Meson-nucleon form factors in a chiral soliton model

Nuclear Physics A484 (1988) 593-619

North-Holland, Amsterdam

MESON-NUCLEON FORM FACTORS

IN A CHIRAL SOLITON MODEL*

N. KAISER, U. VOGL and W. WEISE

Institute of Theoretical Physics, University of Regensburg D-8400 Regensburg, FR Germany

U.-G. MEISSNER

Center for Theoretical Physics, Laboratory of Nuclear Science, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Received 10 March 1988

Abstract: We investigate meson-nucleon form factors within the framework of a non-linear chiral effective

lagrangian which includes pions and vector mesons (p and w mesons) as explicit degrees of

freedom. Nucleons emerge as topological solitons. We find meson-nucleon coupling constants

quite close to their empirical values as derived from phenomenological one-boson-exchange (OBE)

models of the nucleon-nucleon interaction. The range of the form factors corresponds to monopole

cut-off masses A- 1 GeV, not far from those typically used in OBE potentials. Furthermore, we calculate the spectrum, i.e. the mesonic mass distributions of the various meson-nucleon vertex

functions. We find that the spectral distribution of the pion-nucleon form factor is peaked close

to CL*= (m,+m,)Z=(900 MeV)‘. This suggests that the nN coupling invoives dominantly no

intermediate states, a result which is consistent with dispersion theoretic descriptions of the rNN

form factor. Finally, we investigate the spectral functions of our previously calculated nucleon

electromagnetic form factors in comparison with those of the related vector-meson-nucleon vertex

functions and discuss the range of validity of the vector meson dominance principle.

1. Introduction

Boson-exchange (OBE) models have established a remarr ably successful

phenomenology of the nucleon-nucleon interaction I). They OJY* te basically with

pions and vector mesons (the p and w mesons) as relevant degrees of freedom. The

finite size of the meson-nucleon interaction region is commonly parametrized by

monopole form factors F(q’) = (A*- m2)/(A2- q*), where m is the meson mass,

q* the squared four-momentum transfer at the vertex and A a cut-off mass. Precision

fits of OBE models to NN scattering data and deuteron properties ‘) lead to typical

cut-off masses in the range A = 1.0 - 1.5 GeV. For the pion-nucleon vertex function,

somewhat smaller values (A~~~ = 0.8 - 1.0 GeV) are obtained by a detailed analysis

’ Work supported in part by BMFT grant MEP 0234 REA and by Deutsche Forschungsgemeinschaft

grant We 655/9-2, and by US Department of Energy under contract no. DE-AC02-76ER 03069.

0375-9474/88/%03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

594 N. Kaiser er al. / Meson-nucfeon form facrors

of the np 3 pn charge exchange reaction “). Dispersion theoretic approaches ‘s4) have given similar results.

From a QCD point of view, the success of OBE models has often been looked at with a certain suspicious surprise. Whereas the long-range part of the NN interaction is undoubtedly governed by one-pion exchange, it was frequently argued that the physics at distances less than A$,,- 1 fm should be described in terms of quarks and gluons rather than mesons. In particular, the physical meaning of vector meson exchange has been questioned, given that their Compton wavelength h/ mvc -a fm falls well inside the 1 fm length scale which is characteristic of the nucleon size. On the other hand, various attempts to replace the simple and efficient meson exchange picture by seemingly better justified quark-gluon mechanisms have never (so far) really been able to match the quality of the fits to NN data obtained with meson exchange models.

Recent developments have pointed to the possibility of establishing connections between the meson exchange phenomenology and the underlying QCD theory. The guiding principle is the approximate chiral symmetry of the QCD lagrangian in the SU(2) flavour sector with almost massless u- and d-quarks, and its realization in the Nambu-Goldstone mode, with pions as Goldstone bosons. Much emphasis has been given to the derivation of an appropriate non-linear meson theory which approximates low-energy QCD (i.e. the strong interactions of hadrons at energy and momentum transfers up to about 1 GeV) 5,6). Such chiral non-linear lan- grangians, extended to incorporate vector-meson degrees of freedom ‘), and their soliton soiutions which represent baryons, have been used quite successfully to predict baryon properties with only few mesonic parameters as input ‘).

In this paper we shall investigate meson-nucleon vertex functions {the pion- nucleon form factor, the Dirac- and Pauli form factors related to the p-meson- nucleon and w-meson-nucleon couplings) within the framework of a non-linear chiral effective lagrangian including vector mesons. The p- and w-mesons are introduced as hidden gauge bosons ‘). With only three mesonic parameters fixed at their empirical values (the pion decay constant fw = 93.3 MeV, the pm-r coupling constant g = 6 and the pion mass m, = 139.6 MeV) this model accounts successfully for a variety of mesonic processes. It incorporates the vector meson dominance principle (VMD) lo) of electromagnetic hadron interactions which predicts the pion charge form factor quite accurately both for timelike and spacelike 4’ up to I$1 = 1 GeV2/c2. The model has soliton solutions with topological winding number B = 1; these solutions (Skyrme soiitons) are then identified with baryons.

In a recent publication I’) we have demonstrated that this model accurately predicts the nucleon electromagnetic and axial form factors for low momentum transfers 0 s q2 < 1 GeV2/c2. The proton electromagnetic r.m.s radii were found between 0.8 and 0.9 fm, while the r.m.s. radius associated with the axial form factor turned out to be smaller, 0.65 fm, in close correspondence with the empirical values. An important and characteristic feature of this model is the prediction that the r.m.s.

N. Kaiser et al. / Meson-nucleon form factors 595

radius of the baryon number distribution inside the nucleon is significantly smaller

than the electromagnetic radii: it comes out to be

(r~)“2-0.5 fm. (I)

The electromagnetic couplings are subject to VMD, and the propagating vector-

mesons enlarge the spatial extension of the relevant electromagnetic currents with

respect to the baryon number current. For example, the isoscalar charge radius

(I’,):/=~ of the nucleon is related to the radius of its baryon number distribution by

(&,=0=(r3+$, w

with M, = 780 MeV, the w-meson mass. This gives the proper isoscalar charge radius

(r’,):?,, = 0.8 fm which is in excellent agreement with its experimental value.

