Advanced relativisitic quantum mechanics

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Z. Phys. A - A t o m s a n d Nuclei 307, 339-350 (1982) Zeitschrift Atoms f~)r Physik A and Nuclei

�9 Springer-Verlag 1982

Electromagnetic Properties of Relativistic Quarks in Confining Potentials*

R. Tegen and R. Brockmann

Institut fiJr Theoretische Physik der Universit~it, Regensburg, Federal Republic of Germany

W. Weise

Department of Physics, State University of New York, Stony Brook, USA, and Institut ftir Theoretische Physik der Universit~it, Regensburg, Federal Republic of Germany**

Received May 5, 1982

We investigate electromagnetic properties of the quark core of nucleons in a model with massless quarks in confining potentials. We find quark core rms radii of 0.6 fm or smaller to be compatible with form factor data corrected for pion cloud effects. Re- lations between the magnetic form factor and the axial vector or pseudoscalar form factors will also be discussed.

1. Introduction

It is generally believed that the quark core of the nucleon consists of three up or down quarks, with light (current) masses of less than 10 MeV, confined in a volume with a radius of the order of one fermi. One of the standard ways to investigate the validity of such pictures is by probing the charge and cur- rent distributions of quarks inside the nucleon, i.e. by analysing nucleon electromagnetic form fac- tors. There are two basic problems involved in this pro- cedure: first, one faces the difficulty of separating quark core contributions from those parts arising from the charged pion cloud surrounding this core. Bag models which couple a phenomenological pion field to the quarks at the bag boundary put con- siderable emphasis on this point [1, 2]. Secondly, finite momentum transfers to the quark core require an accurate treatment of nucleon recoil, or equiva- lently, a proper description of nucleon center-of- mass motion. This is extremely difficult to achieve in

* W o r k s u p p o r t e d in p a r t by Deutsche Forschungsgemeinschaft, Bundesministerium f'tir F o r s c h u n g u n d Technologie a n d by U S D O E contract No. DE-AC02-76 E R 13001 ** P e r m a n e n t address

standard bag models [6]. It is for this reason that we adopt a picture where bag boundary conditions are replaced by an appropriately chosen confine- ment potential. Attempts to reproduce the nucleon electromagnetic form factors e.g. in the Cloudy Bag model [-2] with- out proper treatment of recoil have so far been of limited success. Approaches of a completely different kind, like in the Geometrodynamical Model [-3] which treats some aspects of recoil correctly, have nevertheless not been able to obtain the proper q2 dependence of the form factors even at small q. The plan of this paper is as follows: after a brief section recalling some of the relevant definitions of electromagnetic form factors, we introduce the quark core of nucleons as three massless quarks moving in a confining potential. The main objective is to see whether such a phenomenology can be made compatible with both electromagnetic form factors at not too high momentum transfers (after introducing pion cloud corrections) and with nu- cleon spectroscopy, while at the same time treating center-of-mass motion correctly to order q2/4M2. Furthermore, we will also discuss quark core contri-

0340-2193/82/0307/0339/$02.40

340 R. Tegen et al.: Electromagnetic Properties o f Relativistic Quarks

butions to magnetic m o m e n t s a n d connections be- twee'n quark core magnetic form factors and other types of couplings, like axial vector a n d pseudo- scalar currents.

2. Relativistic Quarks in Confining Potentials

We assume that the quark core of the nucleon can be described such as to replace the confining forces by a Dirac potential U(r), so that the individual massless* quarks are represented by single particle wave functions satisfying the Dirac equation

Do E~ + i~. v - V(r)] 0~(r) = 0. (1)

Technical details are given in Appendix 1. A dis- cussion of the conditions for reducing the confine- ment problem to a local potential goes b e y o n d the scope of this article, but we note that a transcription of the conceptual frame-work of spontaneous chiral symmetry breaking according to N a m b u and Jona- Lasinio [4] to finite F e r m i o n systems [5] does in fact lead to such a scalar potential picture, given an appropriate effective quark-quark interaction. The potential U should be small in the b a r y o n center (asymptotic freedom) a n d rise infinitely b e y o n d some distance from the center (confinement). W e shall therefore discuss phenomen01ogical scalar potentials of the following type:

U(r)=cr', (2)

with an appropriately chosen constant c a n d power /l,

It will also be useful to consider a relativistic har- monic oscillator potential of the scalar-vector type

U(r) =~ (1 + ~,o)r ~. (3)

Such a potential, t h o u g h probably n o t of immediate physical significance, has the pleasant feature t h a t basically all quantities o f interest can be worked out analytically. We present details in Appendix 2. In particular, the center-of-mass energy of the confined three-quark system can be treated rigorously. Mo- reover, the eigenfunctions corresponding to the po2 tential, (3), can be used as a convenient complete and o r t h o n o r m a l set of b o u n d Dirac spinors for expansions of any given confined q u a r k wave func- tion. N o t e also that scalar-vector potentials propor- tional to 1 + ~o do not produce a spin-orbit splitting in the b a r y o n spectrum.

