Yukawa potential#Relation to Coulomb potential

Prior to Hideki Yukawa's 1935 paper,{{cite journal |author-link=Hideki Yukawa |first=H. |last=Yukawa |title=On the interaction of elementary particles |year=1935 |journal=Proc. Phys.-Math. Soc. Jpn. |volume=17 |page=48}} physicists struggled to explain the results of James Chadwick's atomic model, which consisted of positively charged protons and neutrons packed inside of a small nucleus, with a radius on the order of 10⁻¹⁴ meters. Physicists knew that electromagnetic forces at these lengths would cause these protons to repel each other and for the nucleus to fall apart.{{cite book |title=Understanding the Universe: From quarks to the cosmos |last=Lincoln |first=Don |publisher=World Scientific |year=2004 |isbn=978-9812387035 |location=Singapore |pages=[https://archive.org/details/understandinguni0000linc/page/75 75]–78 |url=https://archive.org/details/understandinguni0000linc|url-access=registration }} Thus came the motivation for further explaining the interactions between elementary particles. In 1932, Werner Heisenberg proposed a "Platzwechsel" (migration) interaction between the neutrons and protons inside the nucleus, in which neutrons were composite particles of protons and electrons. These composite neutrons would emit electrons, creating an attractive force with the protons, and then turn into protons themselves. When, in 1933 at the Solvay Conference, Heisenberg proposed his interaction, physicists suspected it to be of either two forms:

:$J(r)\; =\; ae^\{-br\}\; \backslash quad\; \backslash textrm\{or\}\backslash quad\; J(r)\; =\; ae^\{-br^2\}$

on account of its short-range.{{cite journal |last=Miller |first=Arthur I. |date=1985 |title=Werner Heisenberg and the beginning of nuclear physics |journal=Physics Today |volume=38 |issue=11 |pages=60–68 |doi=10.1063/1.880993 |bibcode=1985PhT....38k..60M}} However, there were many issues with his theory. For one, it is impossible for an electron of spin {{sfrac|1|2}} and a proton of spin {{sfrac|1|2}} to add up to the neutron spin of {{sfrac|1|2}}. The way Heisenberg treated this issue would go on to form the ideas of isospin.

Heisenberg's idea of an exchange interaction (rather than a Coulombic force) between particles inside the nucleus led Fermi to formulate his ideas on beta-decay in 1934. Fermi's neutron-proton interaction was not based on the "migration" of neutrons and protons between each other. Instead, Fermi proposed the emission and absorption of two light particles: the neutrino and electron, rather than just the electron (as in Heisenberg's theory). While Fermi's interaction solved the issue of the conservation of linear and angular momentum, Soviet physicists Igor Tamm and Dmitri Ivanenko demonstrated that the force associated with the neutrino and electron emission was not strong enough to bind the protons and neutrons in the nucleus.

In his February 1935 paper, Hideki Yukawa combines both the idea of Heisenberg's short-range force interaction and Fermi's idea of an exchange particle in order to fix the issue of the neutron-proton interaction. He deduced a potential which includes an exponential decay term ($e^\{-\backslash alpha\; mr\}$) and an electromagnetic term ($1/r$). In analogy to quantum field theory, Yukawa knew that the potential and its corresponding field must be a result of an exchange particle. In the case of QED, this exchange particle was a photon of 0 mass. In Yukawa's case, the exchange particle had some mass, which was related to the range of interaction (given by $\backslash tfrac\{1\}\{\backslash alpha\; m\}$). Since the range of the nuclear force was known, Yukawa used his equation to predict the mass of the mediating particle as about 200 times the mass of the electron. Physicists called this particle the "meson," as its mass was in the middle of the proton and electron. Yukawa's meson was found in 1947, and came to be known as the pion.{{cite journal |last=Brown |first=Laurie M. |year=1986 |title=Hideki Yukawa and the meson theory |journal=Physics Today |volume=39 |issue=12 |pages=55–62 |doi=10.1063/1.881048 |bibcode=1986PhT....39l..55B}}

File:Yukawa m compare.svg

File:Yukawa coulomb compare.svg

If the particle has no mass (i.e., {{math|1=m = 0}}), then the Yukawa potential reduces to a Coulomb potential, and the range is said to be infinite. In fact, we have:

