Electromagnetic form factors of the nucleon in a quark model

IL NUOVO CIMENTO VoL 18 A, N. 2 21 Novembre 1973

Electromagnetic Form Factors of the Nucleon in a Quark Model.

L. MICTY

Institute ]or Atomic Physics - Bucharest

(ricevuto il 12 Diccmbre 1972; manoseri t to revisionato riccvuto i l 10 Aprilc 1973)

S u m m a r y . - - The electromagnetic form factors of the nucleon are evaluated in the inf ini te-momentum frame usiI~g a relat ivist ic generaliza- t ion of the s ta t ic SU6-scheme. The complicat ions due to the concrete, t r ea tment of the spin are avoided by assuming ad hoc cut-off ftmctions. One obtains the scaling proper ty of the Sachs form factors GE(q2)/GM(q 2) ~ ct as q o__> oz. The asymptot ic behaviour of the form factors strongly reflects the form of the cut-off function. SUG-like results are also obtained in the s ta t ic l imi t with free quarks.

l . - I n t r o d u c t i o n .

R e l a t e d to a I l a t u r a l i n t e r p r e t a t i o n of some r e ( e n t d a t a on i ne l a s t i c e l ec t ron

s c a t t e r i n g a n i n t u i t i v e a n d p o w e r f u l m o d e l has b e e n d e v e l o p e d in t h e last, yea r s .

T h e e s sence of t h e m o d e l is t h a t in some c ond i t i ons h a d r o n s b e h a v e an if b u i l t

up f rom p o i n t l i k e c o n s t i t u e n t s cal le( l <~ p a r t o n s ~) b y FEYNMAN, P a r t o n s a r e

s o m e t i m e s i d e n t i f i e d w i t h t h e v i r t u a l mesons a n d n u c l e o n - a n t i n u c l e o n pa i r s

a p p e a r i n g in a f i e l d - p e r t u r b a t i o n t h e o r y , as was p r o p o s e d b y DlZEII., LEVY

a n d Y.~N (1), b u t s o m e t i m e s t h e y a r e i d e n t i f i e d w i t h t h e h y p o t h e t i c a l ob j e c t s

c a l l ed q u a r k s as in t h e i n i t i a l par t .on m o d e l of :BJoRKEN a n d PASCttOS (2).

I n th i s p a p e r we e v a l u a t e t h e e l a s t i c e l e c t r o m a g n e t i c f o rm fac to r s of t h e

n u c l e o n b y w o r k i n g in t, he i n f i n i t e - m o m e n t u m f r a m e a n d us ing t h e h y p o t h e s i s

t h a t nuc leons a r e m a d e of t h r e e q u a r k s t i g h t l y b o u n d t o g e t h e r . ~[he inf in i te-

m o m e n t u m f r a m e is p a r t i c u l a r l y u s e f u l for s t u d y i n g t h e ha ( l ron s t r u c t u r e .

(1) S . D . DR~L~, D. J. LEvY and T. M. YAN: Phys. Rev. , 187, 2159 (1969); Phys. Rev. D, 1, 1035 (1970). (2) J. D. BJORKEN and E. A. PASCHOS: Phys. Rev., 185, 1975 (1969).

327

3 2 8 L. l~iICU

The main advantages of this f rame are re la ted to the fact t ha t the states are

long lived because of relat ivist ic t ime dilatation. Hencefor th in ternal particles are almost real and the violation of energy conservat ion can be ignored. F o l lowing the main hypothesis of the model the photon <~ sees ~> each quark sepa- rately, quarks behaving as free, pointl ike particles.

The calculation is done using Weinberg 's rules (3) for dynamics at infinite momentum.

14 + .= ~ /

, ka P~

c~) b) c)

a) e) f)

Fig. 1. - The time-or4ere4 diagrams contributing to the electromagnetic form factors of the nucleon in the second order of J~qqq coupling.