* We ignore small current q u a r k masses here and in the follow- ing

The nucleon is considered to be the energetically lowest configuration of three massless quarks con- fined by the potential U. The nucleon wave function is then a linear c o m b i n a t i o n of products of the 0~(r~) (i = i, 2, 3), totally symmetric in its spin-flavor-SU [4] content, combined with the totally antisymmetric color S U [ 3 ] wave function to form a color singlet. In calculations of nucleon form factors it is impor- tant to construct translationally invariant wave func- tions. This is done by the following projection pro- cedure: Let

r = f d3 r e-'P'r 0~(r) (4)

be the Fourier transform of the confined q u a r k Di- rac spinor tp~(r). Then

r2, r3)= ~ o[ d3Pl d3pzd3P3 ~(3)(~. Pi - P)

�9 eiY "~ "r~ 4~(pl) 4~(p2) 4,(p3) (5)

is translation invariant in the sense that the center of mass moves as a plane wave with fixed momen- t u m P. To verify this, introduce Jacobian coor- dinates (R, r ~/) with R = � 8 9 a n d the mo- m e n t u m P conjugate to R, to obtain from (5):

Te(r> r2, r3) = e i P ' R ~int" (6)

The intrinsic wave function Ti,t depends on ~, tt a n d o n P, but n o t o n R. W e have left out the time behaviour of the wave function which adds phases of the usual form e -~Et, where E are the energy eigen- values for each of the quarks. The wave function Te does not satisfy L o r e n t z in- variance in the sense that the small component as- sociated with the center-of-mass m o t i o n is not treat- ed properly. This is justified as long as the nucleon as a whole moves nonrelativistically, i.e. if

tPI/(~+M)<I, where M is the nucleon mass. In calculations of nucleon electromagnetic form factors, this limits the applicability of the mo- m e n t u m projected wave functions, (5) to four-mo- m e n t u m transfers _q2 =qZ _q~ ~ 4M z. The problem of constructing fully Lorentz-invariant wave func- tions for three relativistic confined F e r m i o n s has n o t yet been solved satisfactorily in the existing litera- ture [6-9]. Finally, the normalization factor N(P) in (5) is de- termined by the condition

fd3rl d3rzd3r3 7tp+ ( r l , r z , r3) t//Q(r 1 , r 2, r3) = (2 r~) 3 6(3)(p _ Q). (7)

This leads to

N(P) = [~d3re-ie'ap~(r) p~(r)&.(r)] -~ (8)

R. Tegen et aL : Electromagnetic Properties of Relativistic Quarks 341

where

d3k p~(r) = j" ~ e 'k'" ~ + (k) q~(k),

p (0) = L

For three quarks in the lowest 1S1/2 orbits

N(P) = [4rc~drrZjo(Pr)p3(r)] - ~ o (9)

(where we have dropped the index attached to p for simplicity). It is instructive to investigate N(P) as obtained with the harmonic scalar-vector potential (1/2)c(1+~o)r 2. In this case N(P) becomes propor- tional to exp(pE/2E~) times a polynomial in p2 (see Appendix 2), where E o is the lowest eigenvalue in that potential. This behaviour is quite familiar from the nonrelativistic harmonic oscillator.

3. Nucleon Electromagnetic Form Factors

3.1. Definitions

We recall that the nucleon electromagnetic form fac- tors probed by the process, Fig. 1, are defined as

F~:,~(q2)=eums(Pf)' {Fa(q2)yu + Fz(q2) ia. ~ )

(lO)

where u.,(P) are plane wave nucleon Dirac spinors (m denotes the spin projection) and

is the 4-momentum transfer, Pz=Pf=M2. It is con- venient to introduce the Sachs form factors

G~(q 2) = Fl(q 2) -- r/F2(qZ), Gi(q2)=F~(q2)+ F2(q 2) (11) where ~/= -q2/4M 2 with q2 the 4-momentum trans- fer squared. These are normalized as follows:

G~(0) = / 1 proton ( 0 neutron

(12) = ~ f l + g p proton

GM(O) ( g. neutron

where g r = 1.793 and g . = -1.913. The magnetic mo- ments are then given by

e /~p,, = G~"(0) 2 M ' (13)

N f

q 2 ~ Nf

Ni N,

(QI (b) Fig. 1. Nucleon electromagnetic form factor (a) as viewed From the photon interacting with the quark core (b)

In terms of G E and GM, one obtains for the time component of FU(q2), for a nucleon initially at rest:

1 / l + t / 2 F~ 2) =e6m~m, i/ ~ G~(q ) V i d - z ~ l

(14)

while for the space components (k = 1, 2, 3), one de- rives

~t'(q2) = e 1 §

�9 {6mf~Ge(q2)~k+ izL(a x q)kZmGM(q2)} (15)

where Z~ are two component Pauli spinors and =q/lq].

3.2. Quark Core Contributions to G~,M(q 2)

F o r a nucleon consisting of three pointlike quarks, the total current is just the sum of the quark cur- rents, e ~ , j ? , ~ j . Hence the form factor in this model is simply given by

(2g)3 (~(3)(pf _ p / _ q) FU(q2): ~ d3rl d3r2 dar3

�9 Ties(r1 r2 r3)[2~, e~Y~(j) e+ iq"J] ~ / r l r2 r3) (16)

where ej are the individual quark charges and the ~vp are the momentum projected 3-quark wave func- tions, (5). Note that translation and gauge invariance holds in (16). Consider now first the proton in the absence of recoil corrections (i.e. without projection onto fixed total momentum), with two u-quarks and one d- quark occupying l s orbits. The individual quark

342 R. Tegen et al.: Electromagnetic Properties of Relativistic Quarks

wave functions are then of the form

1 ( i g ( r ) / r O(r) = - ~ \a. if(O/r] Z (17)

where Z is a Pauli spinor. In terms of the upper and lower components g and f,, respectively, the follow- ing expressions for the proton form factors G~ and G~t are easily derived to leading order in q = - q 2 / 4 M 2

oo

Gf~(q 2) = ~ drjo(qr) [g 2(r) + f2(r)], 0

oo 4 M ~ drji(qr)f(r)g(r). (18) G f u ( q 2 ) = - [q~ o

Note that G~(O)=~d3rO+(r)O(r)=l, and the mag- netic moment due to the quark core is given by

e ce # p = ~ - ~ G P ( q 2 = 0 ) = --~e ~ drrf(r)g(r). (19)

