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My lecture notes write that the coupling constant for the strong force goes to infinity as the distance goes to infinity.

However the range of the strong force is said to be small.

If the strength of the force (which is represented by the coupling constant) is infinite at large distances, why the short range?

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This is the consequence of Color confinement, which is not yet completely understood. The main reasoning is that when you separate two color-charged particles (i.e. quarks in a hadron) enough, it becomes energetically more favorable to create a particle-antiparticle pair, so it's in practice impossible for particles interacting via strong force to reach the large distances. Consequently, the strong force only 'acts' at a short range.

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  • $\begingroup$ Oh right, so in principle, the range should be infinite but due to colour confinement, it is small? $\endgroup$
    – code noob
    May 16 '20 at 16:29
  • $\begingroup$ Depends on what you call "range". You can compute the strength of the strong force at any distance, but they will (almost?) never happen physically, but it has important consequences outside a hadron, the main example being the nuclear force, also called residual strong force, which binds nuclei together. Usually the effects of the other forces become more significant at higher distances, which is why we don't experience it under normal circumstances. $\endgroup$
    – Renan Nobuyuki Hirayama
    May 16 '20 at 16:39
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The general idea is that is so strong that at microscopic distances all charges get neutralized by opposite charges or by the other color charges (because three different color charges neutralize themselves, like in the proton). This happens to in electric charge, but there are a few phenomena by wich at macroscopic scales it manifest: *Electric charges can be separated from a bunch of matter, so the matter gets charged. This can't happen with nuclear strong force, because it's so strong that the energy needed to separate one charge un-neutralizing a bunch of nuclear matter (for example, taking a quark from a proton) is enough to make a quark-antiquark pair that neutralizes the separated charge (with the antiquark) and the bunch of matter. *Electric charges can radiate photons to infinity. Even if it's not understood how (that is the "mass gap" part of the problem "Yang-Mills existence and mass gap") it's supposed that the carriers of the strong force get massive (and not by a Higgs mechanism), and in some sense they can't get far away from the source of the radiation.

Also, we aren't really sure that it gets stronger with the distance, but it can't get weaker with distance. We only can only say that it gets weaker at shorter and shorter distances, that's called "Asymptotic Freedom". Even, at scales much smaller than the proton and the neutron, they can get weak.

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