In the present work we are primarily interested in the spatial sizes of strong

interaction regions related to the pion-nucleon and vector-meson-nucleon coup-

lings. The paper is organized as follows. After a brief review of the basic non-linear

chiral effective lagrangian and a description of its B = 1 soliton solution, we shall

define the meson-nucleon source and vertex functions. It is important to note that

this can be done entirely in a classical context, in close analogy to the original

Yukawa approach. The resulting meson-nucleon form factors will be given for

spacelike momentum transfers q* < ~(G~V/C)~, and the corresponding source distri-

butions in r-space will be discussed.

Special emphasis will then be given to the investigation of spectral distribution

functions which appear in the dispersion integral representation of the meson-

nucleon form factors. These spectral distributions represent the mass spectrum of

intermediate multimeson states involved in the coupling of a given meson to the

nucleon. At this stage we can make interesting contacts with previous dispersion

theoretic approaches 3*4) b ase d on RN + nrr helicity amplitudes. Finally we examine

the spectral functions of the nucleon electromagnetic and axial form factors and

comment on vector or axial vector meson dominance.

2. Brief description of the model

Our starting point is the following chiral effective lagrangian of the coupled rpw

system:

+$g7. pL2u+1. poU]

596 N. Kaiser et al. / Meson-nucleon form faciors

(for a more detailed discussion see refs. 11,‘2)). Here the SU(2)-valued variable (the chiral field ) U(X) = exp (ir . n(x)/f,> incorporates the isovector pseudoscalar pion field ar(x), which is the Goldstone boson excitation of the spontaneously chirally broken vacuum. The pion decay constant fr = 93.3 MeV is determined by the weak semileptonic decay rr-+ /IV. The auxiliary variable t(x) is related to U(X) by t(x) = v’U(x) in the unitary gauge (5 = 6: = &); the latter gauge condition eliminates “scalar pion” degrees of freedom present in the hidden gauge symmetric lagrangian with independent & and & . (See also refs. 1’,12).) The universal coupling constant (equal for p and w mesons) g = g,,, is determined by the p + mr decay width. An important dynamical prediction of the model is the KSFR-relation m, = m, =

Jzfxg,m which determines the vector-meson masses and leaves us with only three independent parameters (f,,, g, m,), as mentioned before. Our value g = 5.85 is chosen to give m, = 772 MeV. The Wess-Zumino term which accounts for the chiral anomaly of the underlying QCD governs the u meson coupling to the topological baryon current

ePU@ BP=--- 24rr2 tr (a,UU+a,UU+~,UU+) , (4)

with the SU(3)-value of the w-coupling strength g,,, =zg. Furthermore, plLv =

alLoP -&P, + go& XP” and wBv = a,o, - i1,0, are the nonabelian and abelian p and w field strength tensors, respectively. The last term with a parameter A in eq. (3) is also part of the Wess-Zumino action and incorporates additional mpw couplings which are important for anomalous meson decays such as w+3rr in the Cell Mann-Sharp-Wagner model 13) and 7r” + 2y in the vector dominance model.

The lagrangian (3) with h =0 is called the “minimal model” since it couples the o meson to the baryon current B,, only; with A = 1 we refer to the “complete model” which respects more completely the anomaly structure of QCD.

One should note that chiral non-linear effective meson theories such as the one represented by the effective lagrangian (3) have links to the underlying strong- coupling QCD in the context of the large-N, expansion as proposed by t’Hooft and Witten r4). Furthermore, recent developments employing partial bosonization techniques 15) have pointed to the connection between non-linear meson theory, including vector mesons, and an extended version of the Nambu and Jona-Lasinio model 16) which can be regarded as a low energy QCD effective lagrangian involving only quark degrees of freedom.

3. Nucleons as solitons

Nucleons emerge from the effective lagrangian (3) as soliton solutions in the sector with baryon number B = 1. The investigation of such soliton solutions starts from an ansatz of maximal symmetry (the hedgehog ansatz)

U(r) = exp[ iT * W(r)] , (54


G(r) pia(r) = EikaXk_ gr2 ’

COO(r) = w(r), (SC)

for the classical pion, p and w field configurations. The energy functional is then

given in terms of the radial profiles F(r), G(r) and w(r) which are determined by

numerical solution of the coupled field equations. The basic relations are summarized

in appendix A. The baryon number distribution inside the nucleon is expressed in

mesonic terms as

F’(r) sin2 F(r) B,(r)= - 2m2r* .

The boundary conditions F(0) = r and F(a) = 0 for the radial pionic profile (the

chiral angle) make sure that B = j d3rBo( r) = 1. The r.m.s. radius of the baryon

density, RB = ( &Ii2 with

(&= -31” dr r2F’( r) sin2 F(r) , (7) 0

turns out to be RB = 0.5 fm. This number sets the size scale for all subsequent form

factor discussions.

3.1. THE PION-NUCLEON FORM FACTOR

Consider as a specific example the rNN form factor GrNN(f) defined by

(N(P’)KXO)IN(~))= G~NN(~)U(P’)~Y~~~U(P), (8)

where u(p) and u(p’) are the nucleon spinors with in- and outgoing four-momentum

p, and p: and t = q: = (p: -p,)’ is the invariant squared four-momentum transfer.

In the Breit frame with p’* = (E, iq) and pp = (E, -$q), the energy transfer q.

vanishes. In this case eq. (8) reduces to

(N~(fq)JJ”,(O)INi( -b))= -iG+d-q2)x: 2 TaXi t-4

(9)

with two-component Pauli spinors/isospinors x (here MN is the nucleon mass). We

can now identify, at the classical level, the right-hand side of eq. (9) with the Fourier

transform of the static source distribution of the pion field q”(r) given by the pion

field equation

as follows:

(V’- m’,)q”(r) = J”,(r) (10)

a*q -i- TaGmNN(-q2) = j d3r eiprJc(r)

2Miv

= -(q2+mt) I d3r eiq’r(p”(r). (11)

598 N. Kaiser et al. / Meson-nucleon form factors

The proper pion field pa is found from the coupling of the chiral field U to any

isodoublet left- or right-handed fermion $R,L = i( 1 f rs)$:

-_&[&WR+ L~+IcIJ = -iM~Cr tr U+ JIwJI. tr (7Wl. (12)

Using the quantization rule (see also appendix B) for the rotating chiral field

U( r, t) = A(t) U( r)A+( t), namely

tr (AT”A+T~) = -$(+‘T~, (13)

we find for the pion field

cpa(r) = -& tr (A7 * ;A++‘) sin F(r)

=3 g ’ f * W sin F(r)

The rNN form factor becomes

(14)

(15)

The corresponding source function

J”,(r) = -u . WS,( r) (16)

has a radial distribution S,(r) which is explicitly given in appendix C, eq. (C.3)

[see also ref. i7), where a different sign convention has been used].