Our task is to evaluate the contr ibut ion of the tr iangle diagrams in Fig. 1

describing the (( dissociation )> of the initial nucleon into th ree quarks, the point- like quark-photon interact ion and the <~ reconstruct ion ~> of the final nucleon. The first and th i rd vert ices are described by the ~ q q q coupling. We ask for t hem kinemat ical propert ies equivalent to the SU6-symmetry scheme. As far as no consistent theory of strong in teract ion is available, dynamics cannot be explici t ly t aken into account. Dynamics is included by cut t ing the re la t ive momenta inside the nucleon as is suggested by the Bethe-Salpeter equat ion

in the ladder approximat ion. A cut-off of the type 1/k~ gives the desired q-4

asymptot ic behaviour of the electromagnetic form factors. The effect of the cut-

off acts also on the longitudinal momenta requir ing positive longitudinal mo-

menta for all the consti tuents . Complications due to the spin are consequently

avoided. One of the most in teres t ing results of the model is G~(q2)/G~(q ~)-+ct, as

q2_+ c~, which comes from the consistent t r e a t me n t of the spin.

Due to the explicit form of the 2Tqqq coupling, SUe-like predictions are obtained from the final results a f te r replacing each variable under the integral b y i t s mean value. One has for instance (P) (n) _ 3 G~ (O)/G~ (0)-- - - ~ and G~)(0)/-E~P)(0) = 3

(a) S. WEINBERG: Phys. l~ev., 150, 1313 (1966).

E L E C T R O M A G N E T I C FOI~M F A C T O R S OF T H E N U C L E O N I N A Q U A R K M O D E L 329

as compared with the exper imenta l data - -1 .46 and 2.79 respect ively. Note t ha t the resul t P~)(0) = 0 is an ident i ty , not a mean result .

I t is in teres t ing to observe tha t the t r iangle diagram supposed to be the key point of this model is the relat ivist ic version of the older quark-recombinat ion

model (~). The presen t approach appears to be the field aspect of the model for which the s y m m e t r y aspect was discussed recent ly by R o s ~ n (5).

The second Section of the paper deals with the 2Vqqq coupling. The con-

t r ibu t ion of the t r iangle diagram is evaluated and discussed in the th i rd Sec- tion. The last Section is devoted to the discussion of the results.

2. - The AVqqq coupl ing .

The first problem to solve is to find a concrete A~qqq coupling. For lack of a consistent dynamical theory we separate the problem into two parts : the kinemat ical aspect and the pure dynamical aspect.

Taking now the SU6 symmet ry as a basis for our kinematics, we construct a suitable A~qqq coupling.

Consequently quarks are spin-½ objects of positive intrinsic par i ty , described by 4-component wave functioIts obeying Dirac equation. According to the SU6 scheme the coupling of the quark wave fuuctions to the nucleon nmst be total ly symmetr ic at the pe rmuta t ion of spin-isospin indices. Quark wave

functions must heuce sat isfy commuta t ion relations ra the r than anticom- muta t ion relations. This is the famous SU 6 dilemma which was not ye t solved al though various solutions like parastat is t ics or (~ coloured ~) quarks were proposed. We shall not deal fu r the r with solutions to this di lemma and will con-

fine ourselves to writing a completely symmetr icM coupling of quark wave functions.

Fu r th e r on we shall avoid de~ ivat ive couplings because in the noD_relativistic l imit they are customari ly re la ted to orbi tal excitat ions which are not supposed to exist for such ground states like nucleons in the classical SU6 scheme.

The most general nonder iva t ive ~Wqqq coupling satisfying Lorentz invariance and par i ty conservat ion can be wr i t t en in the form

(1) ~x~ (x ) = a~(~v(x)7.7~ ®'c~lx)~(x)Cf' ® ~(c~(x) ÷

q- a3~o(x)I ® I~(x)~v~'(x) C7 5 ® 7:2 yJ(x) q-

q- a4~v~v(x)Y~,Q I~(x)~7(x)C7 t' 75Q "~F(x) q-

q- aSf~o(x)~% ® I~f(x)~'(x)C ® ~ ( x ) + h.c.,

(4) L. Micu: Nucl. Phys., 10 B, 521 (1969). (5) J . L . ROSNER: University of Minnesota preprint (May 1972).

22 - I I Nuovo Cimento A .

3 3 0 L. MICU

where ~vx and W are nucleon and quark wave functions, ~1,2,3 are the 2 × 2 Pauli matrices for isospin and C is the charge conjugation matrix with the following properties:

Cy I, C-1 = - - y~ , C -1 = C z = - - C•

The isospin couplings have been established from symmetry requirements. Accordingly matrices like C, C7~y5, Cy 6 giving rise to antisymmetrical spin coupling must be associated with the antisymmetric coupling of the isospin of quarks• Similarly symmetric couplings of quark spins like those given by Cy~ and Can, require symmetrical isospin coupling• This is the first step in providing the A~qqq coupling with the desired symmetry property.