:$m=0\; \backslash Rightarrow\; e^\{-\backslash alpha\; m\; r\}=\; e^0\; =\; 1.$

Consequently, the equation

:$V\_\{\backslash text\{Yukawa\}\}(r)=\; -g^2\; \backslash ;\backslash frac\{e^\{-\backslash alpha\; mr\}\}\{r\}$

simplifies to the form of the Coulomb potential

:$V\_\{\backslash text\{Coulomb\}\}(r)=\; -g^2\; \backslash ;\backslash frac\{1\}\{r\}.$

where we set the scaling constant to be:{{cite book |title=Introduction to Quantum Mechanics |last=Griffiths |first=David J. |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-17986-8 |location=Cambridge, United Kingdom |pages=415}}

:$g^2\; =\; \backslash frac\{q\_1\; q\_2\}\{4\; \backslash pi\; \backslash varepsilon\_0\}$

A comparison of the long range potential strength for Yukawa and Coulomb is shown in Figure 2. It can be seen that the Coulomb potential has effect over a greater distance whereas the Yukawa potential approaches zero rather quickly. However, any Yukawa potential or Coulomb potential is non-zero for any large {{mvar|r}}.

The Yukawa potential can be thought of as arising from modifying the electromagnetic wave equation to describe a particle with nonzero mass. {{cite book | last1=Feynman | first1=Richard P. | author-link1=Richard Feynman | last2=Leighton | first2=Robert B. | author-link2=Robert B. Leighton | last3=Sands | first3=Matthew | author-link3=Matthew Sands | title=The Feynman Lectures on Physics | volume=2 | chapter=28 | chapter-url=https://www.feynmanlectures.caltech.edu/II_28.html | publisher=Addison-Wesley | year=1964 | isbn=978-0-201-02115-8 }}

The electromagnetic wave equation reads

$\backslash Box\; A^\backslash mu\; =\; \backslash text\{source\; terms\}$.

Where $A^\backslash mu$ is the electromagnetic four-potential and $\backslash Box\; =\; 1/c^2\; \backslash partial\_t^2\; -\; \backslash nabla^2$. The potential will go as $1/r$ for a point source.

This wave equation describes a photon. For the nuclear force, we hope to describe pions. The pion can be described by a scalar field, as opposed to a vector, and we modify the wave equation by adding a multiple of the field (which doesn't affect relativistic invariance):

$\backslash Box\; \backslash phi\; +\; \backslash mu^2\; \backslash phi\; =\; 0$

If $\backslash phi$ depends only on the radius $r$ in spherical coordinates and is time independent, then we have $\backslash nabla^2\; \backslash phi\; =\; \backslash mu^2\; \backslash phi$.

Using the expression for the Laplacian in spherical coordinates $\backslash nabla^2\; \backslash phi\; =\; \backslash frac\{1\}\{r\}\; \backslash partial\_r^2\; (r\; \backslash phi)$ we get

$\backslash partial\_r^2\; (r\; \backslash phi)\; =\; \backslash mu^2\; (r\; \backslash phi)$

which has solutions

$r\; \backslash phi(r)\; =\; Ke^\{-\; \backslash mu\; r\}$ so that

$\backslash phi(r)\; =\; K\; \backslash frac\{e^\{-\; \backslash mu\; r\}\}\{r\}$ which is the Yukawa potential.

For the photon, a wave solution will look like $\backslash exp(i\; (k\; \backslash dot\; r\; -\; \backslash omega\; t)\; )$. We can substitute this in to the homogenous electromagnetic wave equation to obtain

$\backslash frac\{\backslash omega^2\}\{c^2\}\; =\; k^2$,

By using the Planck relation $E\; =\; \backslash hbar\; \backslash omega$ and the de Broglie relation $p\; =\; \backslash hbar\; k$, we get

$\backslash frac\{E^2\}\{c^2\}\; =\; p^2$.

The mass-energy equivalence tells us that $\backslash frac\{E^2\}\{c^2\}\; =\; p^2\; +\; m^2\; c^2$, so this tells us that the photon is massless.

If we repeat this analysis for the pion, we get

$\backslash frac\{E^2\}\{c^2\}\; =\; k^2\; +\; \backslash mu^2\; \backslash hbar^2$, which tells us that the meson has a mass of $m\; =\; \backslash frac\{\backslash mu\; \backslash hbar\}\{c\}$.

$\backslash mu$ can be estimated from the observed range of the nuclear forces. The nuclear force has a range on the order of femtometers ($10^\{-15\}\; \backslash text\{m\}$), and therefore the pion mass should be about $170\; \backslash text\{MeV\}$, close to the actual value of around $140\; \backslash text\{MeV\}$.

