1. Short answer
Yes, there exists the analog of electric force for the other gauge bosons (the electric field is defined by having the electric force, so we need the classical analogs of other fundamental forces to define the corresponding classical field). In fact, for each gauge boson (including weak force carriers $W^{\pm}$ and $Z^0$, gluons in strong interactions, massless/massive gravitons, and even for vector mesons), you can find a corresponding classical effective potential/force. All of them have already been found, but, the range of forces are drastically different. For vector bosons and also vector mesons (3 polarizations), the corresponding force is very short-range, so only detectable at short scales. They obey the Yukawa-potential like law. For the gluons, the color confinement does not allow the long-range behavior of spin-1 massless gluons to appear. For gravitons, even with a tiny mass (i.e., with 5 polarizations), you can have a long-range force.
To avoid confusion with the concept of macroscopic waves, see Section 4 of this answer. Finally, you can skip the details in this answer and see the explicit forms of classical effective forces/fields of the weak/meson and strong forces in Section 6.
2. Longer (intuitive) answer
According to Standard Model of elementary particles (SM), elementary particles interact with each other by the exchange of gauge bosons, in which interactions are described by a gauge theory. The standard model is explained based on quantum field theory (QFT). In QFT, force carriers (also known as intermediate particles) are virtual particles that give rise to forces between other particles. The virtual, force-carrying particles that mediate the electromagnetic, weak, and strong interactions are called gauge bosons, all of them are spin-1 vector fields (remember that the graviton is not part of SM yet, but I discuss this case as well).
So, your question is quite sensible and, in my opinion, important.
In order to answer this question, we shall utilize QFT or classical effective field theory. Degrees of freedom in flat four-dimensional spacetime are particles, classified by their spin. These degrees of freedom (degrees of polarization) are carried by fields (e.g., see Refs. [1-6]). In principle, according to QFT, the exchange of a particle can produce a force, so it is possible to associate a particle with each of the known forces: for example, the photon with the electromagnetic force and the graviton with the gravitational force. This pattern could be generalized to the other particles such as $W^+$, $W^-$ and $Z^0$ gauge bosons with three degrees of polarization, massive gravitons (if exist any, of course) with five degrees of polarization, which leads to finding new/unconventional potentials (and consequently forces), and this already had been applied by Yukawa for describing the interactions between nucleons. This means, these forces can be effectively extracted from the exchange of a particle (a massless/massive gauge boson), such as photon or graviton or weak gauge bosons. There are two methods (that I know) for evaluating the corresponding effective potential/force: 1) path integral formalism of QFT, 2) Green's function approach in classical effective field theory (which, I think, is essentially the result of the first method). For the electromagnetic and gravitational cases, the resulting forces between two point sources obey the $\frac{1}{{{r^2}}}$ law (Newton’s gravitational force and Coulomb’s electric force). The first lesson: two degrees of polarization for any massless, bosonic gauge field implies the $\frac{1}{{{r^2}}}$ law.
If you consider a free field with extra degrees of polarization (more than two polarizations), it means that you are dealing with a massive field which is drastically different from massless fields. The known massive gauge bosons (as force carriers) are $W^+$, $W^-$ and $Z^0$ particles, i.e., the weak force carriers. For these, you can compute the classical, effective potential/force, which is a common exercise in QFT/effective filed theory and you can repeat this for a massive gravitational field theory (with massive spin-2 force carriers), and you always end up with a Yukawa-like potential law. The second lesson: three or more degrees of polarization for any massive, bosonic gauge field implies the Yukawa potential law as $U(r) \propto \frac{1}{{4\pi r}}{e^{ - mr}}$.
Using a rigorous approach from the path integral formalism of QFT (see Ref. [1]) or using the effective field theory (see Ref. [2] and also Ref. [3]) as a simpler way, the corresponding classical effective potential for each gauge boson can be computed in a straightforward way. The case of gluons, because of the color confinement problem, is more complicated and will be discussed at the end (the Quarks and Gluons part, Section 5).
