Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

N U C L E O N F O R M FACTORS IN T H E P R O J E C T E D L I N E A R C H I R A L S O L I T O N M O D E L

P. ALBERTO a,b, E. RUIZ ARR1OLA a, M. FIOLHAIS B, F. G R U M M E R a, J.N. URBANO b and K. GOEKE ac

Institut.l~r Kernphysik, KFA Jiilich, D-5170 Jiilich, Fed, Rep. Germany h Centro de Fisica Teorica (1NIC) and Departmento de Fisica, Universidade de Coimbra, 1"-3000 Coimbra, Portugal " lnstitut.~r Theoretische Physik II, Ruhr-Universitgit Bochum, D-4630 Bochum, Fed. Rep. Germany

Received 17 March 1988; revised manuscript received 25 April 1988

Electromagnetic and axial form factors of the nucleon are evaluated using the lagrangian of the linear chiral soliton model. To this end angular momentum and isospin projected mean field solutions are determined variationally assuming valence quarks and pions in generalized hedgehog configurations. With the proper pion decay constant and after fitting the quark-meson cou- pling constant to the nucleon energy both proton and neutron charge form factors are reproduced as well as the slope of the magnetic ones. The axial form factor agrees less well with experiment. The pion form factor can be approximated by a monopole with a cut-off mass of 690 MeV.

Nucleon form factors provide deep insight into the structure of the nucleon and allow, therefore, for sen- sitive tests o f various subnuclear theories and their range of applicability. Particularly the electric neu- tron form factor is extremely difficult to reproduce because it requires rather detailed and accurate de- scriptions of the various fields involved. Actually, in recent years there is an increasing interest in effective non-topological [ 1-3] and topological [4 -6 ] chiral models because there are indications that they can be derived from QCD in the long-wavelength limit [ 7,8 ] and hence should be applicable to small momen tum transfers. Indeed, these relativistic models are quite good describing static properties of the nucleon and low lying baryon resonances [9,10 ]. With regard to dynamical properties only the topological models have been investigated [ 11,12 ] and they show only moderate success in reproducing the form factors, in particular the neutron electric one, even if vector me- sons are taken into account. The present paper ap- plies for the first time a non-topological soliton model to the form factors and it will turn out that such a calculation is successful.

We use the lagrangian of Gell-Mann and Levy [ 1 ] involving elementary quark-, sigma- and pion-fields

5~= cli y,Ouq+ ½ 0,o0,,(~+ ½0~'n.c3/,rc

- g O ( a + i y~r.n) q - U (o,~) . (1)

The mexican-hat potential U is slightly modified by an explicitly chiral symmetry breaking form

U ( o , ~ ) i 22 (02+71:2 p2)2..bm~Crc(l..l_const. ,

(2)

where

2 2 2 2 = ( m o - r n ~ ) / 2 f ] ,

p 2 = f 2 _ m 2 / 2 2 .

Here m~=0.138 GeV andf~=0 .093 GeV. Since the results are rather insensitive [9 ] to the value o f mo this is chosen to mo= 1.2 GeV and the quark-meson coupling constant g is fitted to the nucleon energy.

In the present approach the above lagrangian is solved by variational techniques involving angular- momentum and isospin-projected mean field states [ 9,13,14 ], the theory of which is described in ref. [ 9 ]. It involves a generalization o f the well-known hed- gehog structure this being necessary in order to fulfill the Goldberger-Treiman relation and the pion virial theorem [ 9,15 ]. Altogether the trial Fock state is ei-

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Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

genstate of angular momentum and isospin and reads as

J T I JTMMT ) = ~ gKKT PMKPMTKT [ ~/'/gh ) , (3)

with the generalized hedgehog mean field state [ ~Ugh ) = I q3h ) IX) I Hgh ). Here

qgh (r) = (u(r) , iv(r ) t~.~) (cos t/I uS ) --sin t/I d? ) ) ,

and IX) and I Hgh) are coherent states in the corre- sponding plane wave basis. The classical pion fields associated to IHgh) have the structure xqh( r ) , y~2(r ) , z~3(r) (with (J~l ~ ( P 2 ) f o r the isospin-1,-2, -3 component, respectively, the mixing coefficients g~,'m are obtained by diagonalization for frozen fields and given mixing parameter t/:

tl~(JT) p~jr),~J'I') ~ ~(Jr) = 0 (4) \ t t K K T K ' K , r - ~ , , K K T K ' K t F ! g K ' K t r ,

K ' K ~

with the hamiltonian overlaps

h ( J T ) t T H P KK' P]CTKr r [ ~//gh ,

and the norm overlap analogously. After the projec- tion the radial shapes of the fields are varied together with the generalized hedgehog parameter r/. This re- suits in a projection-before-variation procedure the details of which can be found in ref. [9].