3.2. VECTOR-MESON-NUCLEON FORM FACTORS

Let us next consider the vertex functions for the coupling of p and w mesons to

nucleons. The vector meson source functions J”,(x) are defined by

(v* - m&J:(x) = J:(x), (174

(V’- m2,)w,(x) = JO,(x), (17b)

with a = 0 for the isoscalar o-meson and a = 1,2,3 for the isovector p-meson. Their

matrix elements between nucleon states define the Dirac and Pauli form factors

F,(t) and F*(f) as follows:

W(p’)lJ;(O)IWp))= Q’) [

F:(f)y,+&F:(f)W’. +W) (18) N 1 (j = p or w) with 7’ = 1 for the w-meson and ~~(a = 1,2,3) for the p-meson. It is

convenient for our purposes to introduce the vector and tensor coupling constants

gv and g-r by the values of F,(t) and F2( t) at t = 0 (note that the usual definition

of the coupling constants is at the meson pole, i.e. at t = rn; or r = m:).


It is often convenient to work with “electric” and “magnetic” vector-meson-

nucleon form factors

GP”(t)=Ff”(t)+-& FY(t), N

G$“(t)= FT”(t)+Fp”(t). (19)

In particular, in the Breit frame with t = -q2, GE and GM are identified with the

Fourier transforms of the time and space components of the source functions Jz;

(Nf(~q)IJ,“(0)INi(-lq))=G,(t=-q2)X:7a~i 7 (204

(Nf(iq)JJ"(O)J Ni(-$a))= G"~~-q2)X;iUX qTaxi. N

(2Ob)

the source functions are re-expressed in terms of the corresponding radial distribu-

tions Sk;‘(r) as follows:

&Y(r) = S:(r) 3

J&(r) = %(~)~cl,

T(r)=S”,(r)axi,

J:(r) = Sh(r)a x ha. (21)

In terms of these radial source distribution, the form factors (19) are expressed as:

J 02

G;“(-q2) = 47r dr r2j,,(qr)S$“‘(r), (224 0

J cc

GGw(-q2) =87T(kf,/q) drr2j,(qr)SGw(r). 0

with q = (q(. The explicit results are given in appendix C.

Wb)

4. Spectral representations of form factors

In the preceding section we have shown how to calculate meson-nucleon form

factors in the Breit frame, with t = -q* < 0, by taking the Fourier transforms of the

corresponding source functions as they emerge in a chiral soliton model. We shall

now establish contacts with dispersion theoretic approaches and discuss the

“mesonic content” of these form factors in terms of their spectral distributions.

Dispersion theory treats the various vertex form factors as functions in the complex

t-plane. On the real axis t = qt is the invariant four-momentum transfer. The t-plane

has a cut along the positive real axis extending from t = t,, to infinity. The cuts

starting at to = 4rn: for the p-meson-nucleon vertex function and at to = 9mt for

the rr and w nucleon form factors reflect the kinematical thresholds for the p + ITT

and n + 3 r, w + 3 n channels, respectively.


In our case, the integral representations (15) and (22) of the meson-nucleon form factors, written for spacelike t = -4’ in the Breit frame, can be followed into the timelike region up to t = to by analytic COntinUatiOn using imaginary q = in. The

branch points are determined by the asymptotic behaviour (r+ CO) of the meson

source functions: we have to = p2 if S(r) - emPr for r + co. Hence the large-distance properties of S(r) directly reflect the thresholds of the multipion continuum. More precisely, this result follows from the asymptotic behaviour of (r + 00) the meson profiles,

(234

G(r) + e-2m-r/ rn”,r2, Wb) o(r) + e-3mJ/ m4,rs, (234

and of the radial profiles of “excitations” as discussed in appendix B.

Cl(r), L(r) -+ e-2m-r/4Cr21 (234

+(r)-,e-3”-‘/m4,r3, We)

Note that the chiral angle F(r) has the well known asymptotic Yukawa form characteristic of the pion field from a static source, whereas the vector-meson fields do not fall off exponentially with their masses m, = mP = m,. The reason is.that the 2~ and 3n source terms dominate the behaviour for large r in the p- and w-meson field equations, respectively. In fact, the non-linear meson theory represented by the lagrangian (3) has just the proper ingredients to account for the relevant p -+ 27r and w + 3~ dynamics.

As analytic functions in the (cut) complex t-plane the form factors* satisfy unsubtracted dispersion relations which lead to the spectral representation

for t ~2 to. (24)

The spectral function F(p*) is one half times the discontinuity of the imaginary part of the form factor G(t) across the branch cut to’l=- t <CO along the real t-axis. This function represents the mass spectrum of multimeson states involved in the meson-nucleon coupling.

The spectral distribution r(p’) is evidently of considerable interest when compar- ing results based on the chiral soliton model (3) with conventional phenomenology of the NN-interaction ‘) and with models of form factors based on dispersion theoretic approaches 3*4). One expects that the mass & at which r(p’) has its maximum determines the characteristic range R - ~6’ of the region over which the meson-nucleon interaction takes place.

l For our model form factors we assume that the analytic behaviour in the upper t-halfplane is such

that eq. (24) holds.


The evaluation of T(P*) requires to invert the relation (24). This inversion

procedure is described in detail in appendix D.

5. Results

In this section we present results for the spectral distributions r(p*) of all relevant

nucleon form factors: the meson-nucleon vertex functions discussed in sect. 3, and

in addition the electromagnetic and axial form factors. It will then be interesting

to locate the dominant parts of the meson mass spectra represented by r(p*) and

discuss their physical interpretation in comparison with other models.