The next step is to symmetrize the coupling with respect to the third quark. This can easily be done by using the Fierz transformation and requiring ~x~Q~ to conserve its form.

One has therefore to replace everywhere in eq. (1) ~p(x)yF(x)C ®T2 by

16 ]

(2) 1 ~ ~ FR ®~i[Tr (F~FR)]-l~7(x)CF~®~2ri~f(x) ' R-1 ~-0

where/'~ are the 16 independent y-matrices and To = I. The ~('qqq coupling will have the form

. , 1 1 ) (3) ~jvQq~(x) = al -- ~ a~ ÷ g aa -- ~ 14 -- ~ 15 •

( 3 9 1 1 1 ) • vS~v(x)y~y5 ® ~yJ(x) yf(x) Cy' @ T2~yJ(x) + -- ~ al ÷ ~ a2 + ~ aa - - ~ a, -- ~ a5 •

• f ox (x )y . y~ ® Z~(x) ~'(x) o r " ® ~. ~(x) -

~ (aa + as)[Cfov(x)a,,y5 ® 'v~(x) ~fl~'(x) Ca "~ ® T2"vyJ(x) ÷

( 3 9 1 1 1 ) + Cfx(x ) at,~y 5 ® I~f(x) yY(x) Ca ~ ® T~ W(x)] ÷ ~ al -- ~ a2 ÷ ~ a3 -- ~ a~ + ~ a5 •

( 1 3 1 1 1 ) • ( f x ( x ) I @ I~f(x)~p~(x) Cy5 (~ v2~(x) ÷ -- ~ al ÷ ~ a2 ÷ ~ a3 -- ~a4 ÷ -~ a5 •

( 3 9 1 1 1 ) • ~ f x ( x ) I ®'c~(x)v ,~(x) Cy~ ® ~ ( x ) + - ~ al - ~ a. - ~ a~ - ~ a~ + ~ a~

( 1 3 1 1 1 ) • ~x(x)y, ® ;~(x) ~(x) cy"y~ ® ~. ~(x) + ~ a~ + ~ a~ - ~ a~ - ~ a, + ~,~ •

- , . ~ ( 3 a 9 1 1 1 )

( 1 3 1 1 1 ) • v~..v(x)y a ® I~(x)~fZ(x) C ® %F(x) ÷ ~ al ÷ -~ a2 ÷ ~ aa ÷ -~ a, ÷ -~ a~ •

• ~x(x)~ ® x~(x) ~(x) C ® ~ ( x ) ÷ b.c.

E L E C T R O M A G N E T I C FOI~M FACTORS OF THE N U C L E O N I N A Q U A R K M O D E L 331

The comple te s y m m e t r y iu spin-isospin indices of quarks in the 5Tqqq

coupling requires the iden t i ty of the expressions (1) and (3).

One obtains in this way a s y s t em of t en homogeneous equat ions for the con-

s tan ts a~, a2, aa, a~, a~. Only four of t h e m arc independen t and allow us to

express the five coupling constants by means of a single one. The comple te

symmet r i ca l 5 ' q q q coupling looks then as follows:

(~) Y f A~ (x) = A [ ~ ( x ) 7 , Y~ Q IW(x)y~(x)CTF' Q T2"cYJ(x) ~-

~- 2(~o(X)I QIy~(x)Y~r(x)C75 (D T2~(x) - - 2(~ov(x)y5 (~ IV(x)YJ~(x)C ~) T2~P(x) - -

- - @ ~ ( x ) ~ ® I ~ o ( x ) ~ ( x ) C 7 ~ 7~ ® ~ ( x ) ÷ b.c .] .

This resul t is equiva len t to the old re la t iv is t ic aproach of the S U e scheme

of S A I ~ A and W~LI (6). I t m u s t be emphas ized t ha t this coupling has only

k inemat ica l significance and says noth ing abou t the dynamica l p roper t ies of

the qqq bound-s ta t e function. We needed such concrete coupling in order to

t r e a t spin ia a cons is tent way and also in order to get plausible results , l ike

for instance, zero charge for neutron. On the o ther hand this model can be

considered as the r ight re la t iv is t ic general izat ion of the old stat ic SU6 scheme.