0

This result can also be obtained using the standard definition of the magnetic moment,

/ , = 1 ~ eJ f d3r 0i(r) (r x 7) Oj(r). (20)

By employing the Dirac equation, (1), for a scalar confining potential, (19) can be cast into the form

e ( 2 i d r f 2 ( r ) ) (21) #P = ~--o 1 - - ~

where E o is the energy eigenvalue of a single quark in the lowest (1 s)-orbit. In the absence of center-of- mass corrections, the nucleon mass is simply given by M = 3E o. For the neutron magnetic moment, the standard quark model ratio

2

follows. We note in passing that #v of (t9) is particularly easy to obtain for a potential U=(c/2)(l+y,o)r 2, in which case

2 e 4(E__~) e (22) # ' = 3 Eoo=3 2M"

The quark core neutron electric form factor G) is identically zero in the absence of couplings to the charged pion cloud, and G~t(q2)= - 2 / 3 G~,t(q2). We would now like to turn to the treatment of recoil corrections, using the translationally invariant wave functions of (5). Transferring a finite momen- tum to one of the quarks induces kinematical cor-

relations with the remaining two quarks in such a way as to conserve overall momentum. Inserting (5) into (16), the quark core form factor is now evaluat- ed as follows:

(27C)3 •(3)(pf _ p / _ q) F,(q2)

= ~ ej ~ d3r e +'qr' q~j(r) ~," ~(r)N(q) N(0) (23) J

where ~ ( r ) now contains the effects of the two re- coiling quarks if the j-th quark is hit by the virtual photon. More precisely,

d 3 ~ j ( r ) = ~ ( ~ 5 " e i p ' r q~j(P) Wkt(P) (24)

where (jkl) refer to quarks (123) or any permutation thereof. Here

Wk,(p ) = 5d3re-iPr~k(r) fi,(r)

with

(251

^ d3p' ~P . . . . . (26) &(r) = 5 ( ~ ) 3 e 4k(p) qSk(p) etc.,

and ~bk(P) is the Fourier transform of the single quark wave function, as in (4). For three quarks in l s orbits, all/3's of (26) are the same and

eo W(p)=4~z .[ drr2jo(pr)~2(r). (27)

0

Introducing upper and lower components of ~ (see (24)):

1 [ ig(r)/r \ ~ ( r ) = ~ - ~a.ff('r)/r~, Z, (28)

one observes that

g(r) 2 - dpp z W(p)Io(P)jo(pr)

r 7c o

f ( r ) = 2 7 dpp2 W(p)Ia (p)jI~r) r 7z o

with

(29)

Io(P) = ~ drrjo(Pr)g(r), II(P)- drrjl(pr)f(r ) 0 0

in terms of the upper and lower components of t h e quark spinors, (17). It is instructive to compare the upper and lower components of the recoil-corrected wave functions, and f, with the uncorrected ones, g and f This is

R. Tegen et al. : Electromagnetic Properties of Relativistic Quarks 343

igl .4=_ r-

.d

U = cr 2

// \..0,:, L - . ,

I I I I i

0.5 1.0 1.5 2.0 2.5 a r [ f m ]

g _d y,

g•\N U = c r 3

\ , < ,

~ . _ . ~ f ( r )

I I I I I

015 1.0 1.5 2.0 2.5 b r [fin] Fig. 2a and b. Upper and lower components (g(r) and f(r), respectively) of ls Dirac wave functions evaluated with different confining potentials. Also shown are the upper and lower com- ponents ~(r) and f(r) obtained in the procedure of recoil cor- rections, according to (29). (a) Confining potential U=crZ; (b) U =cr 3. In all cases, c has been adjusted to obtain an energy eigenvalue E o = 540 MeV for the is orbit

s hown in Fig. 2 for a scalar potential U = cr 2 with c =0.83 G e V / f m 2, a p a r a m e t e r which is later shown to yield an rms radius c o m p a t i b l e with G~z(q2). N o t e t h a t ~ and f extend further out in space t h a n g an d f, so that it might a p p e a r that form factors corrected for recoil fall off faster in q2 t h a n n o n - c o r r e c t e d ones. This is not the case, however, since the n o r m a - lization factor N(q), (8), increases exponentially with q2, so that the net effect o f recoil corrections works

always such as to effectively reduce the rms radius*. This is to be seen in co n t rast with [17], where the recoil c o r r e c t i o n a p p a r e n t l y increases the rms radius o f the hag, b u t corrections for wave function n o r m a - lization have b een o m i t t e d (see also E18] for a criti- cal discussion o f the p r o c e d u r e suggested in [17]). F u r t h e r ev al u at i o n o f the form factors to l e a d i n g o rd er in q2/4M2 yields the following results:

G~(q 2) = fi[ (q) ~ drjo(qr ) [g(r)~(r) + f (r) f (r)], 0

2M Gfu(q2) = _ ~ _ ~r(q) ~ drj~ (qr) Eg(r) f (r) + f (r)~,(r)].

o (30)

N o t e t h at recoil corrections enter t h r o u g h W(p) in (29). Putting W ( p ) - i leads b a c k to the s t a n d a r d result, (18). Th e n o r m a l i z a t i o n N(q) is i n t r o d u c e d such that Gf~(0)= 1 is maintained. This requires

1 ~ , N(q) ~ f f ) ] - i ( q ) = ~ ( ~ [ ! dr(g~, + (31)

where N(q) is given b y (9). E q u a t i o n s (30) su m m ari ze the q u a r k core contri- b u t i o n s to the p r o t o n form factors o n ce recoil is included. Actual comparisons with experiments re- quire first to correct for those parts o f the empirical d at a which are related to m e s o n cloud c o n t r i b u t i o n s s u r r o u n d i n g the q u a r k core. This should ultimately be d o n e in a consistent t h e o r y o f p i o n - b a r y o n cou- pling. It is nevertheless believed t h a t constraints o n the q u a r k core c o n t r i b u t i o n s can already be ob- t ai n ed fro m the following simplified procedure.