3. Mathematically rigorous answer
Here, I briefly show how to compute the classical, effective potentials/forces associated with weak gauge bosons and also vector mesons which are a very short range force. Theoretically, by adjusting the gauge boson's (or vector meson's) mass, one can control the range of force. Since the mass of ${W^ \pm }$ and $Z^0$ bosons are very big, the resulting force is necessarily short-range. After that, I explain that it is even possible to have a long range gravitational force with massive gravitons.
I. The rigorous strategy for finding effective potentials (From path integral formalism of QFT) (this part has been adapted from the great book by Zee, Ref. [1])
- In the path integral formalism, the generating functional (which tells us everything we could possibly want to know about a system) of a free filed theory is given by
$${Z_0}[J] = \int {{\cal D}\phi {e^{i\int {{d^4}x} \left( {{\cal L}[\phi ] + J\phi } \right)}}},$$
where $J$ is a sum of hypothetical sources (this term is added to the potential in order to create a particle, watch it propagate for a while, and then annihilate it somewhere else in space). It can be shown that the generating functional is equivalently written as
$${Z_0}[J] = C{e^{ - (i/2)\int {\int {{d^4}x{d^4}yJ(x)D(x - y)J(y)} } }} \equiv C{e^{iW(J)}},$$
where $D(x)$ is known as the (free) propagator. Propagators, which are analogous to the Green’s function amplitudes, contain the information about excitations of the system.
In the path integral formalism, $Z=C{e^{iW(J)}}$ represents $\left\langle 0 \right|{e^{ - iHt}}\left| 0 \right\rangle = {e^{ - iUt}}$ in quantum mechanics, where $U$ is the energy due to the presence of the two sources acting on each other. So, we have $W=-Ut$, finally. The constant $C$ doesn't have anything to do with our final result, so I will ignore it in the rest of this answer.
Suppose, there are two sources in space, ${J_1}(x)$ and ${J_2}(x)$. $J(x)= {J_1}(x) \pm {J_2}(x)$ (where ${J_i}(x)= {\delta ^{(3)}}\left( {\vec x - {{\vec x}_i}} \right)$) is a sum of sources that are time-independent infinitely sharp spikes located at $x_1$ and $x_2$ in space. see the figure below (From Zee [1]). The force carrier created at a source and will be absorbed by the another source.
- Computing $W(J)=-Ut$ leads to finding $U$ as the effective potential between two sources (two point particles).
Now, let's see a generic example which is valid for weak gauge bosons and also vector mesons: Massive, (electrically neutral) spin-1 vector boson is described by the following Lagrangian density
$${\cal L}(x) = - \frac{1}{4}{F^{\mu \nu }}(x){F_{\mu \nu }}(x) + \frac{1}{2}m_Z^2{A_\mu }(x){A^\mu }(x),$$
where $m_Z$ is its mass (in what follows, I ignore the index $Z$ for the mass). The generating functional is written as
$$\begin{array}{l}
{Z_0}[J] = \int {{\cal D}{A_\mu }{e^{i\int {{d^4}x} \left( {{\cal L}[{A_\mu }] + {J_\mu }{A^\mu }} \right)}}}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \int {{\cal D}{A_\mu }{e^{i\int {{d^4}x} \left( {\frac{1}{2}{A_\mu }\left[ {(\partial ^2 + {m^2}){g^{\mu \nu }} - {\partial ^\mu }{\partial ^\nu }} \right]{A_\nu } + {J_\mu }{A^\mu }} \right)}}} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {e^{ - (i/2)\int {\int {{d^4}x{d^4}y{J^\mu }(x){D_{\mu \nu }}(x - y){J^\nu }(y)} } }} \equiv {e^{iW(J)}}
\end{array},$$