As s t a t ed in the In t roduc t ion , dynamics is included in our model by assuming

definite p roper t i es of the qqq bound-s ta te wave funct ion in m o m e n t u m space.

I t is knowlt (7) t h a t the solution of the Be the-Sa lpe te r equat ion in the ladder

app rox ima t ion for two scalar par t ic les bounded by scalar-meson exchange

behaves like 1/k~.l as ]~,~-~ c~, k 2 be ing the re la t ive m o m e n t u m . Similar tel

resul ts hold for sca la r -par t i c le - sp in- l -par t i c le bound states. Considering the

cut-off oll the re la t ive m o m e n t a of the in te rna l part icles as the character is t ic

f ea tu re of ~ bound s ta te , we suppose t h a t in a many-par t i c le bound s ta te a sim-

ilar cut-off exists for each re la t ive m o m e n t u m .

3. - The triangle diagram.

To evalua te the cont r ibu t ion of the t r iangle d iagram we choose the y-matr ices

as follows:

70 = _ , 7i = -- a~ , 7~ = , %v = 7~7~ - - g~,,,

where ~i are 2 X2 Paul i spin matr ices .

(6) B. SAKITA 3,114 K. C. WALI: Phys. Rev. , 139, 1355 (1965). (7) S . D . DR~LL and T. D. LEE: SLAC-l)UB-997, 1971 (to ~ppe~r i~ Phys. Rev.); D. AM~TI, L. CAz~v.SCHI and R. JENGO: Nuovo Cimento, 58 A, 783 (1968).

332 L. racy

The positive-energy solutions of the Dirac equation are

" " \ E + M ;6

where P~ E are the momentum and energy of the state and Z~ are two-component Pauli spinors, l~ueleon isospinors are denoted by ~. Throughout the paper (E, P) with E = (p2~_ M2)½ and (E', P ' ) will design the initial and final 4-mo- men tum of the nucleon. Momentum _P is directed along the thi rd axis of the co-ordinate system. In te rna l particle 4-momenta ~re denoted by k~=(E~, k, . , %P) where E,----(v~P2+ k~.q- m2) ½. The photon 4-momentum q =(q0, q, 0) is supposed to have zero longitudinal momentum. The kinematical con- ditions of the calculation are

P -+ c~, ]qt-+ oo, qo=q2/2M.P-+O.

According to Weinberg's rules for dynamics at infinite momentum one has to consider the six different t ime-ordered triangle diagrams in Fig. I and to add their contr ibution (~). I t will be pointed out fur ther tha t due to the cova- r iant eut-offs of the relat ive momenta the only nonvanishing contribution in the inf ini te-momentum l imit comes from the diagram 1 a).

Following WEI~]3ERG (~) the contr ibution of the ordered diagrams to the mat r ix e lement of a vector current is up to a constant

( 5 ) 2E1 2E2 2E3 E~ d(3)(k~ + k2 -~ k3-- P)"

• ~ * g(P ' , s ' )F . ( ) , .k~" '+ m)~,.(),.ki')+ m)FR, u(P, s) anian, I, ' 1 . 1 , . o

• Tr [P" ' (~ . k(~ ') + m)F"@, k(3 ') + m)]~0'* T~jTz,~dH,}"

.g(k[') 7-( ' )k( 'h- ,k ' , )k~') k(.~r(E ~ ( ' ) ~ ' " ) ~ ~ l ~1 ~ i n t / ~ ~ ~ i n t ] J

where i denotes the six diagrams in Fig. :[, E ('), E '(') are the energies of the initial and final states respectively, E (') E'(~) are the energies of the int ? - - i n t

in termedia te states, ..~k(')=(--Ej, ks) if the line j is t ime reversed in the diagram i (the momentum of the corresponding antiparticle is k j - (Ej, - -kj)) and k~)-- - k~ if it is not reversed. Indices (RI), (R'I ' ) denote the foul" terms in

I the A~qqq coupling, Ea, /'~, are the corresponding ?-matrices and k 1 -- = ([(kl_ k q)2[_ m2]½ kl±_ k q, •lp)" The matrices ~ with ~o : I describe the isospin coupling and ~j is ~a-k ½ vo for the ease of the electromagnetic current• The function g[al'"(~),,.2/~(~), k~'))3 is the cut-off function of the relative momenta.