3.3. Meson Cloud Corrections

T h e empirical G~(q 2) is k n o w n to be r e a s o n a b l y well r e p r o d u c e d by a dipole form

G~(q 2) = [1 - q2/0.71 G eV 2 ] -2 (32)

(although i m p r o v e d fits show deviations at b o t h low an d high q2 [-10, 11]). T h e p r o t o n rms charge radius o b t a i n e d from the dipole fit is (r2)1/2--=0.81 fill, b u t part o f it has to be assigned t o the coupling of the p h o t o n to the p i o n cloud r a t h e r t h a n the q u a r k core. In o rd er to isolate m e s o n cl o u d effects, it is useful to discuss the isovector formfactors:

v 2 1 p 2 n 2 GE, M(q ) (33) --~[GE,M(q )--G~,M(q )].

Experimentally, G ~ ( q 2) "~ G ~ ( q 2) f o r - q 2 ~ 0 . 4 G e V 2, so t h at G~z~_2G). T h e p o i n t is t h at m e s o n cloud

* This tendency has also been found from a different reasoning by Myhrer, Ref. [193 and by DeTar [201

344 R. Tegen et al. : Electromagnetic Properties of Relativistic Quarks

1.0

0.8

0.6

0 4 �84

0,2

G~ (q2)

X

----- no recoi[ \ ' x ~ - - NIT 1.0 f m ~ . ~

,1, I I _ _ [

0.1 0.2 0.3 0.4 -qZ[GeV 21

Fig. 3. C h a r g e f o r m f a c t o r G~P(q 2) o b t a i n e d w i t h a c o n f i n i n g potential U=cr z with c=0.83 GeV fm -2 following (36) and (30). For comparison, the result omitting recoil corrections, (18), is also shown. Included for comparison is the result with MIT bag wave functions [14] (in absence of recoil corrections) using a bag radius R = 1 fro. Data are taken from the compilation of [11]

c o n t r i b u t i o n s in t h e i s o v e c t o r f o r m f a c t o r s a r e m a i n - ly t w o - p i o n i n t e r m e d i a t e s t a t e s d o m i n a t e d b y t h e d i r e c t c o n v e r s i o n o f t h e p h o t o n i n t o t h e p m e s o n . F o l l o w i n g [11] t h e t w o - p i o n c o n t i n u u m is well a p - p r o x i m a t e d in t h e r a n g e 0 < _ q 2 < 1 G e V 2 b y a fac- t o r [ l - q Z / r n ~ ] -1 i n El(q2), t h e i s o v e c t o r p a r t o f Ft(q2), w i t h m ~ = 7 3 2 M e V . I n F~(q 2) t h e c o r r e s p o n d - ing m a s s m~ t u r n s o u t t o b e o n l y slightly l a r g e r , m~ = 777 M e V . H e r e F ~ v , ~, 2 a r e g i v e n in t e r m s o f G~- M b y

1 F v(~2' + r / i [G~(q )--GE(q )

+ ~1(G~(q =) - G~(q2))]

1 F (q =)= [G#(q =) 2(1 + ,/)

-G~(q2) + G"e(q2)] (34)

w h e r e 0 = - q 2 / 4 M 2 as b e f o r e . G i v e n t h a t G ) - 0 f o r t h e q u a r k c o r e , it is t h e n a g o o d a p p r o x i m a t i o n t o w r i t e

G ~ ( q Z ) = ( 1 2/ 2-1 v 2 --q ;my) [G~(q )]q.a,-k (35)

1.0

0.8

06

0 4

0.2

~ ~ ( q 2 )

\ U=cr ~

recoit " ~ " -, no rec0it

I [ I l

01 0 2 0 3 0 4 -q2[GeV 2]

Fig. 4. As Figure 3, for a confining potential U=cr 3 with c = 1.25 GeV fm - 3

a n d i d e n t i f y [GP(q2)]quark with t h e q u a r k c o r e c o n t r i - b u t i o n (30). A s i m i l a r p r o c e d u r e h a s b e e n s u g g e s t e d in [12]. M o r e precisely, t h e i s o v e c t o r f o r m f a c t o r s F~ a n d F~ c a n b e w r i t t e n as

p 2 5 p 2 [Ge(q )+3~lG~(q )]quark (36a)

F[(q2) = 2(1 + r/)(1 - q2/O. 536 G e V 2)

p 2 5 a p 2 [ - G e ( q ) + 3 u(q )]qu.rk ( 3 6 b )

F~(q2) = 2(1 + ,/)(1 - q2/0.603 G e V 2 ) '

f o l l o w i n g [ 1 1 ] . T o o b t a i n (36), t h e r e l a t i o n G ~ = - 2 / 3 G~t h a s b e e n used.

