Phenomenology and Meson Theory of Nuclear Forces

Abstract

Historically, the first fundamental idea for a theory of nuclear forces was advanced by the Japanese physicist Hideki Yukawa in 1935. He proposed that the exchange of subnuclear particles (eventually called mesons) between nucleons would create the force. The resulting meson theory was the most popular approach to nuclear forces for more than half a century. However, with the advancement of quantum chromodynamics (QCD) to the fundamental theory of strong interactions, meson theory had to be demoted to the level of a model. Yet, among all approaches to the nuclear force, the meson model remains the most insightful as well as the most quantitative one. Nucleon-nucleon potentials based upon meson-exchange are still today in frequent use. Because of its historical and conceptual value as well as its quantitative strength, the meson model is discussed in detail in this chapter. Since meson theory and phenomenology have been entwined throughout history, we also review the phenomenological approach to nuclear forces.

Appendices

Appendix A: The Relativistic One-Boson-Exchange Potential

Popular Lagrangians for meson-nucleon coupling are

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathcal L}_{ps}& =&\displaystyle -g_{ps}\bar{\psi} i\gamma^{5}\psi\varphi^{(ps)} {} \end{array} \end{aligned} $$

(92)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathcal L}_{pv}& =&\displaystyle -\frac{f_{ps}}{\hat m_{ps}}\bar{\psi} \gamma^{\mu}\gamma^{5}\psi\partial_{\mu}\varphi^{(ps)} {} \end{array} \end{aligned} $$

(93)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathcal L}_{s}& =&\displaystyle -g_{s}\bar{\psi}\psi\varphi^{(s)} {} \end{array} \end{aligned} $$

(94)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathcal L}_{v}& =&\displaystyle -g_{v}\bar{\psi}\gamma^{\mu}\psi\varphi^{(v)}_{\mu} -\frac{f_{v}}{4 \hat M} \bar{\psi}\sigma^{\mu\nu}\psi(\partial_{\mu} \varphi_{\nu}^{(v)} -\partial_{\nu}\varphi_{\mu}^{(v)}) {} \end{array} \end{aligned} $$

(95)

with ψ the nucleon and $\varphi ^{(\alpha )}_{(\mu )}$ the meson fields (notation and conventions as in Schwartz (2014)). $\hat m_{ps}$ and $\hat M$ are scaling masses to keep the associated coupling constants dimensionless. For isospin-1 mesons, φ^(α) is to be replaced by τ ⋅ φ^(α) with τ^l (l = 1, 2, 3) the usual Pauli matrices for isospin-$\frac 12$. The sub- and superscripts ps, pv, s, and v denote pseudoscalar, pseudovector, scalar, and vector couplings/fields, respectively.

The one-boson-exchange potential (OBEP) is defined as a sum of one-particle-exchange amplitudes of certain bosons with given mass and coupling. Using the six non-strange bosons with masses below 1 GeV/c², one has

$$\displaystyle \begin{aligned} V_{OBEP}=\sum_{\alpha=\pi,\eta,\rho,\omega,\delta,\sigma} V^{OBE}_{\alpha} {} \end{aligned} $$

(96)

with π and η pseudoscalar, σ and δ∕a₀ scalar, and ρ and ω vector particles. The contributions from the isovector bosons π, δ∕a₀ and ρ include a factor τ₁ ⋅ τ₂.

The above Lagrangians imply the following OBE contributions:⁴

$$\displaystyle \begin{aligned} \begin{array}{rcl} & {}{}{}&\displaystyle {\langle {\mathbf p}^{\prime} \lambda_{1}^{\prime}\lambda_{2}^{\prime}|V^{OBE}_{ps}| \mathbf{p}\lambda_{1}\lambda_{2}\rangle} \\[3pt] & = &\displaystyle - \frac{g^{2}_{ps}}{(2\pi)^{3}} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) i \gamma^{5} u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) i \gamma^{5} u(-{\mathbf p},\lambda_{2})/\\[3pt] & &\displaystyle [({\mathbf p}^{\prime} - {\mathbf p})^{2}+m_{ps}^{2}] ; {} \end{array} \end{aligned} $$