The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transform. One has

:$V(\backslash mathbf\{r\})\; =$

\frac{-g^2}{(2\pi)^3}

\int e^{i\mathbf{k \cdot r}}

\frac {4\pi}{k^2+(\alpha m)^2} \,\mathrm{d}^3 k

where the integral is performed over all possible values of the 3-vector momenta {{mvar|k}}. In this form, and setting the scaling factor to one, $\backslash alpha\; =\; 1$, the fraction $\backslash frac\{4\; \backslash pi\}\{k^2\; +\; m^2\}$ is seen to be the propagator or Green's function of the Klein–Gordon equation.

Image:Momentum exchange.svg

The Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The Yukawa interaction couples the fermion field $\backslash psi(x)$ to the meson field $\backslash phi(x)$ with the coupling term

:$\backslash mathcal\{L\}\_\backslash mathrm\{int\}(x)\; =\; g~\backslash overline\{\backslash psi\}(x)~\backslash phi(x)~\backslash psi(x)~.$

The scattering amplitude for two fermions, one with initial momentum $p\_1$ and the other with momentum $p\_2$, exchanging a meson with momentum {{mvar|k}}, is given by the Feynman diagram on the right.

The Feynman rules for each vertex associate a factor of $g$ with the amplitude; since this diagram has two vertices, the total amplitude will have a factor of $g^2$. The line in the middle, connecting the two fermion lines, represents the exchange of a meson. The Feynman rule for a particle exchange is to use the propagator; the propagator for a massive meson is $\backslash frac\{-4\backslash pi\}\{~k^2+m^2~\}$. Thus, we see that the Feynman amplitude for this graph is nothing more than

:$V(\backslash mathbf\{k\})=-g^2\backslash frac\{4\backslash pi\}\{k^2+m^2\}~.$

From the previous section, this is seen to be the Fourier transform of the Yukawa potential.

{{clear}}

The radial Schrödinger equation with Yukawa potential can be solved perturbatively.{{cite journal |last=Müller |first=H.J.W. |year=1965 |title=Regge-Pole in der nichtrelativistischen Potentialstreuung |journal=Annalen der Physik |volume=470 |issue=7–8 |pages=395–411 |doi=10.1002/andp.19654700708 |language=German |bibcode=1965AnP...470..395M}}{{cite journal |last1=Müller |first1=H.J.W. |last2=Schilcher |first2=K. |date=February 1968 |title=High-energy scattering for Yukawa potentials |journal=Journal of Mathematical Physics |volume=9 |issue=2 |pages=255–259 |doi=10.1063/1.1664576}}{{rp|at=ch. 16}} Using the radial Schrödinger equation in the form

:$\backslash left[\backslash frac\{\backslash mathrm\{d\}^2\}\{\backslash mathrm\{d\}r^2\}\; +\; k^2\; -\; \backslash frac\{\backslash ell(\backslash ell\; +\; 1)\}\{r^2\}\; -\; V(r)\backslash right]\; \backslash Psi\backslash left(\backslash ell,k;\backslash ,r\backslash right)\; =\; 0,$

and the Yukawa potential in the power-expanded form

:$V(r)\; =\; \backslash sum\_\{j\; =\; -1\}^\{\backslash infty\}\; M\_\{j\; +\; 1\}\; \backslash ,\; (-r)^j,$

and setting $K\; =\; jk$, one obtains for the angular momentum $\backslash ell$ the expression

:$\backslash ell\; +\; n\; +\; 1\; =\; -\backslash frac\{\backslash ,\backslash Delta\_n(K)\backslash ,\}\{2K\}$

for $|K|\; \backslash to\; \backslash infty$, where

:$\backslash begin\{align\}$

&\Delta_n(K) = M_0 - \frac{1}{\,2K^2\,}\Bigl[\,n(n + 1)\,M_2 + M_0\,M_1\,\Bigr] - \frac{\,2n + 1\,}{4K^3}\,M_0\,M_2 ~+ \\

&\qquad\qquad\quad + \frac{1}{\,8K^4\,}\,\Bigl[\,3(n-1)n(n+1)(n+2)\,M_4 + 2 (3n^2 + 3n - 1)\,M_3\,M_0 ~+ \\

&\qquad\qquad\qquad\qquad\qquad ~+~ 6n(n + 1)\,M_2\,M_1 + 2\,M_2\,M_0^2 + 3M_1^2\,M_0\,\Bigr] ~+ \\

&\qquad\qquad\quad + \frac{\,2n + 1\,}{\,8K^5\,}\,\Bigl[\,3(n^2 + n - 1)\,M_4\,M_0 + 3\,M_3\,M_0^2 + n(n + 1)\,M_2^2 + 4\,M_2\,M_1\,M_0\,\Bigr]~+ \\

&\qquad\qquad\quad +~ \operatorname{\mathcal O}\Bigl(\,\frac{1}{\,K^7\,}\,\Bigr) ~.