3.4. Results

W e s h o w in Figs. 3, 4 t y p i c a l r e s u l t s o b t a i n e d for G[(q 2) u s i n g m o m e n t u m p r o j e c t e d w a v e f u n c t i o n s c a l c u l a t e d w i t h s c a l a r c o n f i n i n g p o t e n t i a l s o f t h e t y p e U = c r " f o r n = 2 a n d n = 3 . T h e c o r r e s p o n d i n g p a r a m e t e r s a r e c = 0 . 8 3 G e V / f m 2 f o r n = 2 a n d c = 1.25 G e V / f m 3 f o r n = 3. T h e l o w e s t e i g e n v a l u e s i n t h e s e p o t e n t i a l s a r e E o = 540 M e V i n b o t h cases. T h e

R. Tegen et al.: Electromagnetic Properties of Relativistic Quarks 345

1.0

0.8

0.6

0.4

0.2

F1v (q2)

U = c r 2 - - . rec0i[ . . . . no recoit

\ \

\ \ \ .

HIT 1.0 fm \ \ -

1.0

0.8

0.6

F2V(q 2)

l \ \ \ \

0./+-

U = c r 2

recoi[ no recoil

0.2 0, '~'t - 016 0.8 1.0 -q2 [(GeV/r 2 ]

(a)

0.2 MIT 1.0 fm"~'~,,.

~ ' ~

I I I I I

0.2 0./+ 06 0.8 1.0 _ q 2 [(GeV/c) 21

(b) Fig. 5. Isovector form factors F f and F~ according to (36) (with F~ normalized at q2=0), obtained with a scalar confinement potential U = c r 2 as in Fig. 3. A comparison between calculations with and without recoil corrections is shown. The result with MIT bag wave functions (R = 1 fm) is also shown for comparison

rms radii of the q u a r k core are ( r 2 ) l / 2 = 0 . 6 4 f m for n = 2 a n d ( r 2 ) l / 2 = 0 . 6 0 f m for n = 3 , respectively. It might appear t h a t E 0 is quite large; in fact 3E 0 is m u c h larger t h a n the average between nucleon a n d A(1232) masses. On the other hand, a substantial fraction of 3 E 0 is related to spurious center-of-mass oscillations which have to be removed in compa- risons with measured b a r y o n masses. In a potential

c of the type U = ~ ( l + Y 0 ) r z, we show in Appendix 2

t h a t the energy associated with the lowest spurious center-of-mass m o d e is just Eo, so that the average between nucleon a n d A(1232) mass, M = 1085 MeV, should be c o m p a r e d with 2E o rather t h a n 3E o. F o r scalar confining potentials, similar conclusions can- n o t be d r a w n rigorously, b u t one can expand the i s q u a r k wave function of a given scalar confinement potential into the complete set of eigenfunctions de- rived from the (c/2)(l+yo)r 2 potential a n d thereby finds t h a t the energy to be assigned to center-of- mass m o t i o n is of similar order o f m a g n i t u d e as E o (more precisely speaking, it turns o u t to be 20-30 % larger t h a n E0). In a n y case, it appears that q u a r k

core rms radii of a b o u t 0.6 fm are n o t inconsistent with the empirical nucleon mass once the center-of- mass energy is removed. In fact, further (attractive) corrections to the nucleon mass are expected to come from self-energies involving the pion cloud [1, 2-1. Hence E 0 should be somewhat larger to start with, and an rms radius of 0.6 fm is probably to be regarded as a n upper limit. F r o m Figs. 3, 4 the importance of a correct treat- m e n t of recoil becomes evident. The curves with no recoil corrections correspond to (18). Also shown is the result using M I T bag wave functions with a bag radius R = 1 fm. Once corrections for the p meson pole are introduced as in (35), the M I T bag turns o u t to be too large. Recoil corrections as described in [6-1 help somewhat, but do not remove this dis- crepancy. Chiral bag models with a q u a r k core ra- dius reduced by the pressure of the surrounding pion field, appear to be more successful [19]. Using m o m e n t u m projected wave functions, the low q behaviour up to tqt = 5 0 0 M e V is quite well repro- d u c e d with confining potentials which place the low- est q u a r k eigenvalue between 500 a n d 600MeV.

346

I 0

F1V(q 2)

O~ U = c r 3

r e c o i l - - - - - no r e c o i l

04

0 2

\

-\

\ � 9

I, I 1 I I I 0 2 0 ~ 0.6 0 8 1.0

-q 2[(GeVlc}2] (a} Fig. 6. S a m e as Fig. 5, for a c o n f i n i n g p o t e n t i a l U = c r 3 (see Fig. 4)

R. T e g e n et al.: E l e c t r o m a g n e t i c Properties o f Relativistic Q u a r k s

1 0 r - - - - - - - FzV(q z}

0,8 U =or 3

...... r e c o i l - - ' - - no recoil

0.6

0.4

0.2

\

I I ........... I I i

0.2 0 4 0.6 0.8 1.0 _q2 [(GeVlc}~]

(b)

At Iq[ > 1 GeV, the experimentally observed fiat be- haviour in Gf~(q 2) seems to require contributions of even smaller size. Results for the isovector form factors F~(q z) a n d F~(q 2) of (36) are s h o w n in Figs. 5, 6. Characteristic deviations are visible, although the a m o u n t of d a t a is quite limited. One expects that specific m e s o n c l o u d contributions especially to F a should b e signif- icant over a n d b e y o n d the t w o - p i o n correction fac- tors discussed before, since pion dynamics is k n o w n to provide a considerable fraction of the a n o m a l o u s magnetic m o m e n t s for sufficiently small ( r 2 ) of the q u a r k core [12, 20]. In T a b l e 1 we present results for the q u a r k core contributions to p r o t o n magnetic

Table 1. Q u a r k c o r e values o f ( r 2 ) a n d G ~ ( 0 ) c a l c u l a t e d w i t h different c o n f i n i n g potentials. F o r G~(0), results w i t h o u t (18) a n d with (30) C M c o r r e c t i o n s are s h o w n

U(r) (r2>§ Gg,(0)

n o C M corr. incl. C M corr.

c r 2 0.64 1.51 1.47 c r 3 0,60 1.48 1.41

moments, b o t h with and w i t h o u t center-of-mass (CM) corrections, for different confining potentials. The C M corrections turn out to b e less than 5 %, b u t one should also note that they depend sen- sitively o n the ratio of l o w e r to u p p e r c o m p o n e n t s o f the C M - c o r r e c t e d w a v e functions in the surface.