ELECTROMAGNETIC FOI~M FACTOttS OF TI IE NUCLEON IN A QUAI~K IC[ODEL ~ 3 ~

:Note thag due to the symmet ry of the Affqqq coupling the wave functions of the two quarks nonin terac t ing with the photon can always avoid to couple to the nucleon wave funct ion by means of a y-matr ix . The factorizat ion of the numera to r in the expression (5) follows consequently. One sees easily t h a t in eq. (5) the in ter ference between differeng coupling te rms of ~/xa~ is generally

forbidden. The single except iou are the te rms mixing _F R = Cy 5 and Fn, = Cy,. The sum over R and R' reduces then to six germs only.

In ghe inf in i te -momentum l imi t the ma t r i x e lement of the good (~) compo- heats # ----- 0, 3 of a vector cur ren t be tween nucleon states has the following form:

(6) ~(p,, s,)(F,/~(q~)~, + 1 '" ° '\ P ~-,;2"

P ( q ) - e_,~ > "~ Z~*" :~ 2 F(2t)(q~) 2 M ] Z , ,

where I is the isospin index of the cur rent . In order to compute the two scalar form factors/~1 and F2 one has to com-

pare the expressions (5) and (6) a f te r per forming in the first one the infinite-

m om en tum limit . Of course only te rms behaving like P are retained. After a s t ra ightforward calculagion the numera to r in eq. (5) with J , the

electromagnetic cur ren t can be pu t in the form

where ~"), ~(~), 5 z(i) denote the following expressions:

( 8 a ) o][co = 3 2 _ , [ ~ i ) . ( i ) ~ _ 2 m a p ~ i) 4- ~ ( ~ ) - ' ~ ~.(i) ~ ~ (i) (~) .

i 1 (i) y~ --1 (i).x }

(8b) ~co = 64m~(P', ~'~ 1 /-,(t) aafla~(t) A- mK(i) .~ a A - G L ")~.(~)~m°

5~") 64~(P', ! O 3 (i) (i) (i)oca a 'c >t(i)o¢l~(i)fl~,(t)Ya,~, ,w xq ~[p (8c)

where p(O = ~(t) ~ ]Cv(O 0(0 = ~,;(i) k(.0 (o = ~.{i) ~ ~o:)

p. '~ l / J ! l~t ~ ~/* "1/~ - - l / l ~ l # '~21~ ~ '"3/*

(d) __~,(t) kr(i)~_ Eqi) l~(t)~_ 2 t K t t v - "'llt iv | "~llt " ' 1 , - g.~(m - - k l . k l ) , 1~(i) ,(t) (i) ~( t )k( i )

-~ ~ g , ~ g ~ g ~ ) - - (~--+a) ,

and s~e, is the complete an t i symmet r ica l tensor with s012a=l.

3 3 4 L. MICU

Using the well-known paramet r iza t ion of the momenta , one has in the inf in i te -momentum l imi t

(9) d3kl d3~2 d~k3 1

~-7 5(3)(kl -~ k2 ~- k3 - - P ) > E 1 E 2 .E a E i ~-->~

~-~dek~±d2k~±d~ks±~(2)(kl± -~ k~± ~- ks i ) d T ] ~ 2 ~ d~3 - ~ ~(?]l ~- ~2 ~- ~3 - - 1) p.--->co ~

~ 2 ~ 3

We can now discuss the cont r ibut ion of the six diagrams in Fig. I to the form factor and the influence of the cut-off function. Le t us begin with the denom-

inators of the in tegrand in eq. (5). In the inf in i te -momentum frame one has

(~0)

where e~=--i if the line j is t i m e reversed in the diagram i and e( /~=l if it is not . Hence the denominator E (~ E ~) goes like P if a t least one par t ic le

- - in t

in the in t e rmed ia t e s ta te goes backward in t i m e ( s j< 0) or in space (~b < 0) and like P-x otherwise.

On the other hand i t follows f rom eqs. (8) t ha t the behaviour of the numera to r can also be increased up to p8 or p5 when some of the in te rmedia te

part icles go backward in space or in t ime. Hence the contr ibut ion of the diagrams b)-]) in Fig. 1 cannot be a priori excluded in the _P-> co l imit .

