Nucleon-Meson Coupling Constants and Form Factors in the Quark Model

We demonstrate the calculation of the coupling constants and form factors required by effective hadron lagrangians using the quark model. These relations follow from equating expressions for strong transition amplitudes in the two approaches. As examples we derive the NNm nucleon-meson coupling constants and form factors for m = pi, eta, eta', sigma, a_0, omega and rho, using harmonic oscillator quark model meson and baryon wavefunctions and the 3P0 decay model; this is a first step towards deriving a quark-based model of the NN force at all separations. This technique should be useful in the application of effective lagrangians to processes in which the lack of data precludes the direct determination of coupling constants and form factors from experiment.


I. INTRODUCTION
Effective hadronic lagrangians are widely used the description of the interactions of hadrons.In this method a distinct quantum field is introduced for each relevant hadronic species, and interactions are assumed for these fields that are consistent with known symmetries and conservation laws.Although a more physically justified description of hadron scattering amplitudes would employ momentum-dependent coupling constants (form factors), it often suffices near threshold to assume pointlike "hard" hadronic vertices with fixed coupling constants.This approximation may be relaxed by incorporating hadronic form factors as a power series of gradient interactions.Unfortunately, this typically leads to nonrenormalizable ultraviolet divergences, which requires the determination of many coupling constants from experiment.
When using effective lagrangians one typically ignores the existence of hadronic substructure (quarks and gluons), and determines each coupling constant in the effective lagrangian from experiment.Although this procedure is feasible in experimentally highly constrained problems such as the NN interaction, in other processes with little data the coupling constants are simply assigned plausible values.In such cases it would be very useful to have numerical estimates of the coupling constants.Since these coupling constants and form factors are determined by the underlying QCD degrees of freedom, one may evaluate them directly in terms of the interactions of quarks and gluons and the substructure of the hadronic bound states.
In this letter we will investigate this relation between effective lagrangian couplings and quark model bound state hadron wavefunctions.The specific cases we consider are the meson-nucleon couplings, which are usually fitted to data in meson exchange models of the NN force.These NNm vertices are chosen as our initial examples in part because they are among the most important hadronic couplings for nuclear physics applications, and also because the NNπ coupling constant g NNπ ≈ 13.5 is the best determined strong coupling in hadron physics.It is clearly of interest to determine whether this experimental coupling constant is consistent with the value predicted using quark model hadron wavefunctions.
We regard this computation as the first step in an attempt to compute the strong interactions of nucleons at all separations from microscopic quark-based models.In this approach the short ranged NN interactions are dominated by the quark-gluon forces encountered by overlapping nucleon quark wavefunctions, and the long ranged NN interactions are mediated by t-channel meson exchange.The meson-nucleon couplings in t-channel meson exchange, which we compute here, are themselves determined by the overlap of meson and nucleon quark wavefunctions.