In order to extract from I JTMMT >, which is nor- malized to unity, the nucleon form factors [ 16 ] we start from the definition ofF~ (q2) and F2 (q 2 ) ,

( N ( p ' ) I J ~ M ( 0 ) I N ( p ) )

=f i (p ' ) [F~ (q')7~'+ (i/2MN)F2 (q')aU"q~ ] u ( p ) , (5)

taking the electromagnetic current J~M as example. This corresponds to a nucleon-photon vertex. Here q=p' - p is the four-momentum transfer, IN(p) ) is the extended nucleon state with four-momentum p and u(p) is a plane wave spinor. The electric and magnetic form factors are related to F~ and F2 in the standard way as

GE (q2) =Fl (q2) +q2F2 (q2) / (2MN)2,

GM (q2)=F~ (q2)+F2(q2) •

For small momentum transfer [ql << 2 My we will make the static approximation [16] assuming the nucleon at rest before and after the interaction with the photon. Furthermore, we consider the projected

state ]JT) of the soliton a state of zero momentum. If one normalizes the IN ( p ) ) as

( N (p ' ) IN ( p ) ) = (2~)363 (p--p') E/MN ,

with rnN being the nucleon rest mass, one can make the correspondence

IN(0) > = [ (21r) 3 63 (0) 11/2 [ j = T= ½, M, ?kit > .

Thus in the Breit frame, where p~'= (E, ½ q ) ,p~= (e,+ lq) and q~'= (0, q), one can write

<N (½ q) IJEM (0) IN(- ½ q)>

= f exp( - i q.r) (JTMMT IJ~M (r) [JTMMT ) d3r,

(6)

yielding finally for e.g. a neutron with spin up

G~ (q2)=jjo (qr) (n~lJ°M (r) l n t ) d3r, (7)

- [Jl (qr) [r^ (n?IJEM (r) ln'f)]zd3r. G~ (q2)=~3 0 qr

(8)

The evaluation of the projected matrix elements in eqs. (7), (8) is complicated though straightforward and will be presented elsewhere.

The actual calculations are performed using the value g = 5.00 for the quark-meson coupling con- stant, which is obtained by fitting E (Jr) to the energy 0.938 GeV for the nucleon. The sigma mass has only negligible influence on the results and is hence kept fixed at m. = 1.2 GeV. There are no other adjustable parameters and the calculations, given as projected generalized hedgehog (PGH) in the figures, are fully variational. Fig. 1 presents the axial form factor. It is better than the one from the skyrmion model [ 11 ] but worse than the one of skyrmion with vector me- sons [12] (SKV). It may be that the inclusion of vector bosons in the chiral soliton lagrangian will im- prove this. Fig. 2 shows the magnetic form factor of the neutron, the features of the proton one are similar and hence not presented in this short note. Here the situation is in favour of the present calculations which go right through all experimental data up to ]qZ] ~< 0.7 GeV 2. Apparently the Skyrme models [ 11,12 ] (SK, SKV) show slight systematic deviations with SKV reproducing better the general trend. The electric proton form factor is displayed in fig. 3. Here the present values are in good agreement with the data

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Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

1.0

0.8-

J

0.2- --- SK --- SNV - - PGH 0.0 i i

0.0 0.1 0'.2 0[.3 0.4 0'.5 0'.6 017 q [OeV 2]

Fig. 1. The axial form factor of the nucleon: Shown are the results of the present calculations (projected generalized hedgehog, PGH ), of the skyrmion model (ref. [ 11 ], SK) and of the skyr- mion model with vector bosons ( ref. [ 12 ], SKV).

and one notices a systematic i m p r o v e m e n t going f rom the o rd ina ry hedgehog ( P H H ) to the genera l ized one ( P G H ) . In the Skyrme mo d e l there is a no t iceab le cor rec t ion due to the coupl ing of vec tor mesons al- though even the f inal SKV values are all slightly off. The mos t in te res t ing plots, however , are shown in fig. 4 regarding the electric n e u t r o n form factor. This is a

quan t i t y very sens i t ive to the detai ls o f the mode l used. Appa ren t ly the s k y r m i o n ( S K ) a n d the c loudy bag mode l [ 17 ] ( C B M ) are completely off. The SKY

1.0

.~ 0.5 " C L<<<<<<

- SK ..........