(97)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &{\langle {\mathbf p}^{\prime} \lambda_{1}^{\prime}\lambda_{2}^{\prime}|V^{OBE}_{pv}| \mathbf{p}\lambda_{1}\lambda_{2}\rangle} \\[3pt] & = &\displaystyle \frac{1}{(2\pi)^3} \frac{f^{2}_{ps}}{\hat m_{ps}^{2}} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma^{5}\gamma^{\mu}i(p^{\prime}-p)_{\mu} u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{5}\gamma^{\mu}i \\[3pt] & &\displaystyle (p^{\prime}-p)_{\mu} u(-{\mathbf p},\lambda_{2})/ \\[3pt] & &\displaystyle [({\mathbf p}^{\prime} - {\mathbf p})^{2}+m_{ps}^{2}] \\ & = &\displaystyle \frac{f^{2}_{ps}}{(2\pi)^3} \frac{4 M^{2}}{\hat m_{ps}^{2}} \{\bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma^{5} u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{5} u(-{\mathbf p},\lambda_{2}) \\ & &\displaystyle +[(E^{\prime}-E)/(2M)]^{2} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma^{5} \gamma^{0} u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{5} \gamma^{0} u(-{\mathbf p},\lambda_{2}) \\ & &\displaystyle +[(E^{\prime}-E)/(2M)] [\bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma^{5} u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{5}\gamma^{0} u(-{\mathbf p},\lambda_{2}) \\ & &\displaystyle +\bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma^{5}\gamma^{0} u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{5} u(-{\mathbf p},\lambda_{2})]\}/ \\ & &\displaystyle [({\mathbf p}^{\prime} - {\mathbf p})^{2}+m_{ps}^{2}] ; {} \end{array} \end{aligned} $$

(98)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \\ &{\langle {\mathbf p}^{\prime} \lambda_{1}^{\prime}\lambda_{2}^{\prime}|V^{OBE}_{s}| \mathbf{p}\lambda_{1}\lambda_{2}\rangle} \\ & = &\displaystyle -\frac{g^{2}_{s}}{(2\pi)^3} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) u(\mathbf{p},\lambda_{1}) \bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) u(-{\mathbf p},\lambda_{2})/\\ & &\displaystyle [({\mathbf p}^{\prime} - {\mathbf p})^{2}+m_{s}^{2}] ; {} \end{array} \end{aligned} $$

(99)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \\ &{\langle {\mathbf p}^{\prime} \lambda_{1}^{\prime}\lambda_{2}^{\prime}|V^{OBE}_{v}| \mathbf{p}\lambda_{1}\lambda_{2}\rangle} \\ & = &\displaystyle \frac{1}{(2\pi)^3} \{g_{v} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma_{\mu} u(\mathbf{p},\lambda_{1}) +\frac{f_{v}}{2 \hat M} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \sigma_{\mu\nu}i(p^{\prime}-p)^{\nu} u(\mathbf{p},\lambda_{1})\} \\ & &\displaystyle \times \{g_{v}\bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{\mu} u(-{\mathbf p},\lambda_{2}) -\frac{f_{v}}{2\hat M}\bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \sigma^{\mu\nu}i(p^{\prime}-p)_{\nu} u(-{\mathbf p},\lambda_{2})\}/ \\ & &\displaystyle [({\mathbf p}^{\prime} - {\mathbf p})^{2}+m_{v}^{2}] \\ & = &\displaystyle \frac{1}{(2\pi)^3} \left\{ \left( g_{v}+f_{v} \frac{M}{\hat M} \right) \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) \gamma_{\mu} u(\mathbf{p},\lambda_{1}) \right. \\[3pt] & &\displaystyle \left. -\frac{f_{v}}{2 \hat M} \bar{u}({\mathbf p}^{\prime},\lambda_{1}^{\prime}) [(p^{\prime}+p)_{\mu}+(E^{\prime}-E)(g^{0}_{\mu}-\gamma_{\mu}\gamma^{0})] u(\mathbf{p},\lambda_{1})\right\} \\[3pt] & &\displaystyle \times \left\{ \left( g_{v}+f_{v} \frac{M}{\hat M} \right)\bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) \gamma^{\mu} u(-{\mathbf p},\lambda_{2}) \right. \\[3pt] & &\displaystyle \left. -\frac{f_{v}}{2 \hat M}\bar{u}(-{\mathbf p}^{\prime},\lambda_{2}^{\prime}) [(p^{\prime}+p)_{\mu}+(E^{\prime}-E)(g^{\mu 0}-\gamma^{\mu}\gamma^{0})] u(-{\mathbf p},\lambda_{2}) \right\}/ \\ & &\displaystyle [({\mathbf p}^{\prime} - {\mathbf p})^{2}+m_{v}^{2}] . {} \end{array} \end{aligned} $$