Moreover integrals over ~]~ have to be ex tended over the whole plane ~ -~ ~ - ~3 = 1, i.e. in te rna l part icles are no longer constrained to move forward. These unpleasant complications are due to the concrete t r e a t m e n t of the spin (~).

Assuming the following invar ian t form for the cut-off function:

(11) _,~.(,) ~(~) k(i)) (i) ]c,(~)/~(]~(i) ~(t)/~(]c(i) /c(t)/21-1 Y(~i''~2 ' 3 ~[(ki ----2 , ,'-~ ----a, ,'~3 --'-i / J ,

we have in the infinite-!momentum l imit

(12) t J l j

I t is easy to see now tha t , due to the ex t ra powers of P in the denominator

induced by the cut-off funct ion (12), the cont r ibut ion of the diagrams b)-]) in ]~ig. 1 vanishes in the / ) - -> co l imit . Acgng on the contr ibut ion of the diagram 1 a), the cut-off funct ion requires ~ > / ~ / / ) > 0 , where # <<m. As the coefficient o f / ~ in eq. (12) vanishes identical ly for diagram 1 a) with ~ inside

! I

the t r iangle ~ i > 0, the funct ion g(kl, k2, k3) g(k~, k2, k3) acts there as a cut-off for the t ransverse momenta thus giving rise to the q-4-behaviour of the electro-

magnet ic fo rm factors in the q2_~ c~ limit. Of course many other forms of cut-off functions ma y be proposed. How-

ELECTROMAGNETIC FOR:M FACTORS OF THE NUCLEON IN A. QUARK ~IODEL 3 ~

ever it s e e m s to b e p a r t i c u l a r l y use fu l to cons ider i n v a r i a n t func t ions of r e l a t i ve

m o m e n t a b e c a u s e t h e y can se rve to e l i m i n a t e t he s p i n - g e n e r a t e d com-

p l i ca t ions too. I t is i n t e r e s t i n g to n o t e t h a t t he usua l a s s u m p t i o n t h a t p a r t o n s (quarks

and a n t i q u a r k s ) a lways m o v e f o r w a r d in t h e i n f i n i t e - m o m e n t u m f r a m e of t h e

nucleon leads to t h e neglec~ of ~he c o n t r i b u t i o n of t h e d i a g r a m s b)-]). There -

fo re t h e i n v a r i a n t cut-off f unc t io~ m a y be r e p l a c e d b y a cut -off of t he t r ans -

v e r s e m o m e n t a a n d t h e a s s u m p t i o n ~ > 0 for s~> 0 a n d ~ < 0 for e~< 0.

I f we p e r f o r m t h e 19 _~ c~ l i m i t in eqs. (8) for t h e d i a g r a m :1 a) a n d com-

p a r e eqs. (6) a n d (7)~ i t follows t h a t

(13) (I) s ~ , | S F~.~(q ) . . . . 2 d kl±d°L,±d~ka±6 ( ) (k~ ~- k~± + k3±)

~1 ~2 ~3 J

• [2ra s + (k~_ - - k~_) s] [2m s + (ks± - - k~±)°-] z-

• [2m" ÷ ( k l ± " q- -k s±)~][2m2-? (k~±+ q - - ka±) ~] ,

w h e r e

(14a) ~(x 1~ = { m(kl . ks -]- k~ . k3-- ks" ks) -~ 2m ~ q-

1 } mSq s

(14b) ~o~ 1 1 1 m(k~ "ks q-ks " k3 q- k~ " k3) + 6m 3 q- ~ [ m~(k~ . P q- ks . P q- k~ . P) -I-

÷ 2(k~.k~)(kl .P) + 2(k~ .k~)(ks .P) ÷ 2(k~'ks)(k~ "P)]} ~h--

qS 12 M [m2(1 ~ ~71) - - 4~]1(k~.- ks . ) - - 2~/s (kl±" ks.) - - 2~]s(k~±. k~j.)],

(14e) ¢~r(1) v 2 : ( 1 - - r h ) [m(2mS q - kl "ks ~- kl "k3) q- 2~hM(m s -4- ks "ks) -]-