\end{align}

Setting all coefficients $M\_j$ except $M\_0$ equal to zero, one obtains the well-known expression for the Schrödinger eigenvalue for the Coulomb potential, and the radial quantum number $\backslash ,n\backslash ,$ is a positive integer or zero as a consequence of the boundary conditions which the wave functions of the Coulomb potential have to satisfy. In the case of the Yukawa potential the imposition of boundary conditions is more complicated. Thus in the Yukawa case $\backslash nu\; =\; n$ is only an approximation and the parameter $\backslash nu$ that replaces the integer {{mvar|n}} is really an asymptotic expansion like that above with first approximation the integer value of the corresponding Coulomb case.

The above expansion for the orbital angular momentum or Regge trajectory $\backslash ell(K)$ can be reversed to obtain the energy eigenvalues or equivalently $\backslash bigl|K\backslash bigr|^2$. One obtains:{{cite journal |last1=Müller |first1=H.J.W. |year=1965 |title=On the calculation of Regge trajectories in nonrelativistic potential scattering |journal=Physica |volume=31 |issue=5 |pages=688–692 |doi=10.1016/0031-8914(65)90006-6 |bibcode=1965Phy....31..688M}}

:$\backslash begin\{align\}$

&\bigl|K\bigr|^2 ~=~ - M_1 ~+~ \frac{1}{\,4(\ell+n+1)^2\,}\,\biggl\{\; M_0^2 - 4n(n+1)(\ell+n+1)^2 \,M_2\, M_0 + 4(2n+1)(\ell+n+1)^2\frac{M_2}{\;M_0\,} ~+ \\

&\quad +~4\frac{\;(\ell+n+1)^4\,}{M_0^3}\,\Bigl[\,3(n-1)n(n+1)(n+2+3)\,M_4\,M_0 ~+ \\

& \qquad \qquad \qquad \qquad \qquad \qquad - ~ 3 n^2(n+1)^2\,M_2^2 + 2(3n^2+3n-1)\,M_3\,M_0^2 + 2\,M_2\,M_0^3\,\Bigr] ~+ \\

&\quad - ~ 24\frac{\,(2n+1)(\ell+n+1)^5\,}{M_0^4}\,\Bigl[\,(n^2 + n - 1)\,M_0\,M_4 + M_0^3\,M_3 - n(n+1)\,M_2^2\,\Bigr] ~+ \\

&\quad - ~ 4\,\frac{\,(\ell + n + 1)^6\,}{M_0^7}\,\Bigl[~10(n-2)(n-1)n(n+1)(n+2)(n+3)\,M_6\,M_0^2 ~+ \\

&\qquad \qquad \qquad \qquad + ~ 4\,M_3\,M_0^5 + 2\Bigl(\,5n(n+1)(3 n^2 + 3n - 10) + 12\,\Bigr)\,M_5\,M_0^3 ~+\\

&\qquad \qquad \qquad \qquad + ~ 2 (6 n^2 + 6n - 11)\, M_4\,M_0^4 + 2(9 n^2 + 9n - 1)\,M_2^2\,M_0^3 ~+ \\

&\qquad \qquad \qquad \qquad - ~ 10 n(n+1)(3 n^2 + 3n + 2)\, M_3\,M_2\,M_0^2 + 20 n^3(n+1)^3\, M_2^3 ~+ \\

&\qquad \qquad \qquad \qquad - ~ 30 (n-1)n^2(n+1)^2(n+2)\,M_4\,M_2\,M_0\,\Bigr] \quad + \quad \cdots \biggr\} \quad .