(100)

Working in the two-nucleon CMS, the momenta of the two incoming (outgoing) nucleons are p and −p (p^′ and −p^′). $E\equiv \sqrt {M^{2}+{\mathbf {p}}^{2}}$ and $E^{\prime }\equiv \sqrt {M^{2}+{\mathbf p}^{\prime \,2}}$. Using the BbS (Blankenbecler and Sugar 1966) or Thompson (Thompson 1970) equations, the four-momentum transfer between the two nucleons is (p^′− p) = (0, p^′−p). The Dirac equation is applied repeatedly in the evaluations of the pv-coupling, and the Gordon identity (Schwartz 2014) is used in the case of the v-coupling. (Note that in Ep. (100), second line from the bottom, the term (p^′ + p)_μ carries μ as a subscript to ensure the correct sign of the space component of that term.) The propagator for vector bosons is

$$\displaystyle \begin{aligned} i\frac{-g_{\mu\nu}+ (p^{\prime}-p)_{\mu}(p^{\prime}-p)_{\nu} /m_{v}^{2}}{-({\mathbf p}^{\prime} - {\mathbf p})^{2}-m_{v}^{2}} {} \end{aligned} $$

(101)

where the (p^′− p)_μ(p^′− p)_ν-term is dropped which vanishes on-shell, anyhow, since the nucleon current is conserved. The off-shell effect of this term was examined in Holinde and Machleidt (1975) and was found to be unimportant.

The Dirac spinors in helicity representation are given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} u(\mathbf{p},\lambda_1)&=&\sqrt{\frac{E+M}{2M}} \left( \begin{array}{c} 1\\ \frac{2\lambda_1 |\mathbf{p}|}{E+M} \end{array} \right) |\lambda_1\rangle {} \end{array} \end{aligned} $$

(102)

$$\displaystyle \begin{aligned} \begin{array}{rcl} u(-\mathbf{p},\lambda_2)&=&\sqrt{\frac{E+M}{2M}} \left( \begin{array}{c} 1\\ \frac{2\lambda_2 |\mathbf{p}|}{E+M} \end{array} \right) |\lambda_2\rangle \,, \end{array} \end{aligned} $$

(103)

where the helicity λ_i of particle i (with i = 1 or 2) is the eigenvalue of the helicity operator $\frac 12 \mathbf {\sigma }_i \cdot {\mathbf {p}}_i/|{\mathbf {p}}_i|$ which is $\pm \frac 12$.

They are normalized covariantly, that is

$$\displaystyle \begin{aligned} \bar{u}(\mathbf{p},\lambda) u(\mathbf{p},\lambda)=1. {} \end{aligned} $$

(104)

with $\bar {u}=u^{\dagger }\gamma ^{0}$.

To ensure convergence of the scattering equation, it is customary to multiply each meson-nucleon vertex with a form factor, for which the following form may be chosen,

$$\displaystyle \begin{aligned} {\mathcal F}_\alpha[({\mathbf{p}}^{\prime}-\mathbf{p})^2]=\left(\frac{\varLambda^2_\alpha-m^2_\alpha} {\varLambda^2_\alpha+({\mathbf{p}}^{\prime}-\mathbf{p})^2} \right)^{n_\alpha} {} \end{aligned} $$

(105)

with m_α the mass of the meson involved, Λ_α the so-called cutoff mass, and n_α an exponent. Thus, the OBE amplitudes Eqs. (97)–(100) are multiplied by ${\mathcal F}_\alpha ^2$.