5.1. SPECTRAL DISTRIBUTION OF THE rNN FORM FACTOR

Consider first the pion nucleon form factor (15) derived from the source function

S,(r) of eq. (C.3) (appendix C), and given by its spectral representation as

(25)

We show in fig. 1 the result for the form factor GmNN( t) in the spacelike region t < 0

and up to the branch point at t = 9m2, which represents the threshold of the three-pion

continuum. For t > 9mt the form factor becomes complex, with

lim Im GmNN( t + ie) = r,,,(t). E-.0+

(t29mZ,) (26)

The techniques described in appendix D lead to the spectral distribution rxN,.,(t)

as shown in fig. 1. This spectral function has a maximum at about t = p* = 49mt,

not very far from (m, + m,)*. This indicates the important role of rp contributions

to the rNN form factor. This observation is consistent with the calculations of

refs. 18*4) which show that G rNN(t) is dominated by a triangular diagram with rrp intermediate states. In particular, ref. 18) also points out that there is a maximum

in the spectral function at p2=49m’, on top of a broad 3~ continuum. These

phenomenological results find their clear correspondence in the present approach.

In fact one observes from eq. (C.3) (appendix C) that the radial pion source function

S,(r) has the following asymptotic form for large r:

.S,(r)+~F~(r)+bF(r)G(r), (27)

where a and b are constants of the same order of magnitude. Note by comparison

with eqs. (5) and (16) that F3 and FG represent 3 7~ and rp field configurations. It

is not meaningful, however, to disentangle the 3~ and rp components, since the

p-meson itself is a two-pion composite in our model.

The shape of the pion source function S,(r) in fig. 2 illustrates that most of the

pion-nucleon interaction strength is located in the region r < 0.6 fm. This figure also


ntW ONPLING

LO

30

20

-r-

,'

t 0

1 2.0

1 _I

Fig. 1. The pion-nucleon form factor G,,, (t) for values of t below the branch-point at 1, =9mt and

the spectral function Im G,,,(t). The dot-dashed curve shows the corresponding real part for t > 9mf,.

nNN FORM FACTOR

Gnd:*) 20 - --

PION SOURCE FUNCTION

lfdl S,(r)

L-

-9m;O. 0.5 10 gi21GeV21 0 1.0 rifml

Fig. 2. Left: The pion-nucleon form factor G,,, q ( *) (full line), compared with the monopole fit obtained

in ref. ‘) (dashed line) with g,,, = 13.7 and A, = 890 MeV. Right: Profile of the radial pion source

distribution S,(r).

shows a comparison between our predicted GmNN(-q’) and a typical phenomeno-

logical result ‘) deduced from the analysis of the proton-neutron charge exchange

reaction.

For small values of t the form factor can be parametrized in the monopole form

GnNN( t) = g,,,,(R~ - mt)/(A’, - t). The empirical values A, = 0.9 - 1.3 GeV corre-

spond to root mean square radii (r*)~~ =&/A, in the range 0.4-0.6 fm. Our


predicted value for A, obtained from a monopole fit to the low q2-region (A,, =

0.85 GeV) is within the range of uncertainty of the empirical form factor. The range

parameters and the coupling constant GTNN(0) = g,,,( 1 - m”,/A’,) are summarized

in tables 1 and 2.

5.2. VECTOR-MESON-NUCLEON FORM FACTORS

The first interesting observation about the calculated vector-meson-nucleon form

factors is that the relative strengths of vector and tensor couplings follow the general

systematics required by NN-interaction phenomenology: the w-meson-nucleon

TABLE 1

Meson-nucleon coupling constants at t = 0, and tensor-to-vector coupling ratios ~~ =

F$(O)/Ff(O) and K, = F;(O)/F;(O) for the p- and w-mesons

Quantity Minimal model

(A =O)

Complete model

(A = 1)

Empirical value

[ref. ‘)]

GNN(O) 14.74 14.05 13.3

Ff(O) 2.67 3.24 2.3

K(O) 14.37 14.11 13.9

% 5.38 4.36 6.1

F;(O) 8.78 8.78 11.5

F;(O) -1.85 -0.59 -0

a” -0.21 -0.07 -0

The empirical values correspond to the OBE potential ‘).

TABLE 2

Size parameters of the most important meson-nucleon form factors

Quantity Minimal model Complete model

(A =O) (A = 1)

Empirical value

[ref. ‘)]

&[GeVl 0.85 0.86 0.89 ‘) (1.3)

A,[Gevl 0.91 0.95 (1.4)

&[GeVl 0.98 0.86 (1.5)

(r2)%13ml 0.57 0.56 0.54 2)

(r2)$.Xfml 0.53 0.5 1

V)%bl 0.49 0.52

The A,(i = 7, p, o) are cut-off parameters in equivalent monopole fits (1 -t/Af)-’ to

the form factors G,+&t)/GmNN(0), F$‘(t)/FP(O) and F;(t)/F;(O) around t =O. The

* corresponding r.m.s radii are given by (r ), “z=~/Ai. Empirical values in brackets are

from the OBE potential fit ‘), apart from the A, taken from ref. *).

604 N, Kaiser er al. / Meson-nucleon form factors

vertex function is dominated by the -y*-type vector coupling, whereas the p-meson- nucleon vertex function is dominated by the a,,,q”-type tensor coupling.

The results for the oNN vector and tensor form factors F:(t) and F;(t) and their spectral functions are shown in figs. 3 and 4. The spectral distributions have

wNN VECTOR COUPLING

10 -\ I5 : i. tiGeV*/czl

Fig. 3. The o-meson vector coupting form factor F;( 1) below the branch point at q,=9m~ and the

spectral function Im F;( 1). The dot-dashed curve shows the corresponding real part Re F;(t) for I 3s 9m$

+l / / t [G&c*

WMI TENSOR COUPLING I i

IIm F;(t) 1

Fig. 4. The w-meson tensor coupling form factor F;(t) below the branch point at t,=9m: and the

spectral function Im F;(t). The dot-dashed curve shows the corresponding real part Re F;(t) for f > 9,:.


their maxima around t = 1 GeV2 which corresponds to a characteristic radius of

about 0.5 fm. Note that the ratio of oNN tensor to vector coupling at t = 0,

K, = FF( t = O)/FP( t = 0), is negative and small (between about K, = -0.1 and -0.2

for the “complete” and “minimal” model) and close to the empirical anomalous

isoscalar magnetic moment of the nucleon, K~ = -0.12.