1 -}-~h(k~ .P)-}- w(k2 -P ) - - (1 --~7~)(ka-P)]-}- (1 q- w ) m ( k s , k3-4- ~ h m M ) + 5~ (k~z" kaz) ,

1{ (14a) ~~°' = 5 2m~(1-3~1) + ~ r [ 2 ( 1 - ~ ) ( k s .k~ + m s ) - ms(1 + ~ ) -

--2~s(~i" k s ) - 2~(k~-k~)] + m[ (z -~)(k~-k~. + k~ .k~)- (1 + ~1)(k~-k~)] +

+ + . P ) -

336 L. MICU

I f we consider now the definition of the Sachs electromagnetic form factors G~(q ~) =t~(q ~') -[- (q2/4M~)F2(q2), Gx(q 2) =~(q~) -k F2(q~), the scaling p rope r ty

GB(q2)/G~(q ~) q.-W-~-+ct follows immedia te ly f rom eqs. (13) and (14). After a slight

t ransformat ion of the expressions (14a) and (14b) following from the s y m m e t r y

proper t ies of the integral (13) in the q = 0 l imi t one obtains

(15) /7~1)(0 ) __ (6) --~1 (0),

which accounts for the zero electric charge of the neutron. However expressions (14) allow a dis t r ibut io~ of charge inside the neutron.

Replacing in eqs. (14) each ~ by its mean value ½, using the relat ion M = 3m

which would hold if quarks were free and neglecting the t ransverse motion of

the quarks, one gets the SUe-like resul ts

(16) (p~ , (n) ih) ) ( v ) o~,(o)/(~(o)=-~, q~,(Ol/F~ (o)=3.

4 . - D i s c u s s i o n o f t h e r e s u l t s .

The results obtained in the preceding Section show tha t the scaling p roper ty

of the e lectromagnet ic form factors GB(q2)/G~(q2 ) ~.-Tz--~-~ct can arise only in a model t ha t takes spin into account consistently. 1Vforeover the present model

emphasizes the fact t ha t the asymptot ic behaviour of the scalar form factors F~ and F2 strongly reflects the na tu re of the nucleon as a bound state.

The SU6 scheme appears now clearly to be the s tat ic l imit with free quarks

of the relat ivis t ic model. On the other hand the present model seems to be the relat ivist ic general-

izat ion of the older one (~) which supposes tha t a ve r t ex is formed with the par t ic ipat ion of a q~l pair with the quan tum numbers of the vacuum which

mixes with the init ial quarks; recombinat ion follows in order to yield the final

state. Indeed , if we use Wick's theorem, the ma t r ix e lement of a cu r ren t

t aken in the second order of the ~ q q q coupling may be t ransformed as follows:

(17) <N(P')]J.(0)IN(P)} = ½ f d 4 x d ' y ( N ' l T ( , . ) f ~ x ~ ( x ) j ~ , ( O ) S x ~ ( Y ) ) l N } ~

R o R '~

I f quarks 123 are considered to come f rom the initial s ta te and quarks 1'23 f rom the final state, t hen the contract ion (V~l,(0)~l,(y))0 stands for the part ic-

ipation of the q~ pair with the quan tum numbers of the vacuum.

:ELECTROMAGNETIC FORSI FACTORS OF TtI]~ NVCLEON IN A QUARK I~OD~L 3 3 7

I t is impor tan t to no te t ha t with slight modifications the model curt be t ransformed so as to include contributions from the quark-ant iquark sea. t t av ing the quantum numbers of the vacuum, these pMrs do not affect the charge or the magnetic moment of the nucleon. Of course, they modify the shape of the charge ~nd magnet ic-moment form factors, but do not change their asymptot ic propert ies as q2 _> o+.

The suggestion of the referee to discuss more carefully the problems included

in the Appendix is grateful ly acknowledged.

flxPPEiNDIX

This Append ix in t ends to discuss more careful ly the resul t G~(q2)/G~(q 0-) ~ 2 ct and its dependence on the cut-off funct ion.