II. TECHNIQUE
Our method for defining effective hadronic coupling constants and form factors is to equate specific hadron emission amplitudes predicted by the effective hadron lagrangian to the corresponding decay amplitude in the quark model.Applied to NNm couplings in particular, we require that N m|H ef f |N = N (q 3 )m(q q)|H decay |N (q 3 ) . ( This general approach was used previously by the Orsay group of LeYaouanc et al. [1], who considered ρππ, NNπ and NNρ couplings; the relation between this earlier reference and our results will be discussed.A similar quark model approach was more recently applied to the determination of HQET electroweak form factors by Isgur et al. [2]. Ideally one would impose this relation using a relativistic quark model, in which case there would be no difficulty in identifying the effective lagrangian matrix element with the quark model result.As the nonrelativistic quark model formalism is much better established, we will instead use nonrelativistic quark model wavefunctions, and apply our defining relation Eq.( 1) between matrix elements near threshold.Since the quark model wavefunctions we use are nonrelativistic, the form factors we extract by equating quark model and (relativistic) field theory matrix elements have an ambiguity in how we relate results in different frames.In this initial study we will simply assume a particular frame, the "decay frame" of a rest initial hadron, as in the earlier work of the Orsay group [1].The complication of implicit frame dependence will be investigated in detail a subsequent study.
It is important to note that these frame ambiguities are resolved in the weak binding and nonrelativistic limits, in which the quark model states approach the transformation properties specified by the Lorentz group.Thus our quark model predictions of form factors are best justified at zero three-momentum, where we define the coupling constants.Of course the model assumption is that the predicted form factors remain useful estimates at moderate nonzero momentum.
The specific quark model strong decay Hamiltonian we use is the 3 P 0 model [1,3,4], which has seen very wide application, and is known to give reasonably accurate numerical results for both meson [5,6,7] and baryon [8] strong decays.
The NNπ coupling constant we find by equating these expressions at threshold is (We have suppressed an overall phase factor of +i, which is normalization convention dependent.)The numerical value of this coupling constant as a function of α and β is shown in Fig. 1.Evidently the range of typical wavefunction length scale parameters α and β leads to a factor of two variation in the theoretical g NNπ ; it ranges between 7.0 − 12.2 for the parameters shown in the figure.
The experimental value of ≈ 13.5 evidently corresponds to values of α and β near the lower end of their respective ranges, provided that we fix the q q pair production amplitude at the meson decay value of γ = 0.4.Since the "experimental" value of g NNπ is not actually based on a direct observation of pion emission, it is prudent to carry out an independent calculation of a closely FIG.1: The theoretical pion-nucleon coupling constant gNNπ (Eq.5) as a function of the quark model Gaussian wavefunction length scales α (baryons) and β (mesons).The range of values of α and β typically found in the quark model literature is indicated (see text).The 3 P0 model q q pair production amplitude γ = 0.4 was taken from studies of meson decays.The experimental gNNπ ≈ 13.5 is also shown.related decay process that does involve a detected pion in the final state.The transitions ∆ → Nπ are useful in this regard because the matrix elements are related to the NNπ coupling by SU(6) symmetry, assuming identical spatial wavefunctions.Specializing to ∆ ++ → pπ + , the quark model result for the total width is This theoretical quark model decay rate is shown in Fig. 2 for the same range of wavefunction parameters α, β as g NNπ in Fig. 1, and the experimental width of 110 MeV is also shown.Evidently the parameter constraints due to g NNπ and Γ(∆ → Nπ) are approximately consistent.Alternatively, since the dimensionless pair production amplitude γ represents poorly understood nonperturbative physics, it may have a different strength in baryon decays than in meson decays.In Fig. 3 we show the ratio of theory to experiment for Γ(∆ ++ → pπ + ) and g NNπ for our "standard" quark model baryon and meson wavefunction length scales α = β = 0.4 GeV, varying the pair production amplitude γ.Evidently with these wavefunctions a value of γ ≈ 0.7 is favored for pion emission from light baryons, whereas γ ≈ 0.4 − 0.5 gives the best description of light meson decays.
Nucleon couplings to the other ground state pseudoscalars (η and η ′ ) are simply related to g NNπ , provided that we assume identical spatial wavefunctions and pure q q states.Given the effective lagrangian and taking the |η and |η ′ flavor states to be the maximally mixed linear combinations (|nn ± |ss )/ √ 2, the NNη( ′ ) coupling constants and form factors are related to the NNπ results by If we use g NNπ = 13.5 as input, this gives g NNη = 11.5 and g NNη ′ = 15.3.Although these appear to be rather large NNη( ′ ) couplings, their effect on NN scattering is suppressed by the larger η and η ′ masses in propagators and in the 1/ √ 2E external line normalizations.