In the “minimal model” with A = 0 (see eq. (3)), the source function Sg( I) of the

w-meson field shown in fig. 5 is just the baryon number density (6) multiplied by

the wNN coupling constant g,,, = G:(O) = sg, hence the radius ( r2)yi = ( r$“2 =

0.5 fm. Correspondingly, an equivalent monopole fit to the leading w NN form factor

F:(t) at small t yields a cut-off mass A, close to 1 GeV.

If,+31 VECTOR MESON SOURCE FUNCTION!

4-----

0 0.5 r[fml 1.0

Fig. 5. Radial distributions of electric vector-meson sources for the w-meson SE(r) (full line) and the

p-meson SE(r) (dashed line).

The p-meson-nucleon form factors Fy( t) and F$Y( t) together with their spectral

distributions are shown in figs. 6 and 7. As mentioned, the tensor coupling represen-

ted by F; dominates by far. Its spectral function has a maximum just below

t = 1 GeV2 and represents the dynamics of the VT-continuum together with field

configurations of higher mass such as 47r, WV, pp etc. The tensor-to-vector coupling

ratio ~~ = F,P(O)/Ff(O) = 5.4 (for the minimal model) is not far from the empirical

K~ -6.1. The radius of the pN interaction region has again the typical size (r’)$ =

0.5 fm. The precise values of pNN coupling constants and range parameters are

summarized in tables 1,2. Finally, the comparison of the “electric” pNN source

function S;(r) with the corresponding one for the W-meson in fig. 5 shows again

the dominance of the o-meson vector coupling over that of the p-meson.

In addition to the imaginary parts Im F(t) of the meson-nucleon vertex functions

we have calculated their real parts ReF( t) for t 2 to in our model, by using a similar

method of analytic continuation as explained in appendix D.

N. Kaiser et al. / Meson-nucleon form factors

VECTOR COUPL~MI II

Fig. 6. The p-meson vector coupling form factor F:(t) below the branch point at t, = 4rn: and the

spectral function Im Fy( t). The dot-dashed curve shows the corresponding real part Re Fy( t) for t > 4mi.

I

I 1 I 1 I 1 I

-2.0 -1.5 -1.0 -0.5 OA 0.5 1.0 \ brnf, .GieVl/c?

Fig. 7. The p-meson tensor coupling form factor F;(r) below the branch point at t,=4m: and the

spectral function Im F%(I). The dot-dashed curve shows the corresponding real part Re F;(r) for t > 4mi.

The dot-dashed curves in figs. 1, 3, 4, 6 and 7 show the results for the real parts

of the pion and vector-meson form factors in the region above the branch point to.

They have the general feature that they pass through zero close to values t = p2

at which the corresponding imaginary parts r(p2) have their maxima. Such a

behaviour is somewhat reminiscent of an approximate Breit-Wigner shape with a

resonance-like structure located at the maximum of the spectral function r(p2).


5.3. SPECTRAL DISTRIBUTIONS OF ELECTROMAGNETIC AND AXIAL FORM FACTORS

The electric and magnetic form factors of the nucleon have been evaluated and

discussed in details already in ref. ‘l) where it was found that the proton and neutron

charge and magnetic radii are well reproduced as a consequence of the underlying

vector meson dominance of electromagnetic couplings: the photon sees the nucleon

primarily via the vector-meson content of the chiral soliton.

The VMD principle manifests itself in an exact form in the isoscalar sector, by

the current-field identity:

which relates the isoscalar electromagnetic current directly to the w-meson field. As

a consequence, the isoscalar nucleon charge form factor becomes

G;(f)= m’ ’ - - G;(t) m’,-t 3g

m2 1 =w-- m2,-t 3g

E(t) . I

and the isoscalar magnetic form factor is

G&(t)= mt, ’ - - G;(t) mt-t 3g

m2 =A +F;(t)+F;(t)].

m2,-t 3g

(29)

It is then instructive to investigate the spectral distribution of the isoscalar electro-

magnetic form factors, defined by

G&,(t) =’ J cc

d t, %4< t’) t’-t ’

(ts cl) = fo

(31)

with

Z$,( t’) = lim Im G&,,( t'+ is) , E+O+

(32)

in comparison with the spectral functions of the corresponding oNN form factors:

their difference reflects the role of the propagating vector meson which connects

the photon with the soliton. The results shown in fig. 8 clearly demonstrate the way

in which the maximum in the spectral function of the electromagnetic form factors

is shifted towards lower masses. This feature translates into the characteristic

difference between the baryon radius ( ri)“2 = 0.5 fm and the isoscalar charge radius

(&“=0.8 fm (see eq. (2)).


I ISOSCALAR SPECTRAL F~CTI~S

Fig. 8. Left: Spectral functions of the nucleon isoscalar electric form factor Im G:(t) (dashed line) and

of the w-meson “electric” form factor (1/3g) Im GE(t) (full line). Right: Spectral functions of the

nucleon isoscalar magnetic form factor Im G&(t) (dashed line) and of the w-meson “magnetic” form

factor (lf3gf Im GE(t) (full line).

In the isovector sector, the VMD current-field identity

J;,e.m.(X) = -tip,(x)+. . * g holds to leading order, but not exactly, with corrections involving at least 27rp or 471 terms. Nevertheless, the behaviour of the electromagnetic isovector form factors at 1 t1-c 0.5 GeV2/ c2:

and

m2 1 =P- m2,-t g F:(f)+-& F;(r) ,

N 1 (34)

(35)

is governed by the leading p-meson dominated terms in eqs. (37) and (38). The spectral functions of G&.,(t) in fig. 9 show again the characteristic shift to lower masses as compared to the spectral distributions of the pNN vertex functions. They also illustrate once again the dominance of the magnetic over the electric couplings in the isovector channel. The electromagnetic tensor-to-vector coupling ratio K~ =

F:(O)/ F:(O) which determines the isovector anomalous magnetic moment comes out to be K” = 4.93 for the “minimal model” and K” = 3.60 for the “complete model”, to be compared to the empirical K~= 3.7 [see also ref. ‘I) for a more detailed


I ISOVECTOR SPECTRAL FUMCTICM 1 l-

2-

6-

I/ ‘-.,Irn G:(t)

\ \

5- ' \ I

\ \ \

0.5 1.0 '\ 1.5 20 Lrng ’ ytiGeV% ‘1

%N_ 1

0.5 1.0 A 4m;

1.5 2.0 t[GeV% 21

Fig. 9. Left: Spectral functions of the nucleon isovector electric form factor Im G:(t) (dashed line) and

of the p-meson “electric” form factor (l/g) Im GE(t) (full line). Right: Spectral functions of the nucleon

isovector magnetic form factor Im G;(r) (dashed line) and of the p-meson “magnetic” form factor

(l/g) Im G&(t) (full line).

discussion]. We note that the large empirical ratio Kp/ K” = 1.7 cannot be reproduced

in the present approach: we find K,/K” = 1.1- 1.2 only.