4. C o n n e c t i o n s B e t w e e n GM(q2) and Pseudoscalar or A x i a l V e c t o r F o r m F a c t o r s

In this section we w o u l d like to discuss relations b e t w e e n the q u a r k core c o n t r i b u t i o n to the magnetic form factor G~(q 2) and other types o f spin depen- dent couplings. Consider for example the isovector pseudoscalar form factor t a k e n b e t w e e n m o m e n t u m projected nucleon w a v e functions:

(2 ~)a 5(3)(pf _ Pi - q) F~ ~(q)

= J'SI d3r; d3r2 d3r3 7JP,(ri r2r3)

�9 [ ~ e+iq'rJys(j)zz(/)] 7'm(h r2r~). (37) J

This D r m factor is relevant for couplings of the q u a r k core to fields carrying the q u a n t u m n u m b e r s

R. Tegen et al.: Electromagnetic Properties of Relativistic Quarks 347

of a pion. It is easily shown that F~ can be reduced to

,oo - .f drj~ (qr)[f(r)~,(r) + g(r)f(r)] (38)

0

where the normalization is the same as in (31). Here au and z u now refer to the spin and isospin of the nucleon (not to the individual quarks), and the brack- ets denote matrix elements to be taken between nucleon two-component Pauli spinors and isospin wave functions. By comparison with (30), one ob- serves that

F~ (q) = ~ ; 5 ( a u -2Mqz~) [Gfu(q 2)] quark (39)

M q + 1 ], (42)

where we have indicated explicitly that this relation holds for the quark contributions to both gA and the proton magnetic moment. The constraints implied by (42) are of some interest. Following Chiral Bag model calculations in which the pion is strictly kept outside the bag [14, 15], the pion cloud contribution adds to ga such that gA=3/2(gA)qua~k. The pion cloud contribution is reduced considerably if the pion enters the bag, as in the Cloudy Bag model [23. It is nevertheless meaningful to assume that the quark core contribution to ga is smaller than or at most equal to the empirical value:

(gA)quark =< g a = 1.25.

where G~t is the quark core contribution to the proton magnetic form factor. Equation (39) is useful for translating pseudoscalar isovector couplings at the quark level to pseudoscalar quantities involving nucleon variables. The factor 5/3 is the standard SU(4) result, while nucleon structure effects scale with Gfu(q2). Note that G~(qZ=O) provides only the fraction of the magnetic moment related to the quark core. In actual applications to pion-nucleon coupling, one must keep in mind though that the pion-quark cou- pling peaks strongly in the surface [1, 13], so that an additional rapidly varying function of r appears under the integral in (38) which makes comparisons between pion-nucleon and magnetic form factors less direct. Next, we examine the axial vector coupling constant gA defined by

(2~z)3 ~(3)(Pf - P/) �9 gA I~rN ~ )

= lim ~yySd3rld3r2d3r3 Tvs(rl r2 r3) q~0 [

[~j ~(j)y 5 (j)~ 32(/')e + iq' rs ] Te,(ri r2 r3) } (40)

Consider gA first in the absence of center-of-mass projections. In this case (40) can easily be worked out to obtain

From (42) it then follows that

M (GP(0))quark =~ 1.75 2E~-~" (43)

For the scalar confining potentials discussed here, we find (see Table 1):

1.5,

compatible with (43). The constraints implied here show most obviously that there must be contri- butions to G~(0) other than those coming from the quark core alone. Our value for gA obtained from (41) with a cr 2 potential is gA = 1.21. In the presence of center-of-mass corrections, (41) is replaced by

gA=~ (1--~ ~drf(r)f(r)o/i dr[g(r)~,(r)+ , ( r ) f ( r ) ] ) (44)

using recoil corrected upper and lower components g and f of (29). In this case a simple relation of the form, (42), cannot be obtained. The value for gA is increased by about 10-15 %, depending on the con- fining potential as compared to the result without center-of-mass corrections. Remaining problems can only be approached in an extended model which consistently incorporates the pion cloud.

ga=~ ( 1 - ~ ~o drf2(r) ) (41)

where f(r) is the lower component of the l s quark wave function, see (17). For a scalar confining poten- tial, (41) can be related to the magnetic moment by comparison with #v of (21):

5. S u m m a r y

We have investigated electromagnetic properties of the quark core of nucleons in a model with massless quarks in confining potentials. We found quark core rms radii of about 0.6fm or smaller to be corn-

348 R. Tegen et al.: Electromagnetic Properties of Relativistic Quarks

patible with form factor data at q2~<0.3GeV2 cor- rected for pion cloud effects. A proper treatment of recoil corrections in terms of momentum projected quark wave functions turned out to be important, 1 quark core of this size gives about one half of the empirical proton magnetic moment. According to [12, 20, 21], the major part of the other half should be due to the pion cloud.

We thank E.Werner for many helpful comments. One of us (W.W.) is grateful to G.E. Brown and the Nuclear Theory Group at SUNY, Stony Brook, for their kind hospitality and stimulating discussions.