The coupling constants for the two different couplings for ps particles are related by

$$\displaystyle \begin{aligned} \frac{g_{ps}}{2M} = \frac{f_{ps}}{\hat m_{ps}}\,. \end{aligned} $$

(106)

Further developments, like partial-wave decomposition and the application of the OBEP in appropriate relativistic three-dimensional scattering equations, are given in Machleidt (1993).

Appendix B: Nonrelativistic Approximations and Position-Space Potentials

The momentum-space expressions for the OBE amplitudes given in Appendix A depend on two momentum variables, namely, the incoming and outgoing relative momenta p and p^′, respectively. A Fourier transformation of these expressions into position space would yield functions of r and r^′, the relative distance between the two in- and outgoing nucleons, i. e. a non-local potential. Because of the complexity of the expressions, this Fourier transformation cannot be done analytically. Both features mentioned are rather inconvenient for position-space (r-space) calculations. Therefore, it is customary to simplify the momentum-space expressions such that an analytic Fourier transformation becomes possible. This is achieved by using Dirac spinors in the representation

$$\displaystyle \begin{aligned} u({\mathbf p},s)=\sqrt{\frac{E+M}{2M}} \left( \begin{array}{c} 1\\ \frac{{\mathbf \sigma} \cdot {\mathbf p}}{E+M} \end{array} \right) \chi_{s} \end{aligned} $$

(107)

(with χ_s a Pauli spinor) for the evaluation of the OBE amplitudes of Appendix A and defining the momentum variables

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf q} & = &\displaystyle {\mathbf p}^{\prime} - {\mathbf p} \end{array} \end{aligned} $$

(108)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf k} & = &\displaystyle \frac{1}{2}( {\mathbf p}^{\prime} + {\mathbf p}) \,. \end{array} \end{aligned} $$

(109)

By dropping χ_s, the resulting potential is an operator in spin space, as customary. The relativistic energies are expanded in powers of q² and k² keeping the lowest order. In this way, one obtains the following simple momentum-space expressions:

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{ps}({\mathbf q}) & = &\displaystyle -\frac{g_{ps}^{2}}{4M^{2}} \frac{ (\mathbf \sigma_{1} \cdot {\mathbf q}) (\mathbf \sigma_{2} \cdot {\mathbf q})} {{\mathbf q}^{2}+m_{ps}^{2}} \end{array} \end{aligned} $$

(110)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{s}({\mathbf q},{\mathbf k}) & = &\displaystyle -\frac{g_{s}^{2}}{{\mathbf q}^{2}+m_{s}^{2}} \left[ 1-\frac{{\mathbf k}^{2}}{2M^{2}}+\frac{{\mathbf q}^{2}}{8M^{2}} -\frac{i}{2M^{2}} {\mathbf S} \cdot ({\mathbf q} \times {\mathbf k}) \right] \end{array} \end{aligned} $$