where, in the second line, the method of integration by parts has been used. And, as usual in QFT path integral, ${D_{\mu \nu }}(k)$ is the solution (propagator) of
$$\left[ {({\partial ^2} + {m^2}){g^{\mu \nu }} - {\partial ^\mu }{\partial ^\nu }} \right]{D_{\nu \lambda }}(x) = \delta _\lambda ^\mu {\delta ^{(4)}}(x).$$
By converting the above equation to momentum space via a Fourier transform, ${D_{\mu \nu }}(k)$ in momentum space is obtained as
$${D_{\mu \nu }}(k) = \frac{{ - {g_{\mu \nu }} + {k_\mu }{k_\nu }/{m^2}}}{{{k^2} - {m^2} + i\varepsilon }}.$$
Yes, this is definition of Green function for QFT field equations, here for massive vector fields! (In this Web Site you can find a nice pedagogic note about this method). Now, we have $W(J)$ with the explicit form of
$$\begin{array}{l}
W(J) = - \frac{1}{2}\int {\frac{{{d^4}k}}{{{{(2\pi )}^4}}}{J^\mu }{{(k)}^*}\left( {\frac{{ - {g_{\mu \nu }} + {k_\mu }{k_\nu }/{m^2}}}{{{k^2} - {m^2} + i\varepsilon }}} \right)} {J^\nu }(k)\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2}\int {\frac{{{d^4}k}}{{{{(2\pi )}^4}}}{J^\mu }{{(k)}^*}\frac{{{g_{\mu \nu }}}}{{{k^2} - {m^2} + i\varepsilon }}} {J^\nu }(k)
\end{array},$$
where, in the second line, it's assumed that the current is conserved (${\partial _\mu }{J^\mu }(x) = 0 \Leftrightarrow {k_\mu }{J^\mu }(k) = 0$). Now, like step 3., suppose that $J(x)= {J_1}(x)+ {J_2}(x)$ is a sum of two identical sources where ${J_i}(x)= {\delta ^{(3)}}\left( {\vec x - {{\vec x}_i}} \right)$). Having this and by converting the above equation to momentum space by use of Fourier transforms, and doing the integral over $d^3x$ and $d^3y$, one obtains
$$\begin{array}{l}
W(J) = \int {\int {d{x^0}d{y^0}} \int {\frac{{d{k^0}}}{{2\pi }}{e^{i{k^0}{{(x - y)}^0}}}} } \int {\frac{{{d^3}k}}{{{{(2\pi )}^3}}}} \frac{{{e^{i\vec k.({{\vec x}_1} - {{\vec x}_2})}}}}{{{{\vec k}^2} - {m^2} + i\varepsilon }}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = -\underbrace {\int {d{x^0}} }_{ = t}\int {\frac{{{d^3}k}}{{{{(2\pi )}^3}}}\frac{{{e^{i\vec k.({{\vec x}_1} - {{\vec x}_2})}}}}{{{{\vec k}^2} + {m^2}}}}
\end{array},$$
in which I omitted many details (see a complete derivation in Zee's book, Ref. [2]). It's time to apply the step 4, i.e., $W(J) = - Ut$. So, the effective potential between two point particles is obtained as
$$U(r) = \int {\frac{{{d^3}k}}{{{{(2\pi )}^3}}}\frac{{{e^{i\vec k.({{\vec x}_1} - {{\vec x}_2})}}}}{{{{\vec k}^2} + m_Z^2}}} = \frac{1}{{4\pi r}}{e^{ - mr}},$$
which is the Yukawa-like potential. We assumed that the two sources are identical (i.e., they are two like point particles). So, the above effective potential predicts that those particles repel each other. If you consider a sum of currents as $J(x)= {J_1}(x)- {J_2}(x)$, you will get an attractive force.
This computation is also valid for vector mesons since they are spin-1 massive particles with odd parity. In addition, for ${W^ \pm }$ gauge bosons, the Lagrangian is given by
$${\cal L}(x) = - \frac{1}{4}{F^{\mu \nu }}F_{\mu \nu }^\dagger + \frac{1}{2}m_W^2W_\mu ^\dagger {W^\mu },$$
and the final result is the same, again but, this time, the gauge boson mass is $m_W$.
II. The simplest strategy for finding effective potentials is the method of Green's function in classical effective field theory. This has been nicely/completely explained in chapter 3. of Schwartz's book, Ref. [2] (especially see exercise 3.6) and also in Chapter 17 of Ref. [3].