The difference between this resul t and previous ones (7) is due essentially to the different t r e a t m e n t of the cons t i tuent spins. Thus due to the relativistic SU6 coupling of the quark spins the ensemble of the two quarks nonin teract ing with the electromagnetic field looks a l te rnate ly like a seMur, pseu- doscMar, vector or axial vector particle. Their corresponding contributions are summed up in the equations (14) which define the expressions ~ ) , ~-~'). I t is impor tan t to note tha t ~ ) does not contain any q2-dependent term. The contr ibut ion to ~ ) may be divided into two parts. The first one comes from the scMar and pseudoscMur par t ic ipat ion and does not contain any q~-dependent term. Considering only contributions of this kind, one has as previously (7)

G 2 G~(q )/ ~(q ) - - - ~ Cq 2 . qa--->co

The second par t comes from the vector and axial vector part icipation. Their contr ibution contains a q2-dependent te rm which is essential in the derivat ion of the result

G~(q~)/G~(q 2) ~ ct .

The comparison between ~1 and ~-2 in eqs. (14a)-(14d) shows th:~t ~-1 contMns explici t ly an extra qO. The condit ion to be imposed on the cut-off funct ion in order to preserve this sMut~ry difference may be be t t e r understood by anMysing the following iden t i ty :

co vo

F(A) ~ ](x) 1 ¢0 ----J x + A dx ~ (x) dx-- ~ 7 + x/A

0 0 o

- - d x .

338 L. MICU

I f we t a k e now t h e l i m i t A--> ~ , i t fo l lows t h a t F ( A ) goes l i ke 1 /A ~s co

A - + c¢ i f u n d o n l y i f f ] (x )dx ex i s t s . 0

Thus , t h e cut -off f u n c t i o n m u s t d e c r e a s e su f f i c ien t ly f a s t in o r d e r to e n s u r e t h e c o n v e r g e n c e of t h e i n t e g r a l s ove r t r a n s v e r s e m o m e n t ~ i n eq. (:12) ~ f t e r t h e f ~ c t o r i z u t i o n of (q2)~ in t h e d e n o m i n u t o r . A p ~ r t f r o m th i s , t h e r e s u l t does n o t d e p e n d on t h e p a r t i c u l u r s h a p e of t h e cut -off f u n c t i o n .

• R I A S S U N T 0 (*)

Si ealeola~o i fa t to r i di forma elet t romagnet ic i del nucleono nel s is tema di impulso infinito impiegando una generalizzazione rela t ivis t ica dello schema SU6 stat ico. Si evitano le eomplieazioni dovute al t r a t t amen to eoncreto dello spin assumendo ftmzioni di taglio ad hoe. Si ott iene la propriet~ di (( scaling ,> dei fa t to r i di forma Ge(q2)/GM(q ~) di Sachs tendent i a c t per q ~ ~o. I1 compor~amento asintotieo dei fa t tor i di forma rispeechia for temente la forma della flmzione di taglio. Si ottcztgono anche r i su l ta t i di t ipo SU¢ he1 l imite stat ieo con quark liberi .

(*) Traduzione a eura della tCedazione.

~.rleKTpOMaltlHTHblI! ~OpM-~aKTop Hy~toua ~ MO~eJIH FdlapKOB.

Pe3mMe(*). ~ B CnCTeMe 6ecKoHennoro nMnynbca BI, I~rrICJanIOTC~ 3J]eKTpoMarrraTrmie qbOpM-qbaKTopbI HyKnoHa, ~cnonb3yn penaTnBHCTCKOe o6o6ulem~e CTaTa~ec~o~ CXeMbI S U 6. YCJIO~HeHrlfl, oSycJIOBneHHBIe KOIdI~peTm, IM paccMoTpemdeM canna, yjIaeTca n36e)KaTb 3a CneT npeJ~nonomermn llnn / lanuoro cnynaa qbyrrrtlm~ o6pe3artaa. I]onyqaeTcn CBOHCTBO NO~lO6rIa ~nn qbopM-qba~TopoB Caeca Ge(q~)/G~(q2)--~ ct, ro r~a q2~oo. Aci~MnrOrH~ecKoe noBe~ieime qbopM-qbaKTopOB CHnLnO 3aBaCHT OT qbOpMbI qbyrn~tina 06pe3arma. B CTaTn- ~IeCKOM npe~Iene co CBO6OjIHI, IMII KBap~aM~ nony~aIOTCS pe3ynbTaTl, I, CXOJIHBIe C ~qU 6.

(*) IIepeae()euo pe3a~tlueft.

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Electromagnetic form factors of the nucleon in a quark model