A p p e n d i x 1

Single Particle Dirac Equation with Scalar Confinement

We consider the Dirac equation

[?oE~ - m + i y . V - U(r)] 6=(r)= 0 (A 1)

[ ig"u(r)/rq~u"~(r) ~ (a2) r \~. t L u ( r ) t r eu,.(t)/,

where

~PUm(t) = Y~ (lmr�89 ?tl 1 rtts

with the normalization

oo

5d3r ~/+ (r) 0~(r)= 5 d r [ f ~ z +g~] = 1 (A3) 0

The coupled equations for g and f are (~:= +(]+�89 forj=l )

= (U(r) + E + m) f ( r ) - ~ g(r) g'(r)

f ' ( r ) = (U(r) - E + m) g(r) + ~ f(r) (A 4)

Elimination of the lower component f yields [16] with m = 0 and g = g / ] / ~ E

g,, [ v" 3 [ v' 2 ( E + U) t-~ LE+ UJ

z- l ( t + 1)-1 u' ,, (E _uj+ w_jg q E + U r

(AS)

This equation has been used in the actual calcu- lations of wave functions [16].

A p p e n d i x 2

Scalar-Vector Harmonic Confinement

Consider a potential

U(r) = �89 + Yo) cr2 (A6)

Although not of immediate physical interest, this potential yields a particularly useful complete or- thonormal set of Dirac eigenfunctions for confine- ment problems. In this case the Dirac equation (A1) reduces to the following simple second order equa- tion for the upper component

. l ( l + 1)-1 g"(r) = [(E + m ) ( c r 2 - - 1 ~ + m ) + ~ ] g(r)

f ( r ) = E ~ ~ - g'(r)+ g(r) (A7)

solution of this equation for m = 0 yields the energies E n l = [ 2 l / / 7 ( 2 n + t - � 8 9 2/3 ( n = l , 2 .... ; l = 0 , 1, ...) and eigenfunctions (with r 2 = (E,l. c)- 1/2):

i~ t r 2 r 2

g"u(r)- N"z ( ~ o ) r exp ( - ~ ~)/~+J:l ( ~ ) (18)

f o r j = t + � 8 9

r - \E.1.rol \roi exp - ] 7 ~

" [/~.+-~, (r~) +/~+ ~2 ( ~ ) ] (19a)

1 for j = / 2

fn,j(r)__ = ( N.t ] ( r _ ] ' - i exp (_.!_ r2] r ~E.t.roi \roi \ 2 r 2 7

with N , , = 1 / 2 [ 2(n--1)! ]-~

V 3 tr(n+ Z+�89 ro ]

Note that the length scale parameter r o is state de- pendent, unlike the situation known from the non- relativistic harmonic oscillator. For 1 s orbits we ob- tain

r 2

(;0t g(r) = N e r 2

(t Al0) f ( r ) = - 1 ~ \ t o ! e [ 8 ]*

with N = [ ~ ] , 2 2 r 6 . E o = 3, c = E3o/9

R. Tegen et al.: Electromagnetic Properties of Relativistic Quarks

with g, f from (A 10) we o b t a i n the following simple expressions for the p r o t o n magnetic m o m e n t , axial charge a n d rms radius:

,up= - 2 e ~ drrg(r) f(r)=~-~ ~ dr( 2 2e 0 o r2- g - 3 f 2 ) = 3 E - - 7 '

( A l l a )

5 ( 1 4 ~ 1 7 6 = ~ = 0 9 3 , g a = 3 - - 3 ! ) z " ( A l l b )

< r 2 ) � 8 9 1 1 / ~ - - = 1 2.345 VY (A12) Eo Eo

C o m b i n i n g (A 1I a) a n d (A 12) gives the relation

#p = e ( r 2 ) � 8 9 2 ~ 1 = 0 . 2 8 4 e ( r 2 ) "~ (A 13)

w i t h o u t recoil the form factors calculated with (A 10) are simply given b y

G~(q 2) = exp( - q2 t.2/4) (1 - q2 rg/18)

2 4 M , - Gfa(q ) = 3 ~ o l / 1 + q 2 / 4 M 2 e x p ( - q 2 r ~ / 4 ) ( 1 1 4 )

Recoil corrections using m o m e n t u m projected wave functions as in (5) can be w o r k e d out analytically in this model. One obtains for the quantities of (8, 25, 26):

p(r) = exp( - r2/4 4 ) (1 - r2/18 r 2) ( 1 15)

t3(r) = exp( - r2/4 r~) �89 (1 + r2/6 r~) ( 1 16)

W(p) =0.269 (2~c) } ro 3

�9 exp(-p2rg/2)(1-O.253pZrg+p4ro4/87) (A17)

N(q)= ]

�9 exp(q2r2o/6){Co+qq:r2+c2q4r4+c3q6r6o} (A18)

~(r) = (77N/432) 7r~,~(r/ro) -exp(--rZ/4rg) (1 + b I rZ/r2o + bzr4/r~)

f (r) = ( - 163 N/2592) rc~ t~/l/3(r/ro) z �9 e x p ( - r e / 4 , ~ ) (1 + a I r2/r 2 + a2r4/r2) (A 19)

G~(q 2) =0.183 ~-(q) rc}ro 3

- e x p ( - q2 ro2/3 ) (d o + d 1 q2 4 q- d2q4ro 4 § d3qaro 6)

af (q2) = 0.07 &q) /5 G7)