B. NNσ
The NNσ coupling may be the most important nucleon-meson coupling in all of nuclear physics.In meson exchange models the exchange of a light scalar I=0 "sigma" meson is held to be the dominant mechanism underlying the intermediate ranged attraction, which is responsible for nuclear binding.The balance between this attraction and the short distance repulsion in the nuclear core determines the equilibrium density of n 0 ≈ 0.16 nucleons/fm 3 in bulk nuclear matter.(Although pions are lighter and hence longer-ranged, and the g NNπ coupling constant is quite large, the fact that pions are emitted in a relative P-wave suppresses their contribution to the near-threshold interactions of nucleons.) Although the intermediate ranged attraction plays a crucial role in nuclear physics, the σ meson itself is a dubious concept in meson spectroscopy.In I=0 ππ Swave scattering one sees a very broad positive (attractive) phase shift, which if interpreted as an s-channel resonance would imply a mass of ca. 1 GeV and a comparably large width.There are arguments from the quark model against a q q state with these properties; for example, an I=0 0 ++ nn resonance (n = u, d) at this mass would have a rather large two-photon width of Γ γγ ≈ 2 keV, and no such state is evident in γγ → π 0 π 0 .There is instead evidence in this reaction for a moderately broad scalar enhancement near 1.3 GeV, with about the expected two-photon width [10]; this broad f 0 (≈ 1300) is often identified with the I=0 0 ++ nn quark model state.
The explicit σ meson included in meson exchange models has been explained as a parametrization of "correlated two-pion exchange", so that its fitted strong coupling to NN and low mass need not correspond to the properties of a physical P-wave nn meson.The picture of the "sigma meson" as a strongly mixed (nn)-ππ state is supported by the large coupling predicted between these channels in the 3 P 0 model; the analogous S-wave kaon system is discussed in Ref. [11].
We can test the plausibility of sigma meson exchange models of the intermediate ranged NN attraction by calculating the NNσ coupling for a pure nn σ meson, using the same techniques we applied above to the NNπ system.If the sigma is dominantly a physical nn scalar meson, we would expect approximate agreement between the calculated and fitted g NNσ coupling constants.If the sigma is instead a parametrization of two-pion exchange, agreement between the theoretical and fitted coupling constants would be fortuitous.
The calculation of the NNσ coupling differs from the NNπ case through the meson spin, space and isospin wavefunctions and the effective lagrangian.We assume the form where the basis states are |L, L z |S, S z .The P-wave momentum space q q wavefunctions are given in Ref. [4].
On equating the effective lagrangian and quark model h f i = pσ|H|p matrix elements, we find (A normalization convention dependent overall phase of (−1) in our result is suppressed here.)The NNσ form factor is the quadratic (1+[(1+r 2 /4)/9(1+r 2 /3)](P/α) 2 ) times the Gaussian found in the NNπ case in Eq.( 4).The numerical g NNσ predicted by Eq.( 10) is shown in Fig. 4 as a function of α and β (assuming m σ = 500 MeV).Evidently a value in the range 3 − 7 is predicted by the quark model given this m σ , with g NNσ ≈ 5 preferred.Although it is of great interest to compare our calculated g NNσ coupling constant with the values reported in meson exchange model fits to NN scattering data, there is unfortunately no single consistent value reported for g NNσ in these models.The three best known mesonexchange models of the NN force in the literature are the Paris [12,13], Nijmegen [14,15,16] and Bonn [17] models, and their NNm couplings are given in Table I, together with our quark model results.The Paris model did not consider a σ meson.In the recent "CD-Bonn" model [17], different g NNσ coupling constants and σ masses are assumed in different NN channels; in the I=0 3 S 1 NN channel a σ mass and coupling constant of m σ = 350 MeV and g NNσ ≈ 2.5 are used, whereas in I=0 1 S 0 , m σ = 452 MeV and g NNσ ≈ 7.3 are used.The g NNσ coupling is allowed to vary with L and I in the L > 0 NN channels, and ranges from 1.9 to 9.9 (see Table XVI of Ref. [17]).In the Nijmegen model [15] a larger value of g NNσ = 17.9 is quoted.Thus, although our quark model result g NNσ ≈ 5 is similar to the mean S-wave NN value quoted in the CD-Bonn model, the scatter in the fitted values of this parameter precludes an accurate comparison between theory and experiment at present.
One may similarly evaluate the quark model prediction for the NN coupling of the a 0 I=1 partner of the σ.Given the effective lagrangian we find Although I=1 scalar exchange contributes a somewhat smaller amplitude to NN scattering than I=0 exchange, it may nonetheless be possible to test for the presence of both of these scalar meson exchange amplitudes through their interference, for example by comparing the I=0 and I=1 S-wave NN scattering amplitudes discussed above.