III. Results for massless/massive gravitons:
By treating general relativity as a quantum effective field theory, it can also be verified that the $\frac{1}{{{r^2}}}$ force law is valid for gravitation [3], and the force is always attractive (see the proof of being attractive in Zee, Ref. [1], Section I.5). This is the case with two polarizations for gravitons. If you consider an explicit tiny mass term for gravitons according to the Fierz-Pauli theory or (more better) according to the de Rham-Gabadadze-Tolley massive gravity, you again arrive at a Yukawa-like potential as $U(r) \propto \frac{1}{{4\pi r}}{e^{ - m_gr}}$ (see Ref. [5] and also Ref. [6]). In this case, massive spin-2 gravitons have 5 degrees of polarizations. There are a number of ways to add extra degrees of freedom to general relativity without considering an explicit mass term for spin-2 fields, e.g. by adding some Lorentz invariant terms such as $R^2$, and the resulting effective potential could be a a short-ranged Yukawa potential, but the functional form is different (see Ref. [4] for these kinds of Lagrangians). For an extremely tiny graviton mass ($m_g << 1$), the effect of the exponent ($e^{-m_gr}$) could be even undetectable in solar scales and also in larger scales if one demands that $m_g {r_ \odot } \sim 0 $ (se more details in Ref. [5]).
4. In summary:
For massless boson force carriers, the classical effective force obeys the $\frac{1}{{{r^2}}}$ law.
For massive boson force carriers, the classical effective potential obeys the Yukawa potential law as $U(r) \propto \frac{1}{{4\pi r}}{e^{ - mr}}$.
In both the massless and the massive cases of vector fields, depending on the nature of the electric or weak charges, the corresponding force could be repulsive (for two like charges) or attractive (for two unlike particles). For the gravitational force (massless/massive spin-2 force carriers), it is always attractive.
Furthermore, adding extra degrees of freedom (polarizations) to a massless field theory leads to deviation from effective $\frac{1}{{{r^2}}}$ law, and this prevents the existence of coherent states for having a macroscopic massive wave. (see below, for more information)
A potential source of confusion: The effective potential/force for weak or strong interactions should not be confused with the weak or strong waves (which do not exist). In interactions between two point elementary particle, virtual (off-shell) gauge bosons are present, but in a macroscopic wave like electromagnetic wave, real (on-shell) gauge bosons like photons are present, so electromagnetic wave are coherent states of photons but this cannot happen for weak or strong gauge bosons. Besides the confinement problem for strong force and the large mass of gauge bosons in weak interactions, the weak/strong force-carrying particles interact with each other, which ruled out the existence of weak or strong macroscopic waves. For gravitons, it is completely possible to have coherent states since interaction between gravitons is negligible (the weak force is $10^{24}$ times stronger than gravity, the hierarchy problem!)
5. Quarks and Gluons
The gluon propagator is basically the same as photon propagator. Therefore, we expect to have a $1/R$ potential law for gluons as well, implying a long-range force. But, the quarks, gluons and consequently the strong force are confined (the color confinement problem) and they cannot be directly observed in normal conditions below the Hagedorn temperature. For this reason, the low energy limit of QCD is one of the greatest mysterious in theoretical physics since it cannot be understood by means of the usual perturbative methods in QFT. Therefore, since we cannot write an effective (classical) field equation for gluons, we cannot repeat the previous argument for finding the effective potential/force. But this not the end of story. In fact, using Lattice field theory and the Wilson's loop argument together with the path integral formalism, the confinement of quarks can be understood which implies an effective potential as $U(R) \propto \sigma R$. In conclusion, we should have potential like this:
$$U(R) = A - \frac{B}{R} + \sigma R,$$
where $A$, $B$, and $\sigma$ are some constants (historically, $\sigma$ is called string tension). This one was my exercise in QFT course a few years ago, and the relevant references can be found separately at the end of this answer (maybe useful for you or anyone else). Especially, for a confirmation of this potential, see Ref. [18]. (I deliberately used $R$ instead of $r$ to remind us of the issue of color confinement.)
6. Final remarks: On the analogs of electric field for the other forces
There are four fundamental forces in nature and all other forces in nature derive from these four fundamental interactions. These forces are described by the force-carrying particles in QFT, which are spin-1 and spin-2 gauge bosons. So, the analog of electric field associated to photons is only reasonable for the other gauge bosons.