Finally, consider the axial form factor GA(t), the one related to the matrix element

of the nucleon axial current A:(x) in the Breit frame as follows

=xf’ ; G,(-q2) u,+(Gkq2)-& G(-q2)h N N I +Xi, (36)

with uL = u - i(u * tf), uL= $(u * Q) and E = J+q’+ ML. The induced pseudoscalar

form factor is dominated by its pion pole contribution and directly related to the

7rNN form factor according to the PCAC relation MN[ GA( t) + ( t/4Mk)Gp( t)] =

[(m’,f~)l(m’rr-f)lG?r~N(f). Th e evaluation of the axial form factor in terms of the

meson profiles of the soliton gives (see ref. ‘I)):

STMN co G,(-q2) = -T E

I dr r2Udqr)A,(r) +

0

‘y A2(r)] (37)

with

A,(r)=‘$(2(G+I)sinF-fsinZF)-s$(sin’F--hGcos’F), (3ga)

A2(r) = -A,(r)+f:F'(r) . (38b)

The present model does not have the A, meson as an explicit degree of freedom,

610 N. Kaiser et al. ,I Meson-nucleon form factors

l-

/ SPE~RAL F~CT~N OF AXIAL ,FORM FACTS /

I I 1

0.5 1.0 1.5 2.0 9mf tlGeV2/cz1

Fig. 10. Spectral function of the nucleon axial form factor Im G,.,(t).

but it appears implicitly, through the coupled rp system carrying A, quantum numbers, i.e. Jp = l+ and isospin I = 1. It is therefore interesting to accompany the (successful) calculation of GA(-q2) reported in ref. I’) by an analysis of its spectral distribution r,(t), with

J co

G*(I)=~ - dt’ r*(f)

r 94 f--t ’ W.4

r,( t’) = lim Im G,( t’-t- is)

E+O+ Wb)

Note that the one pion contributions to GA(t), present in the densities A,(r) and A,(r) of eqs. (38), cancel upon integration by parts. They ,give rise to the pion pole in GP( t) which is also expressed in terms of A,(r) and A,(r). The threshold at which the cut on the real t-axis starts in eq. (41a) is then correctly located at to= (3m,,)*. The resulting spectral dist~bution shown in fig. 10 has a broad peak with a maximum slightly above t’ = (0.9 GeV)2, not far from the mass rnA = 1.05 GeV of the A, as it has been found recently in the analysis of T + Al + neutrino decays 19).

6. Summary and conclusions

We have investigated pion-nucleon and vector-meson-nucleon vertex functions within the framework of a non-linear chiral meson theory in which nucleons emerge as solitons. Such an approach is motivated by the idea that the underlying non-linear effective lagrangian (3) approximates QCD at energy and momentum transfers up


to about 1 GeV. The physics of baryons, including their various form factors for

electromagnetic, weak and strong couplings, is then described in terms of only three

mesonic’parameters: the pion decay constant f= = 93.3 MeV, the pion mass m, =

139 MeV and the universal vector-meson coupling constant g = 6 (the latter one is

actually predicted as e = g = 27r from the coefficient e of a Skyrme 4th-order term

in a scheme which derives the effective meson lagrangian by partial bosonization

from a chiral fermion lagrangian such as the Nambu Jona-Lasinio model, see e.g.

ref. “)).

Our main interest here is in the following question: to what extent can such a

unified theory of mesons and baryons account for the established phenomenological

meson-nucleon form factors derived either by dispersion theoretical methods or

from a best fit to nucleon-nucleon data using boson exchange models?

Our findings can be summarized as follows:

(i) The predicted meson nucleon form facors have typical r.m.s. radii in the

range 0.5-0.6 fm, close to the radius (ri)“’ = 0.5 fm of the baryon number distribution

inside the nucleon, which is in fact the source function of the o-meson field in this

model. Monopole fits to these form factors have therefore typical cut-off masses

around A - 1 GeV.

(ii) The resulting pion and vector-meson couplings to the nucleon have all the

relevant features familiar from the phenomenology of the NN interaction. In

particular, the pNN interaction has a dominant tensor coupling, whereas the wNN

interaction is governed by its vector coupling.

We have been primarily interested here in the calculation of the spectral functions

of meson-nucleon form factors which represent the mass distributions of multimeson

intermediate states involved in these form factors. In fact, non-linear chiral meson

theory provides an appropriate framework in which the relevant multimeson

dynamics is treated non-perturbatively.

We find that the mass spectrum of the rNN vertex function is dominated by

interacting pn field configurations. The maximum of the spectral distribution occurs

slightly above fi = rnrr + m,, the sum of the n- and p-meson masses. This result is

in close correspondence with dispersion theoretical approaches, such as the one by

Durso et al. “) and Nutt et al. ‘*), which use the pseudo-physical TS-+ NN helicity

amplitudes of Hohler and Pietarinen 2’) as input. The fact that our predicted rNN

form factor compares well with phenomenological analysis is quite encouraging,

given that this prediction uses minimal input entirely from the meson sector. It

suggests that the successful meson exchange picture of low energy strong interactions

does in fact have a natural basis in QCD, via the bosonic effective lagrangian (3).

The vector-meson-nucleon vertex functions should be seen in close contact with

the electromagnetic ones. For example, the leading vector coupling term of the

w NN form factor, which coincides with the form factor of the baryon number

distribution, is related to the isoscalar charge form factor by the vector meson

dominance (VMD) principle. Whereas VMD is exact in the isoscalar channel, it is


however, only approximate in the isovector case. The isovector electromagnetic

form factors as compared to the pNN form factors show a leading VMD behaviour

at small ItI, but modifications arising from higher order terms are substantial, as

already pointed out in ref. I’). The ratio K,/K” between K~ = g$INN/giNN (where

ggNN and gyNN are the p-nucleon tensor and vector coupling constants taken at

t = q* = 0) and the anomalous isovector magnetic moment K” comes out to be too

small compared to the empirical K~/K”= 1.7.

