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For magnetism:

$$ \oint \vec{B} \cdot \mathrm{d} A = 0 $$

For electricity:

$$ \oint \vec{E} \cdot \mathrm{d} A = \frac{Q}{\epsilon_0} $$

For gravity:

$$ \oint \vec{g} \cdot \mathrm{d} A = -4\pi G m $$

We know magnetic dipoles (single electrons) exist and yet there is no magnetic charge.

Now electric dipoles composed of multiple particles exist but I don't know of any fundamental electrical dipoles.

Do any fundamental particles with gravitational or electric dipoles exist?

I can't see anything that forbids them.

I guess more complicated cases such as photons also exist also which have oscillating electric and magnetic fields. I'm not sure how to properly describe more complicated cases like these. I'm not sure if it is right to call a photon a monopole or a dipole but I guess it's important to note that things can be more than just monopoles or dipoles.

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    $\begingroup$ The search for the electron's electric dipole moment is an ongoing experimental effort. So far the results are consistent with zero. Any future non-zero result would constitute an example of a fundamental particle (within the Standard Model) possessing an electric dipole moment. $\endgroup$ Commented Jun 4, 2016 at 19:32
  • $\begingroup$ Whether a "particle" (really a field) is fundamental or not depends on the energy of our accelerator facilities. There is no way to tell if today's "fundamental field" will also be tomorrow's. Having said that, once there is a dipole, it can't be "undone" (or at least so I believe intuitively), so one would have to ask why we can't see microscopic dipoles at the scale of the standard model. $\endgroup$
    – CuriousOne
    Commented Jun 4, 2016 at 19:46
  • $\begingroup$ @CuriousOne I'd also assume that black holes can't be undone and yet Hawking radiation exists so I'd be cautious about assuming that a dipole would be indivisible. $\endgroup$ Commented Jun 4, 2016 at 20:18
  • $\begingroup$ You mean that electric dipole (and magnetic monopole) fields would be unstable? Yes, that is a very real possibility that is taken very seriously in high energy searches. I was merely talking about stable dipoles, but you are absolutely correct that one can not and shall not rule out that on some energy scale these fields are major players. $\endgroup$
    – CuriousOne
    Commented Jun 4, 2016 at 20:21

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Fundamental particles like the electron are well-known to have magnetic dipole moments related to their spins. Moreover, the standard model also predicts that they should have small electric dipole moments (EDMs) as well. However, the EDMs predicted in the standard model are very small; they will probably be too small to observe directly for quite some time. The reason is that having a permanent EDM violate time-reversal symmetry (T). The standard model does have T violation, but it can only appear through quantum processes that involve at least three different types of virtual particles, and these higher-order quantum corrections are generally strongly suppressed.

There is quite a bit of interest in the idea that there may be new physics beyond the standard model that involves much stronger sources of T violation. These would, correspondingly, produce much larger EDMs through quantum corrections. Searches for electron and neutron EDMs are some of the best way to test these theories of new physics.

With gravity, the situation is different. It is not actually possible to have a gravitational dipole. An electric dipole involves a separation of the positive and negative charges in an object. However, there are no negative gravitational sources; all masses are positive. The lowest-order multipole for a gravitational field is the quadrupole. For a fundamental particle to have a quadrupole moment, it must have a spin of at least 1. (In contrast, to have a dipole moment, a particle must have a spin of at least 1/2.) A spin-1 particle like the deuteron (a bound state of a proton and neutron) does indeed have an electric quadrupole moment. The deuteron should also have a (T-violating) magnetic quadrupole moment and a gravitational quadrupole moment, although these would both be extremely tiny.

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  • $\begingroup$ I don't follow. Why can there be gravitational quadropoles but not dipoles? Also, there is no such thing as magnetic charge but there do exist magnetic dipoles. For the low mass approximation before tensors are introduced a gravitational dipole would be induced by a gravitational effect analogous to magnetism. Does the math work out such that only quadropoles are allowed? $\endgroup$ Commented Jun 4, 2016 at 21:35
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    $\begingroup$ When people talk about gravitational multipoles, they generally mean in Newtonian (or post-Newtonian) gravity. There are also gravitomagnetic multipoles. Anything with angular momentum (e.g. intrinsic spin) will possess a gravitomagnetic dipole moment. Back in the Newtonian case, you do not need dipoles to construct a quadrupole. The sun has a quadrupole moment, because it is oblate instead of spherical. Similarly, the electric quadropole moment of the deuteron comes from the fact that the proton wave function is not a pure spherical S state, but also has a D state component. $\endgroup$
    – Buzz
    Commented Jun 4, 2016 at 22:19

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