(111)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{v}({\mathbf q},{\mathbf k}) & = &\displaystyle \frac{1}{{\mathbf q}^{2}+m_{v}^{2}} \left\{ g_{v}^{2} \left[ 1+\frac{3{\mathbf k}^{2}}{2M^{2}} - \frac{{\mathbf q}^{2}}{8M^{2}} +\frac{3i}{2M^{2}} {\mathbf S} \cdot ({\mathbf q} \times {\mathbf k}) \right. \right. \\[3pt] & &\displaystyle \left. \left. - \frac{{\mathbf q}^{2}}{4M^{2}} \, {\mathbf \sigma}_{1} \cdot {\mathbf \sigma}_{2} +\frac{1}{4M^{2}} \, (\mathbf \sigma_{1} \cdot {\mathbf q}) (\mathbf \sigma_{2} \cdot {\mathbf q}) \right] \right. \\[3pt] & &\displaystyle \left. +\frac{g_{v}f_{v}}{2 \hat M} \left[ - \frac{{\mathbf q}^{2}}{M} +\frac{4i}{M} {\mathbf S} \cdot ({\mathbf q} \times {\mathbf k}) - \frac{{\mathbf q}^{2}}{M} \, {\mathbf \sigma}_{1} \cdot {\mathbf \sigma}_{2} +\frac{1}{M} (\mathbf \sigma_{1} \cdot {\mathbf q}) (\mathbf \sigma_{2} \cdot {\mathbf q}) \right] \right. \\[3pt] & &\displaystyle \left. +\frac{f_{v}^{2}}{4 \hat M^{2}} \left[ - {\mathbf q}^{2} \, {\mathbf \sigma}_{1} \cdot {\mathbf \sigma}_{2} + (\mathbf \sigma_{1} \cdot {\mathbf q}) (\mathbf \sigma_{2} \cdot {\mathbf q}) \right] \right\} \end{array} \end{aligned} $$

(112)

with ${\mathbf S}=\frac {1}{2} {\mathbf \sigma }_{1}+{\mathbf \sigma }_{2}$ the total spin of the two-nucleon system.

These expressions contain nonlocalities due to k² and (q ×k) terms. The latter leads to the orbital angular momentum operator L = −ir ×∇ in r-space, whereas the former provides ∇²-terms.

A quadratic spin-orbit term, $\frac {1}{2} ( \mathbf \sigma _{1} \cdot \mathbf L \, \mathbf \sigma _{2} \cdot \mathbf L + \mathbf \sigma _{2} \cdot \mathbf L \, \mathbf \sigma _{1} \cdot \mathbf L )$, is obtained when terms up to the fourth power in the momentum are retained, leading to substantially more comprehensive potential expressions. However, within a consistent meson model, these quadratic spin-orbit terms as well as the other terms of higher momenta do not improve the fit to the NN data but cause serious mathematical problems. If substantial improvements over the above expressions are desired, the use of the full unabridged momentum-space expressions of Appendix A is recommended. The role of the quadratic spin-orbit term is different if it is used as a phenomenological term to be fitted to the data. Then, particularly, an improvement of the $^{1}\!D_{2}$ and $^{3}\!D_{2}$ phase shifts can be achieved (Hamada and Johnston 1962).

The Fourier transform, $V({\mathbf r})=(2\pi )^{-3}\int d^{3}k e^{i{\mathbf q \cdot \mathbf r}} V({\mathbf q})$, which can now be performed analytically, yields:

(113)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{s}(m_{s},{\mathbf r}) & = &\displaystyle -\frac{g^{2}_{s}}{4\pi} m_{s} \left\{ \left[ 1-\frac{1}{4} \left(\frac{m_{s}}{M} \right)^{2} \right] Y(m_{s}r) +\frac{1}{4M^{2}} \left[ {\mathbf \nabla}^{2}Y(m_{s}r)+Y(m{s}r){\mathbf \nabla}^{2} \right] \right. \\[3pt] & &\displaystyle \left. +\frac{1}{2}Z_{1}(m_{s}r){\mathbf L \cdot S} \right\} {} \end{array} \end{aligned} $$