The reason is that the predicted K” is somewhat too large as compared to the

experimental K” = 3.7, whereas the calculated value of K~ (see table 1) comes out

not far from the empirical K~ = 6.1. An important observation is that the rNN and

pNN form factors cooperate in just such a way that the resulting isovector tensor

NN potential compares favourably 17) with the one deduced from a detailed NN

phase shift analysis.

In summary, we argue that basic features of the meson exchange phenomenology

of the NN interaction do have their foundation in QCD, in the sense that they

reflect the fundamental symmetries of QCD as they are realized in effective non-linear

chiral meson lagrangians. On the other hand, it should be noted that other important

properties such as the intermediate range attraction in the nucleon-nucleon potential

due to two-pion exchange, cannot be understood in the present framework. This is

mainly believed to be a consequence of the limitation implied by the classical

treatment and simplifications in the quantization procedure used so far. Considering

the dispersion theoretical relationship between the 27r exchange NN interaction

and the rrN scattering amplitude, the first obvious step is to understand p-wave

pion-nucleon scattering from the point of view of non-linear meson theory with

chiral solitons. Work along this line is in progress 22).

Appendix A

ENERGY FUNCTIONAL AND EQUATIONS OF MOTION

We present here the static energy functional and the equations of motion derived

from the chiral non-linear meson lagrangian 9 of eq. (3). Using the ansatz of

maximal symmetry (5) (the hedgehog ansatz) for the meson fields, the energy

functional for the “minimal” (A = 0) and “complete” (h = 1) model becomes

MH= - d3x9’=4m I [a

E(F, G, w) dr (A-1) 0

with the integrand given as

E(F, G, w) =fi($r*F”+sin* F+2(G+ 1 -cos F)*) r2

+f2,mz,r2(1-cos F)+E+ g2

~(G+2)2-~r2~f2-fr2&02

+$ (4wF’ sin* F + hw’G( G + 2) sin 2F) . (A.3


The equations of motion for the meson profiles follow from the usual Euler-Lagrange

variation of (A.2). We have

F”=-~F’+~sinF(G+l-fcosF)++-- 3g 0f[hG(G+2)cos2F-2sin’ F] 87r”f’, r2

+ m’, sin F, (A-3)

G”=~(G+l)(G+2)+rn~(G+l-cos F)+$Iw’(G+l)sin2F, (A.4)

2 3g @‘:= -_w’+m2” -- 0 r 47r2r2

F’ sin’ Ft +-&[G(G+2)F’cos2F+Gr(G+I)

sin 2F], (A.9

These nonlinear coupled equations* are solved numerically. The relevant boundary

conditions to ensure baryon number B = 1 are F(0) = n- and F(m) = 0. For the

vector-meson field we have to impose G(0) = -2, G’(0) = 0, G(K)) = 0, o’(O) = 0,

w(m) = 0 in order to obtain solutions with finite energy and free of singularities.

The detailed form of the meson profiles can be found in ref. I’).

Appendix B

MOMENT OF INERTIA AND VECTOR MESON EXCITATIONS

The B = 1 soliton solution of eqs. (A.3-5) has neither good spin J nor good

isospin I; only the sum K = I+ J = $7 - ir x V is a good quantum number. The

construction of states with good spin and isospin is done by following a semiclassical

quantization procedure. Inserting the ansatz

U(r, t)=A(t)U(r)A+(t), (B.1)

T+P(~, t)=A(t)~*pi(r)A+(t), 03.2)

A(t)=exp[i~.fit] (B.3)

for the adiabatically rotating hedgehog, with a small frequency /al, into the effective

lagrangian and introducing the profiles l,(r), 12(r) and 4(r) of the vector-meson

excitations by

~.p.(r,1)=~A(f)r.[ni,(r)+;n.G(r)]a’(r), (B.4)

W(r,t)=mlxP, r

’ Here we have corrected several misprints which appeared in ref. I’).


one finds the Lagrange function of the spherical rotator:

L(t) = -Mu+ 0 tr (AA’),

The energy becomes

03.6)

withI=J=$,s,.. . ; the mass splitting between nucleon and A-isosbar is MA - MN =

3/20. The moment of inertia 0 is a functional of the classical profiles F, G, o and of the vector meson excitation l,, &, 4;

I

cc @=47r A(F, G, w; CC,, f;, 4) dr (B.7)

0

with the integrand*

A(F,G,w;f‘,,52,4)=~~fz,r2(sin2F+8sin4(~F)+8sin2(~F)(I,+35:+25152+522))

+-$ (bF’(2 sin’ F - Al, - h12)

+A&4’(G-GS1-<r)sin2F-~ 412+$42+rn2y42 ( >

+& (3r2c:‘+2r25:l;+ r2~;‘+4G2(~f+~,12

-2&-52+1)+2(G2+2G+2)5;).

The equations of motion for the vector-meson excitations are determined as the Euler-Lagrange variational equations which minimize the moment of inertia 0:

2 t: = --I I; + rns{, -4g’f ‘, sin’ ($F)

-,I&4’(G+l)sin2F+$(G2(I,-I)-2(G+l)&), (B.9)

[;= -~&+m:5,+4g2~~sin2(~F)-~$&4F’

+,I&4’(G+1)sin2F+~(G2(~,-1)+2(G2+3G+3)~2), (B.lO)

2 4”=;i4+m:4-~F’(sin2F-A~,-h~2)

+~$-$(~F’(G-G~,-~,)cos~F+(G’-G’~’,-G&-~:)~~~~F). (B.ll)

l Here we have corrected some misprints which appeared in ref. ‘I).


These equations are solved numerically subject to the following boundary conditions:

G(0) = 0 = 51(a) 9

5x0) = 0 = 52(a) 9

4(O) = 4’(O) = 0 = 44~) 9 (B.12)

and the constraint at r = 0

25,(O) + L(O) = 2 * (B.13)

See ref. rl) for further details and for plots of the classical meson profiles and of

the vector meson excitations.

Appendix C

MESON-NUCLEON SOURCE FUNCTIONS

In sect. 3 we have defined meson-nucleon source functions by rewriting the

coupled non-linear meson field equations (A.3-5) in the usual laplacian form

(V’- m*)rp = J. The source terms J on the right-hand side are non-linear functions

of the meson fields. They display the meson interactions in the soliton configuration.

