIN, J~. C. BIZOT, ~[. J~UON, J. HAISSINSKI, I). LALANNE, P. MARIN, ~. PERJ~Z-Y-JORBA, F. RUMPF, ]~. SILVA and S. TAVERNIER: Phys. Rev. Lett., 20, 126 (1968); J. E. AUGuS'rlN, J. C. BIZOT, J. Buoy, J . HAISSINSKI, D. LA- ],ANNE, P. ~]'ARIN, H. XGL~YEN N(~OC, J. PEREZ-Y-JORBA, F. RUMPF, E. SILVA and S. TAVERNIER: Phys. Lett., 28B, 508 (1969); D. BENAI(SAS, G. COSME, B. JEAN-MARIE, S. Jt~LLIAN, F. LAPLANCHE, J. LnVRASrqOIS, A. D. LIBERlUA~, G. PARROUR, d. P. REPELLIN, G. SAUVAGE and G. SZKLARZ: presented at the International Symposium on Electron and Photo~ Interactions at High Energies (Ithaca, N.Y., 1971).

[126] See, for instance, N. KROLL, T. D. LEE and B. ZtrMINO: Phys. Rev., 157, 1376 (1967).

[127] d. C. BIZOT, J. BUON, Y. CHATELUS, J. ,lEAN JEAN, D. LALANNE, H. NGUYEN NGOC, J. PEREZ-Y-,]'ORBA, P. PETROFF, F. RICHARD, P. RUMPF and D. TREILLE: Phys. Lett., 32 B, 416 (1970).

[128] V. E. BALAKIN, G. I. BUDKER, E. V. PAKHTUSOVA, V. A. SIDOROV, A. N. SKRINSKY, G. )/[. TUMAIKIN and A. G. KHABAKIIPASHEV: Phys. Lett., 34 B, 328 (1971).

[129] V. A. SIDOROV: Proceedings o] the International Symposium on Electron and Photon Interactions at High Energies (Ithaca, N.Y., 1971), p. 66; and private communication.

[130] J. E. AUGUSTIN, D. BENAKSAS, J. BUON, F. FULDA, V. GRACCO, J. HAISSINSKI, D. LALANNE, F. LAI"LA~CHE, J. LEFRANCOIS, P. LEHMANN, P. C. MArnE, 5. PEREZ-Y-JORBA, F. Ru~'~" and E. SILVA: Lett. Nnovo Cimento, 2, 214 (1969).

302 B. BARTOLI, F. I~ELICETTI and v. SILVESTRINI

[131] B. ]~A.RTOLI, :B. COLUZZI, F. FELICETTI, V. SILYESTRINI, C~. GOGGI, D. SCANNIC- cmo, G. MARINI, F. ~ASSA and F. VANO~I: International Con]erence on High- Energy Physics (Kiev, 1970); Nnovo Cimento, 70 A, 615 (1970).

[132] G. BARBI]~LLINI, IV[. CONV~RSI, IVY. GRILLI, A. ~ULACHI~, 1~. I~IGRO, L. I:)AOLUZI, P. SPILLANTINI, ]~. VISENTIN and G. T. ZORN: Frascati Report LNF 70/38 (1970).

[133] B. BARTOLI, F. ~ELIC]ETTI, tt. 0GREN, Y. SILVESTRINI, G. MARINI, A. NIGRO, N. SPIN~LLI and F. VANOLI: Phys. Lett., 36 B, 598 (1971).

[134] C. BACCI, ]~. BALDINI CELIO, G. CAPON, C. MENCUCCINI, G. P. MURTAS, G. PENSO, A. REALE, G. SALVINI, M. SPINETTI and B. STELLA: Phys. Lett., 38 B, 551 (1972).

[135] V. ALLEs-BORxLLI, M. B]~RN~DINI, D. BOLLINI, P. L. BRUNINI, E. FIORENTINO, T. MASSAM, L. MONARI, F. PAL•ONARI and A. ZlCmCm: Proceedings o/ the EPS Con/erence on Meson Resonances and Related Electromagnetic Phenomena (Bologna, 1971).

[136] B. BORGIA, M. CONVERSI, M. GRILLI, E. IA~OCCI, M. NIGRO, L. PAOLVZI, P. SPIL- LANTINI, L. TRASATTI, V. VALENTE, ]~. VISENTIN and G. T. ZORN : International Symposium on Electron and Photon Interactions at High Energies (Ithaca, N.Y., 1971); L:NF 71/62.

[137] B. BAR'rOLI, F. ~]~LICETTI, G. MA~INI, A. NIG~O, H. 0GREN, V. SILV~STRINI and F. VANOLI: LNF 71/91, to be published in Phys. Rev.

[138] G. B~BARINO, ~. CERADINI, ~V[. CONVERSI, M. GRILLI, ]~. IAROCCI, M. NIGRO, L. PAOLUZI, R. SANTONICO, P. SPILLANTINI, L. TRASATTI, V. VALENTE, R. VISENTIN and G. T. ZORN: LNF 71/96; Lett. £Vnovo Cimento, 3, 689 (1972); same authors, LNF 72/43, to be published.

[139] Results of the ~-7: group (same authors as ref. [132]) presented by G. B~_RBInL- LINI at the In]ormal .Fraseati Meeting, September 1970.

[140] J. LAYSSAC and 1~. M. :R]~NARD: Lett. lYnovo Cimento, ], 197 (1971); IYiontpellier prelorint P.M. 71/2 (1971).

[141] A. BlCA~ON and M. T. GRECO: Lett. -STucco Cimcnto, l, 739 (1971); 3, 693 (1972). [142] V. ALLES-Bo~ELLI, M. BERNARDINI, D. BOLLINI, P. L. BRUNINI, ]~. FIORENTINO,

T. MASSAM, L. MONARI, ~. PALMONARI and A. ZICHICHI: presented at the In/ormal .Frascati Meeting, May 1972.

[143] Results of the ~-r: group (same authors as ref. [132]) presented at the In]ormal .Frascati Meeting, May 1972.

[144] J'. L]~F~ANpOIS: Proceedings o] the International Symposium on Electron and Photon Interactions at High Energies (Ithaca, N.Y., 1971), p. 59.

[145] V. A. SIDOROV: presented at the In]ormal .Frascati Meeting, May 1972. [146] V. SILVESTRINI: Rapporteur talk at the X V 1 International Con]erence on High-

Rnergy Physics at Batavia, 1972.

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Electromagnetic structure of the hadrons