(114)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{v}(m_{v},{\mathbf r}) & = &\displaystyle \frac{g^{2}_{v}}{4\pi} m_{v} \left\{ \left[ 1+\frac{1}{2} \left(\frac{m_{v}}{M}\right)^{2} \right] Y(m_{v}r) -\frac{3}{4M^{2}} \left[ {\mathbf \nabla}^{2} Y(m_{v}r)+Y(m_{v}r){\mathbf \nabla}^{2} \right] \right. \\[3pt] & &\displaystyle \left. +\frac{1}{6} \left(\frac{m_{v}}{M} \right)^{2} Y(m_{v}r) \mathbf \sigma_{1} \cdot \mathbf \sigma_{2} -\frac{3}{2}Z_{1}(m_{v}r) \mathbf L \cdot \mathbf S -\frac{1}{12}Z(m_{v}r)S_{12}(\hat r) \right\} \\[3pt] & &\displaystyle +\frac{1}{2}\frac{g_{v}f_{v}}{4\pi} \left( \frac{M}{\hat M} \right) m_{v} \left\{ \left( \frac{m_{v}}{M} \right)^{2} Y(m_{v}r) + \frac{2}{3} \left( \frac{m_{v}}{M} \right)^{2} Y(m_{v}r) \mathbf \sigma_{1} \cdot \mathbf \sigma_{2} \right. \\[3pt] & &\displaystyle \left. -4Z_{1}(m_{v}r) \mathbf L \cdot \mathbf S -\frac{1}{3} Z(m_{v}r)S_{12}(\hat r) \right\} \\[3pt] & &\displaystyle +\frac{f_{v}^{2}}{4\pi} \left( \frac{M}{\hat M} \right)^2 m_{v} \left\{ \frac{1}{6} \left( \frac{m_{v}}{M} \right)^{2} Y(m_{v}r) \mathbf \sigma_{1} \cdot \mathbf \sigma_{2} -\frac{1}{12}Z(m_{v}r)S_{12}(\hat r) \right\} {} \end{array} \end{aligned} $$

(115)

with

$$\displaystyle \begin{aligned} \begin{array}{rcl} Y(x) & = &\displaystyle e^{-x}/x \end{array} \end{aligned} $$

(116)

$$\displaystyle \begin{aligned} \begin{array}{rcl} Z(x) & = &\displaystyle \left( \frac{m_{\alpha}}{M} \right)^{2} \left( 1+\frac{3}{x}+\frac{3}{x^{2}} \right) Y(x) \end{array} \end{aligned} $$

(117)

$$\displaystyle \begin{aligned} \begin{array}{rcl} Z_{1}(x) & = &\displaystyle -\left( \frac{m_{\alpha}}{M} \right)^{2} \frac{1}{x} \frac {d}{dx} Y(x)\\ & = &\displaystyle \left( \frac{m_{\alpha}}{M} \right)^{2} \left( \frac{1}{x} +\frac{1}{x^{2}} \right) Y(x) \,. \end{array} \end{aligned} $$

(118)

Similar to the σ₁ ⋅σ₂ part of the ps potential, there are δ⁽³⁾(r) function terms in the central force and spin-spin part of the vector potential which are left out (since they drop out anyhow, see below). A form factor, Eq. (105), with n_α = 1 can be taken into account by using for each meson potential

$$\displaystyle \begin{aligned} V_{\alpha}({\mathbf r})=V_{\alpha}(m_{\alpha},{\mathbf r}) -\frac{\varLambda_{\alpha,2}^{2}-m_{\alpha}^{2}}{\varLambda^{2}_{\alpha,2} -\varLambda^{2}_{\alpha,1}} V_{\alpha}(\varLambda_{\alpha,1},{\mathbf r}) +\frac{\varLambda_{\alpha,1}^{2}-m_{\alpha}^{2}}{\varLambda^{2}_{\alpha,2} -\varLambda^{2}_{\alpha,1}} V_{\alpha}(\varLambda_{\alpha,2},{\mathbf r}) \end{aligned} $$

(119)

where Λ_α,1 = Λ_α + ε and Λ_α,2 = Λ_α − ε with ε∕Λ_α ≪ 1.

The easy way to understand the effect of a cutoff is to consider the case of n_α = 1∕2 in Eq. (105), i.e., a factor ${\mathcal F}_\alpha ^2=(\varLambda ^2_\alpha -m^2_\alpha )/ (\varLambda ^2_\alpha +{\mathbf {q}}^2)$ is applied to an OBE contribution. The effect of such a cutoff is that from an OBE potential, Eqs. (113)–(115), the same expression is subtracted with the meson mass replaced by the cutoff mass (and using the same coupling constant), i. e. V_α(r) = V_α(m_α, r) − V_α(Λ_α, r). This removes the δ⁽³⁾(r)-function terms as well as the r⁻³ singularities, and, in terms of momentum-space language, it damps the potential at high momenta.