Here we shall summarize the results for the radial source functions S(r) defined

in eqs. (16), (21). Using the quantization rules fi = a/40, tr(Ar . CBA+T) = -r/20

and tr(Ar*A+rP) = -fu*/ (see also ref. “)). Together with the ansatz (B.l-5), the

static pion and vector-meson fields become

d(r) = - j3idr)+$C*(r)l,

G(r) pa(r)=aXiTa- 3gr ’

The corresponding meson source functions

J:(r) = -u * WS,( I) )

J&(r) = T&(r),

Jz( r) = u x &,SL( r) ,

-G(r) = f%(r) ,

J”(r) = u X is;(r),

(C.1)

(C-2)

616 N. Kaiser et al. / Meson-n~eleon form factors

have the following radial distributions:

S,(r)=~f,(-(sinF)“-(2/r)(sinF)‘+(2/r2)sinF+m2,sinF)

=ff,{sin Fe F12+(2/r2) sin F(l-2 cos F(G+l-4~0s F))

+3g 0’ -cosF(2sin2F-hcos2F*G(G+2))+m~sinF(l-cosF)}; f&r2 r’fi

(C-3)

1 ZZ-

2go gg’fc sin* (+F) -t A 22 16tr ~$‘(G+l)sin2F

-t-A &$F’+$G’(2-X,-L)

-G”+%3+m;G r ’

(C-4)

2g2fz,(1-cos F)+A---y :,“: u’(G+ 1) sin 2F+$ (G+3) ; (C.5)

Sz(r)=w”+(2/r)w’-m2,w

3g -- = 4v2r2

‘cos2F. G(G+2)+sin2F* G’(G+l));

(C-6)

F’(2 sin’ F-h& - Aid

+A~(2cos2F~F’(G-G~,-~,)i-sin2F~(G’-G’~,-G~~-~j)) . I

(C.7)

Here we have used the equations of motion for the meson fields, eqs. (A.3-5), (B.9-11), to eliminate the second derivatives. Obviously the vector-meson fields in eq. (Cl) and their sources (C.4-7) represent conserved currents as they should. A misprint in ref. I’) concerning the w-meson contribution to S,(r) has been corrected in eq. (C.3).

N. Kaiser et al. / ~esoo-nucleon form factors

Appendix D

617

EVALUATION OF SPECTRAL FUNCTIONS OF THE MESON-NUCLEON FORM FACTORS

We show here how to invert the spectral representation of the form factors

(I< to) (D.1)

in order to obtain the spectral distribution T(JA’). We follow the procedure of

Bowcock et al. *O). With the transformation

2t’- t() - = cash x,

zt-t,

to -=-cash z,

to

and the definitions

y(x) = 1 r(t’(x)) forx>O

-r( t’(x)) for x < 0

the dispersion integral (D-1) becomes

y(x) sinh x

dxcoshx+coshzS

(D-2)

(D.3)

(D-4)

Note that g(z) = G( t(z)) is an even function whereas y(x) is odd. Taking the Fourier

transform of g(z) we find

C(p)= dz e-“‘“g(z)

dx y(x) sinh x dz cash z

cash z-t cash x ’ (D.5)

The last integral is evaluated using standard residue calculus. One finds

if(p)=sinh (~*)~{~),

where f(p) turns out to be the Fourier transform of f(x):

(D-6)

+m jf(g) =

I dx eipXy(x). (D.7)

-m

This relation can then be inverted, and the spectral function r(t’) is obtained from

y(x) by substituting x = arcosh [(Zt’- to)/ to]. However, the numerical treatment of

this inversion procedure requires a cut-off in the p-integration. The reason is that

the relation (D-6) can be inverted numerically only if G(p) decreases as fast as

exp( -pi) asymptotically. This is in principle guaranteed by the fact that the spectral

representation (D.l) exists. However, the input form factor G(t) is given only

numerically over a limited interval in -ai < t G 0, so that the asymptotically required


decay of 6(p) is not automatically implied. The cut-off is introduced by multiplying

eq. (D.6) by a function e(p):

I I I I ir(p)C(p) = sinh (pr)G(p)C(p) . u3.8)

For the actual calculation we considered two choices for e: (a) gaussian cut-off:

t(p) = exp [ - Gap)‘1 ; (D.9)

and (b) “Box’‘-type cut-off:

1 for-aSpSa

otherwise ’ (D.10)

where we used typical values for the width parameter of (Y = 1.2 and a = 2. In

practice it turns out that these two choices lead to almost identical results.

Following similar steps we find an analogous relation for the real part of the form

factor Re G( t’) = G+( t’) in the region t’ > to:

G+(P)‘%P) = cash (P&(P)&), (D.ll)

where G+(p) is the Fourier transform in the variable x of the function

g’(x) = G+(t’(x)) forx>O

-G+(t’(x)) forx<O. (D.12)

Relations (D.8) and (D.ll) can now be inverted using the convolution theorem:

I

+@Z G,( t’(x)) = dx’C(x-x’)G(t’(x’))=

I +mdyC(x-y-i?r)*g(y). (D.13)

-cc -c0

Here C(x) is the Fourier transform of C(p). From eq. (D.13), which defines a

smoothed vertex function G,( t’> to), and hence a smoothed spectral function

r, = Im G,). The nature of this extrapolation procedure becomes now obvious. The

analytically continued complex vertex function is determined only to the extent of

its integral weighted with a smoothing function C(x - x’). The numerical input

G( t < to) fixes a parameter which determines the width of the cut-off function C

and therefore the “resolution” of the smoothed spectral density and the smoothed

real part of the form factor. The nonlinearity of the transformations (D.2) has now

the desirable effect, that the “resolution” in the low-energy range (t’J = jq21 <

1 GeV2/c2 is high, whereas for higher momentum transfer the resolution decreases.

We can therefore say that the smoothed r,( t’) and Re G,( t’) properly represent the

true complex vertex function over a range of values t’ compatible with the range

over which the input form factor G( q2 < 0) is determined reliably. For further details

of this method of analytic continuation we refer to ref. 20).

N. Kaiser et al. / Meson-nucleon form factors

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Documents

Meson-nucleon form factors in a chiral soliton model