\end{align}

The above asymptotic expansion of the angular momentum $\backslash ell(K)$ in descending powers of $K$ can also be derived with the WKB method. In that case, however, as in the case of the Coulomb potential the expression $\backslash ell(\backslash ell+1)$ in the centrifugal term of the Schrödinger equation has to be replaced by $\backslash left(\backslash ell+\backslash tfrac\{1\}\{2\}\backslash right)^2$, as was argued originally by Langer,{{cite journal |year=1937 |last1=Langer |first1=Rudolph E. |title=On the connection formulas and the solutions of the wave equation |journal=Physical Review |volume=51 |issue=8 |pages=669–676 |bibcode=1937PhRv...51..669L |doi=10.1103/PhysRev.51.669}} the reason being that the singularity is too strong for an unchanged application of the WKB method. That this reasoning is correct follows from the WKB derivation of the correct result in the Coulomb case (with the Langer correction),{{cite book |first=Harald J.W. |last=Müller-Kirsten |year=2012 |title=Introduction to Quantum Mechanics: Schrödinger equation and path integral |edition=2nd |publisher=World Scientific |place=Singapore |isbn=978-9814397735}}{{rp|page=404}} and even of the above expansion in the Yukawa case with higher order WKB approximations.{{cite journal |last1=Boukema |first1=J.I. |year=1964 |title=Calculation of Regge trajectories in potential theory by W.K.B., and variational techniques |journal=Physica |volume=30 |issue=7 |pages=1320–1325 |doi=10.1016/0031-8914(64)90084-9 |bibcode=1964Phy....30.1320B}}

We can calculate the differential cross section between a proton or neutron and the pion by making use of the Yukawa potential. We use the Born approximation, which tells us that, in a spherically symmetrical potential, we can approximate the outgoing scattered wave function as the sum of incoming plane wave function and a small perturbation:

:$\backslash psi(\backslash vec\{r\})\; \backslash approx\; A\; \backslash left[(e^\{ipr\})\; +\; \backslash frac\{e^\{ipr\}\}\{r\}\; f(\backslash theta)\; \backslash right]$

where $\backslash vec\{p\}\; =\; p\; \backslash hat\{z\}$ is the particle's incoming momentum. The function $f(\backslash theta)$ is given by:

:$f(\backslash theta)\; =\; \backslash frac\{-2\backslash mu\}\{\backslash hbar^2\; \backslash left|\backslash vec\{p\}-\backslash vec\{p\}\text{'}\; \backslash right|\}\; \backslash ,\; \backslash int\_\{0\}^\{\backslash infty\}\; r\; \backslash ,\; V(r)\; \backslash ,\; \backslash sin\backslash left(\backslash left|\backslash vec\{p\}-\backslash vec\{p\}\text{'}\backslash right|\; r\; \backslash right)\; ~\; \backslash mathrm\{d\}r$

where $\backslash vec\; p\text{'}\; =\; p\; \backslash hat\; r$ is the particle's outgoing scattered momentum and $\backslash mu$ is the incoming particles' mass (not to be confused with $m,$ the pion's mass). We calculate $f(\backslash theta)$ by plugging in $V\_\backslash text\{Yukawa\}$:

:$f(\backslash theta)\; =\; \backslash frac\{2\backslash mu\}\{\backslash hbar^2\; \backslash left|\backslash vec\{p\}-\backslash vec\{p\}\text{'}\backslash right|\}\; \backslash ,\; g^2\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash alpha\; m\; r\}\; \backslash ,\; \backslash sin\; \backslash left(\backslash left|\backslash vec\{p\}\; -\; \backslash vec\{p\}\text{'}\backslash right|\; \backslash ,\; r\; \backslash right)\; \backslash ,\; \backslash mathrm\{d\}r$

Evaluating the integral gives

:$f(\backslash theta)\; =\; \backslash frac\{2\; \backslash mu\; g^2\}\{\backslash hbar^2\backslash ,\backslash left[(\backslash alpha\; m)^2\; +\; \backslash left|\; \backslash vec\{p\}\; -\; \backslash vec\{p\}\text{'}\; \backslash right|^2\backslash right]\}$

Energy conservation implies

:$\backslash bigl|\backslash vec\; p\backslash bigr|\; =\; \backslash bigl|\backslash vec\; p\text{'}\backslash bigr|\; =\; p~$

so that

:$\backslash left|\backslash vec\; p\; -\; \backslash vec\; p\text{'}\backslash right|\; =\; 2\backslash ,p\backslash ,\backslash sin\backslash left(\backslash tfrac\{1\}\{2\}\backslash theta\backslash right)~$

Plugging in, we get:

:$f(\backslash theta)\; =\; \backslash frac\{2\; \backslash mu\; g^2\}\{\backslash hbar^2\; \backslash left[(\backslash alpha\; m)^2\; +\; 4\backslash ,p^2\backslash ,\backslash sin^2\backslash left(\{\backslash frac\{1\}\{2\}\backslash theta\}\backslash right)\backslash right]\}$