- e x p ( - q2 rg/3) (h o + h~ q2ro2 + h2 q4ro 4) (A20)

with N(q) given b y (3t) in terms o f N(q) a n d

c o = 0 . 7 2 3 , q = 0 . 0 5 0 , c2=0.001 , c a = l . 5 x 10 - s

b 1 =0.074, b 2 = 0 . 0 0 1 , a 1 =0.092, a 2 = 0 . 0 0 2

349

d o = t + 2(b 1 + k) + 20/3(b 2 + ka 0 +(280/9)ka2, k = 0 . 1 1 8 - d ~ = ~ ( b 1 + k ) + ~ ( b 2 + kal) +s2-~ka 2 d2= 4~-og~5(b2 + ka~)+ 2s-~ka 2, d3= -7-~9ka 2 h o = 1 + 3k + ~ ( b 1 + 3 k a l ) + ~ - ( b 2 + 3 k a 2 ) - h 1 = ~ ( b 1 + 3 kal) +t2A~-(b 2 + 3ka2) h, - 16 _ -- gf(b 2 + 3 ka2)

The normalization function N(q) shows the be- haviour as expected from non-relativistic oscillator models. N o t e the b r o a d e n i n g o f the recoil l s orbits (A19) as c o m p a r e d to (A10) (see also Fig. 2, where the same quantities are s h o w n for a scalar confining potential)�9 Finally, we w o u l d like to show that for three mass- less quarks in their lowest l s orbits, with energy eigenvalues E 0 for each quark, the spurious center- of-mass energy o b t a i n e d with the potential U = 1/2c(1 + ? o ) r a is j u s t E 0, as in the non-relativistic h a r m o n i c oscillator. F o r that purpose, we note that the energy E 0 is completely determined by the equa- tion for the u p p e r c o m p o n e n t g(r), of the D i r a c wave function�9 W e rewrite ( 1 7 ) as

r (g(ri) t =Eo(cr~2--Eo)( g(ri)] (A21)

for each o f the three quarks (i = 1, 2, 3), where I = 0 is understood. W e can sum the three equations (A21) to obtain

[ 1 3 ~ ] [ I (g(r~)l=0 (A22) i= I i= i i = l

a n d i n t r o d u c e J a c o b i a n coordinates. E q u a t i o n (A22) then separates into three identical equations, one o f which determining the zero-point m o t i o n o f the cen- ter o f mass R = �89 1 + r 2 + r3):

( _ ~ o V 2 ' 1 + c p 2 - E o ) g(P) = 0 , p (A23)

where p = I / 3 R .

F r o m there it is o b v i o u s that the center-of-mass energy is o n e third o f the total 3E o. H e n c e the nucleon mass in this m o d e l is M = 2Eo after r e m o v a l o f the c.m. energy. The s a m e is easily s h o w n to be true if the three quarks are given finite, b u t equal masses.

R e f e r e n c e s 1. Brown, G.E., Rho, M.: Phys. Lett. 82B, 177 (1979)

Vento, V., Rho, M., Nyman, E.M., Jun, LH., Brown, G.E.: Nucl. Phys. A345, 413 (1980)

350 R. Tegen et al.: Electromagnetic Properties of Relativistic Quarks

2. Th6berge, S., Thomas, A.W., Miller, G.A.: Phys. Rev. D22, 2838 (1980); Phys. Rev. D24, 216 (1981)

3. Margaritisz, T., Szeg/5, K.: Budapest: (preprint) 1981 4. Nambu, Y., Jona-Lasinio, G.: Phys. Rev. 122, 345 (196I) 5. Broekmann, R., Weise, W., Werner, E.: Preprint (1982) 6. Barnhill, M.V.: Phys. Rev. D20, 123 (i979) 7. Fujimura, K., Kobayashi, T., Namiki, M.: Prog. Theor. Phys.

43, 73 (1970) 8. Lipes, R.G.: Phys. Rev. D5, 2849 (1972) 9. Feynman, R.P., Kislinger, M., Ravndal, F.: Phys. Rev. D3,

2706 (1971) 10. Borkowski, F. et al.: Z. Phys. A - Atoms and Nuclei 275, 29

(1975) 11. H~hler, G. et al.: Nucl. Phys. Bl14, 505 (1976) 12. Brown, G.E.: Nucl. Phys. A358, 39 (1981); see also: Brown,

G.E.: Lectures at the Int. School on Quarks and Nuclear Forces, Erice 1981. Nuel. Part. Phys. (to be published)

13. Weise, W., Werner, E.: Phys. Lett. 101B, 223 (1981) 14. Jaffe, R.L.: Lectures at the Int. School on Quarks and Nucle-

ar Forces, Erice 1981. Prog. Nucl. Part. Phys. (to be pub- lished)

15. Vento, V.: Thesis, Stony Brook (1980); see also [1] 16. Brockmann, R.: Phys. Rev. C18, 1510 (1978) 17. Donoghue, J.F., Johnson, K.: Phys. Rev. D21, 1975 (1980) 18. Wong, C.W.: Phys. Rev. D24, 1416 (1981)

19. Myhrer, F.: Phys. Lett. ll0B, 353 (1982) 20. de Tar, C.: Phys. Rex,. D24, 752, 762 (1981) 21. Brown, G.E., Myhrer, F.: Preprint (1982)

R. Tegen R. Brockmann Institut fiir Theoretische Physik Universitiit Regensburg Universitgtsstrasse 31 D-8400 Regensburg Federal Republic of Germany

W. Weise Department of Physics State University of New York Stony Brook, NY 11794 USA

Institut ftir Theoretische Physik Universit~it Regensburg Universit~tsstrasse 31 D-8400 Regensburg Federal Republic of Germany