PHH

0.0 I PGH

0.0 0.1 0.2 0.3 0.4 0.5 -->2 q [c~v ~]

Fig. 3. The electric form factor for the proton: Displayed are the results of the present calculations (projected generalized hedge- hog, PGH), of the skyrmion model (ref. [11], SK) and of the skyrmion with vector mesons (ref. [12], SKV). The curves for the projected ordinary hedgehog (PHH) are also given.

yields a t r e m e n d o u s i m p r o v e m e n t , bu t still shows large dev ia t ions f rom exper iment . The projec ted gen- eralized hedgehog ( P G H ) , however, goes more or less right through the exper imenta l points. Thus one mus t say that the projec ted general ized hedgehog ( P G H ) reproduces very well the nuc l eon electric form fac-

tors and the slopes of the magne t ic ones. To our knowledge this accuracy is no t met by o ther relat iv- istic models . The ques t ion arises why the present

1.0

0 . 8 ~

0.6

0.4 " ~ ' ~ _ 0.2-

--- SK --- SKV -- PGH

0.0 0.0 011 012 0[3 014 015 0'.6

q [GeV 2]

Fig. 2. The magnetic form factor of the neutron: Shown are the results of the present calculations (projected generalized hedge- hog, PGH), of the skyrmion model (ref. [ 11 ], SK) and of the skyrmion with vector bosons (ref. [ 12], SKV). The curves for the proton look very similar.

0"10 f , "

o.o 1 /

i I .' 0.06 I , .. . . . . . . . . - . . . . . . . . . . . . . . . . . [ ~cr t " / /

- - P G H

o.o o'.1 0'.2 o[a 0'.4- q [OeV 2]

Fig. 4. The electric form factor of the neutron: Given are the re- sults of the present calculations (projected generalized hedgehog, PGH), of the skyrmion model (ref. [ 11 ], SK), of the skyrmion with vector bosons (ref. [ 12], SKV) and of the cloudy bag model (ref. [17], CBM).

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Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

projected calculations are generally superior to those of the Skyrme model. The reason lies probably in the semiclassical quantization method of the latter one. In a non-topological model one knows [ 9,13 ] that the overlap functions are too broad to justify a semiclass- ical quantization and it is plausible that similar defi- ciencies occur also in the topological models [ 9 ].

The pion-nucleon-nucleon form factor has been evaluated as well. Since the pion virial theorem is ful- filled [ 15 ] this can be done in an unambiguous way. The result is well approximated by a cut-off mass (monopole) of A=690 MeV, to be compared with the one of the Skyrme model [18] (582 MeV) and the vector Skyrme model [ 19 ] (850 MeV). All these values are of the order of magnitude needed in charge- exchange reactions [20 ] and estimated, using rather general arguments, by Schfitte and Tillemans [21 ] ( < 690 MeV). They are, however, smaller than those extracted phenomenologically from OBEP studies by Machleidt et al. [22] yielding about 1.3-1.5 GeV. This discrepancy is probably an indication that a proper description of the NN-force requires the con- sideration of processes more complicated than the NN n-vertex as e.g. genuine quark-gluon exchange, mutual polarization of the colliding solitons, etc.

We can summarize our points: Using projected generalized hedgehog states and the chiral soliton la- grangian we can well reproduce the electric and mag- netic form factors for both the proton and neutron. The results are surprisingly good and it remains to be studied how far center-of-mass corrections or the in- clusion of vector mesons [23] will change the con- clusions. Furthermore, one would like to know the extent to which the results are affected by the use of coherent states bases on localized meson quantum states rather than on plane waves. Another problem consists in the polarization of the Dirac sea, however, if one adopts the picture [ 8 ] that this creates the ki- netic energy of the mesons, the influence should not be too large. A more detailed account of the results presented here will be given elsewhere.

The work is supported partly by the NATO grant RG 85/0217, by the Bundesministerium ftir For- schung und Technologie, Bonn, and by the JNICT, Lisbon.

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