C. NNω and NNρ
The NNV couplings are interesting in that the shortranged repulsive core in the NN interaction has previously been attributed to vector meson exchange, and the existence of the ω meson was regarded as support for this picture.This mechanism now appears less plausible, since the short range of vector meson exchange (R ∼ 1/m ω ∼ 0.25 fm) implies extensive overlap of the NN quark wavefunctions.
Evaluation of the NNV couplings and form factors uses the same procedure as the scalar and pseudoscalar couplings discussed above, although there are complications due to the presence of two form factors and the nontransverse components of the vector field.
As above we assume a term in the effective lagrangian for each coupling.For NNω this lagrangian is where We then equate nearthreshold Hamiltonian matrix elements h f i found from this effective lagrangian to the corresponding 3 P 0 decay model matrix elements.There is a complication in relating the relativistic effective lagrangian and nonrelativistic quark model matrix elements; we find that one must assume a vector meson polarization vector and fourmomentum of the form ǫ µ = (0, ǫ) and q µ = (0, q ) to equate these expressions.The NNω γ µ and σ µν form factors may be separated by equating h f i matrix elements with different spin states.The transitions p(+1/2) → p(+1/2) ω(0) and p(+1/2) → p(+1/2) ω(−1) are useful in this regard, since they receive contributions from only the γ µ and σ µν terms respectively.The resulting form factors are proportional to the NNπ result Eq.( 5), since they involve the same Gaussian overlap integrals.The NNω coupling constants (and form factors) satisfy the relations and The NNρ form factors, defined through the generalization of the NNω effective lagrangian to an I=1 ρ meson, (where F µν = ∂ µ ρ ν −∂ ν ρ µ ) are related to the NNω results by and The NNρ vector (γ µ ) coupling constant was previously evaluated by LeYaouanc et al. [1].Their Eq.(3.13) for g NNρ is consistent with our Eqs.(5,17), provided that i) their factor of 3R 2 N R 2 ρ is actually 3R 2 N R ρ (their result as written is dimensionally incorrect), ii) their factor of m 3/2 ρ should instead be m 1/2 ρ m N , analogous to their g NNπ result, and iii) the factor of 1  2 τ in their ρ effective lagrangian Eq.(2.17) should be τ , as in their π effective lagrangian Eq.(2.12).LeYaouanc et al. did not evaluate the σ µν term, and did not consider the NNω case.
It is interesting to compare our theoretical NNV couplings with the fitted values found in meson exchange models of NN scattering.If the 3 P 0 model is reasonably accurate in describing the coupling between nonstrange baryons and vector mesons (which is currently being tested at TJNAF in their search for missing baryon resonances decaying to Nω and Nρ), and the meson exchange models are correct in assuming that vector meson exchange is the dominant mechanism underlying the short-ranged NN force, we would expect to find approximate agreement between these couplings.
The fitted NNV couplings found in the three wellknown meson exchange models are given in Table I, together with our quark model results.Evidently we do not find good agreement.Note in particular that the fitted strength of the dominant NNω γ µ coupling in the meson exchange models is about a factor of 2 smaller than the quark model prediction.The ratio of the NNρ to NNω γ µ couplings is rather similar in the two approaches; the meson exchange models quote a ratio of ≈ 0.2−0.4,whereas the theoretical ratio (an SU (6) symmetry factor rather than a detailed test of the quark model predictions) is +1/3.Although the ratio "κ" of magnetic (σ µν ) to vector (γ µ ) couplings does not yet appear to be well determined for both vectors in the meson exchange model fits, there does appear to be agreement that |κ ρ | >> |κ ω |.This is inconsistent with our quark model prediction of equal magnitudes for these NNV "strong magnetic" couplings, κ (ω,ρ) = (−, +) 3/2.Since these ratios are simple SU(6) factors and do not involve uncertainties in the spatial wavefunctions, this disagreement may imply that vector meson exchange is not the dominant short ranged NN interaction mechanism.This will be addressed in detail in a future study of the NN scattering amplitudes and phase shifts due to meson exchange, augmented by quark model constraints on the nucleon-meson vertices.

III. SUMMARY AND CONCLUSIONS
In this paper we have developed a formalism for determining hadron strong vertices and form factors, "threepoint functions", in the context of the quark model.We apply this approach to the evaluation of meson-nucleon vertices, several of which are important in meson exchange models of nuclear forces.The quark model expression we find for the NNπ coupling confirms an earlier Orsay result.With standard quark model parameters, this g NNπ about half the experimental value.Our quark model expression for the theoretical g NNσ strong coupling of nucleons to scalar mesons is a new result, and is numerically similar to the isospin-mean fitted NN S-wave value in the CD-Bonn model.Our quark model result for the NNρ vector (γ µ ) coupling is consistent with an earlier Orsay result (after correcting typographical errors), although we find a nonzero magnetic (σ µν ) coupling.The strengths of the fitted NNV γ µ couplings in meson exchange models are rather smaller than our quark model predictions.The NNV σ µν couplings are also not in good agreement with quark model predictions, although they may not be well determined in the current fits to NN scattering data.
In future we plan to carry out calculations of the NN phase shifts predicted by meson exchange models, assuming quark model constraints on the NNm couplings and form factors as derived here.This should allow a determination of the sensitivity of the data to parameters such as the g NNω /g NNρ and κ ω /κ ρ ratios, for which we have definite quark model predictions.b Assumes g NNπ = 13.5.c This "CD-Bonn" model introduces different g NNσ coupling constants for (I=0; I=1) NN channels, which would not be expected for an isosinglet σ.In addition these g NNσ couplings and the σ mass are allowed to vary with L (S-wave is quoted here), and a higher-mass σ with large g NNσ couplings is also assumed.
d Assumes mρ = mω .e This value is cited in Ref. [12] but is not actually used in the Paris model, which does not incorporate ρ exchange.

TABLE I :
[17]13]ry of NNm coupling constants.Our calculated values are shown in the middle columns, and fitted or assumed values in the meson exchange model literature are shown at right.Values in square brackets were fixed input.Coupling This ref. a This ref. b Paris[12,13]Nijmegen[15]Bonn[17] a Assumes "standard" quark model parameters α = β = 0.4 GeV and γ = 0.4 (see text).