The electromagnetic force acts between electric charges. In classical physics, assuming the electric charges do not move in space, we can define an electrostatic field around the charge. The other charge feels the electric field and reacts to it. As a result, depending on the electric charges, the two charges repel or attract each other. The idea of field for describing the corresponding force also works for the gravitational force, which acts between masses. It is always attractive as QFT easily predicts it (See Zee, Ref. [1], Section I.5).
We were already familiar with gravitational and electric fields in classical physics. But this story could be extended to the other fundamental forces, provided that we can find classical effective potentials for them. The strong and the weak forces act only at very short distances and, here, we showed they could also have classical effective potentials.
Thus, to make it more clear: The electric field is defined as the electric force per unit charge, so it is defined by having the electric force as
$$U(r) \propto \pm \frac{1}{r}\, \Rightarrow \,F = - \nabla U(r)\, \Rightarrow \,{\rm{Electric}}\,{\rm{Field}} \propto \pm \frac{1}{{{r^2}}}.$$
Therefore, the weak/meson field analog of the electric field is given by
$$U(r) \propto \pm \frac{1}{r}{e^{ - mr}}\, \Rightarrow \,{F_W} = - \nabla U(r)\, \Rightarrow \,{\rm{Weak/Meson}}\,{\rm{Field}} \propto \pm \frac{{{e^{ - mr}}\left( {mr + 1} \right)}}{{{r^2}}},$$
which, depending on the nature of particles (see Section 3, in this answer), could be repulsive or attractive.
And, finally, the strong field analog of the electric field is obtained as
$$U(R) \propto A - \frac{B}{R}\, + \sigma R \Rightarrow {F_G} = - \nabla U(R)\, \Rightarrow {\rm{Strong}}\,{\rm{Field}} \propto - \left( {\frac{B}{{{R^2}}} + \sigma } \right)$$
which always is attractive, as expected from lattice simulations. (I deliberately used $R$ instead of $r$ to remind us of the issue of color confinement.)
These are classical descriptions of the strong and weak/meson fields, which are well-known in particle physics but this statement does not mean that they are macroscopic, as the context of our discussion indicates (see more information in Section 4, in this answer).
References
[1] A. Zee, Quantum field theory in a nutshell, Princeton university press (2010).
Chapters: I.1, I.2, I.3, I.4, I.5, and I.6
[2] M.D. Schwartz, Quantum field theory and the standard model, Cambridge University Press (2014).
Chapter 3 (effective field theory) and also Chapter 13 (for generating functional and Green's function)
[3] T. Lancaster and S.J. Blundell, Quantum field theory for the gifted amateur, Oxford University Press (2014).
Chapter 17 for deriving Yukawa potential from Green's function method
[4] J.F. Donoghue, Introduction to the effective field theory description of gravity." Advanced school on effective theories, arXiv:gr-qc/9512024 (1995).
[5] C. de Rham, Massive gravity, Living Rev. Relativity 17, 7 (2014).
[6] K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84, 671
(2012)
References for "Lattice Gauge Theory, Wilson Loop and Confinement of Quarks":
[7] C. Gattringer and C.B. Lang, Dynamical fermions: Quantum chromodynamics on the lattice, Springer Berlin Heidelberg, (2010).
[8] M.D. Schwartz, Quantum field theory and the standard model.” Cambridge University Press (2014). Section 25.5
[9] K.G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).
[10] R.D. Klauber, Student friendly quantum field theory, Sandtrove Press (2013)
[11] M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, CRC Press (1996).
[12] M. Natsuume, AdS/CFT duality user guide, Springer (2015). Chapter 8
[13] G. Münster and M. Walzl, Lattice gauge theory-A short primer, arXiv: hep-lat/0012005 (2000).
[14] G.P. Lepage, Lattice QCD for novices, Strong Interactions at Low and Intermediate Energies, World Scientific (1998).
[15] J. Greensite, An Introduction to the Confinement Problem, Springer Berlin Heidelberg (2010).
[16] Y.M. Makeenko, Brief introduction to Wilson loops and large N, Physics of Atomic Nuclei 73, 878 (2010).
[17] J. Schwinger, Gauge invariance and mass. II, Phys. Rev. 128, 2425 (1962).
[18] A.P. Trawiński et al, Effective confining potentials for QCD, Phys. Rev. D 90, 074017 (2014).