We thus get a differential cross section of:

:$\backslash frac\{\backslash mathrm\; d\; \backslash sigma\}\{\backslash mathrm\; d\; \backslash Omega\}\; =\; \backslash left|f(\backslash theta)\backslash right|^2\; =\; \backslash frac\{4\; \backslash mu^2\; g^4\}\{\backslash hbar^4\backslash \; \backslash left[\; (\backslash alpha\; m)^2\; +\; 4\; p^2\; \backslash sin^2\backslash left(\backslash frac\{1\}\{2\}\backslash theta\backslash right)\; \backslash right]^2\}$

Integrating, the total cross section is:

:$\backslash sigma\; =\; \backslash int\; \backslash frac\{\backslash mathrm\{d\}\; \backslash sigma\}\{\backslash mathrm\{d\}\; \backslash Omega\}\; \backslash mathrm\{d\}\backslash Omega$

= \frac{4 \mu^2 g^4}{\hbar^4} \int_0^\pi \frac{2\pi \sin(\theta) \mathrm{d}\theta}{\left[(\alpha m)^2 + 4 p^2 \sin^2\left(\frac{1}{2}\theta\right) \right]^2}

= \frac{4 \mu^2 g^4}{\hbar^4} \frac{4\pi}{(\alpha m)^2 \left[(\alpha m)^2 + 4p^2 \right]}

The potential outside of an infinitesimally thin, uniform spherical shell with total scaling constant $G$ and radius $R$ is also a Yukawa potential, but in general the scaling contstant for the equivalent point source is larger than for the shell.{{cite web |last=Kuhn |first=Paulo |title=Debye-Hückel interaction, or Yukawa potential, in different geometries |url=https://wp.ufpel.edu.br/pskuhn/files/2024/09/debye-huckel-interaction-or-yukawa-in-different-geometries.pdf |access-date=14 February 2025}}{{cite web |last=McDonald | first=Kirk |orig-date=April 17, 1984 |date=December 20, 2021 |title=A Naïve Estimate of the Coupling Constant in Yukawa Theory |url=http://kirkmcd.princeton.edu/examples/yukawa.pdf |access-date=14 February 2025}}{{cite web |title=Shell theorem for a general potential |url=https://math.stackexchange.com/questions/296180/shell-theorem-for-a-general-potential |website=Mathematics Stack Exchange |access-date=14 February 2025 |language=en}} More specifically, the potential of a point with scaling constant $g$ outside of the shell is

:$V(r\; >\; R)\; =\; G\; g\; \backslash frac\{e^\{-\backslash alpha\; m\; r\}\}\{r\}\; \backslash frac\{\backslash sinh\; \backslash alpha\; mR\}\{\backslash alpha\; mR\}.$

which is the same as replacing the shell with a point source with magnitude $G\; \backslash frac\{\backslash sinh\; \backslash alpha\; mR\}\{\backslash alpha\; mR\}$.

The interior potential is

:

If $m\; =\; 0$, then one recovers the shell theorem for the inverse square potential.

A consequence of this is that in modified gravity theories where the graviton has nonzero mass, the weak equivalence principle would be violated and the gravitational acceleration of a body in free fall would depend on its composition.{{cite journal |last1=Ciufolini |first1=Ignazio |last2=Matzner |first2=Richard |last3=Paolozzi |first3=Antonio |last4=Pavlis |first4=Erricos C. |last5=Sindoni |first5=Giampiero |last6=Ries |first6=John |last7=Gurzadyan |first7=Vahe |last8=Koenig |first8=Rolf |title=Satellite Laser-Ranging as a Probe of Fundamental Physics |journal=Scientific Reports |date=4 November 2019 |volume=9 |issue=1 |page=15881 |doi=10.1038/s41598-019-52183-9|pmid=31685911 |arxiv=1907.00395 |bibcode=2019NatSR...915881C }}

• Yukawa interaction
• Screened Poisson equation
• Bessel potential

{{reflist|25em}}

• {{cite book |author1-link=Gerald Edward Brown |first1=G.E. |last1=Brown |first2=A.D. |last2=Jackson |title=The Nucleon-Nucleon Interaction |year=1976 |publisher=North-Holland Publishing |place=Amsterdam |isbn=0-7204-0335-9}}

Category:Gauge theories

Category:Scattering theory

Category:Quantum mechanical potentials