Why protons/neutrons are not destroyed in nuclear reaction?

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In a nuclear reaction, protons and neutrons are separated from each other. My question is, why can protons and neutrons separate, but quarks can not separate from each other.

When the nucleus is intact, the gauge bosons for the strong nuclear force, gluons are responsible for holding quarks within a single proton or neutron together. They are also responsible for holding protons to neutrons.

So why do quarks within a particular proton (or neutron for that matter) get separated? Don't they "feel" the tearing force that is the sum of the protons repelling each other?

Does it have something to do with "glueballs" (don't laugh, but someone suggested this, hopefully they weren't messing with me) that don't apply between protons and neutrons.
 
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Well, I'm not sure I entirely understand your question, but I have two clarifications to make. Yes, gluons hold single nucleon together, but no, they don't bind protons to neutrons. This occurs via the "residual strong force", which is mediated by pions, not gluons.

No, glueballs likely have nothing to do with the binding of nucleons, if they even exist.
 
If you draw three parallel quark lines for a proton and three parallel quark lines for a neutron you can "exchange" two quarks between the nucleons; this exchange lines look somehow like a quark-antiquark pair, a pion.

The residual force is rather well-described by pion-exchange, but in principle other meson will contribute. The residual force is used in the context of an effective low energy theory that describes nucleon forces via the exchange of color-neutral objects. This effective theory breaks down at high energy and/or high density. E.g. in a quark gluon plasma the nucleons do break up and the effective low-energy theory is no longer valid.

The fact that quarks are never seen as isolated, colored objects (but appear only in color-neutral bound states) is called color-confinement. It is well-established in lattice gauge theory which allows one even to calculate nucleon masses, form factors etc. But the theoretical reason for color confinement is not completely understood. There are several approaches all showing some qualitative facts or hints indicating confinement, but a sound proof is still missing.

Glueballs do not play any role in confinement, but could be additional states to be taken into account in low-enery effective theories.
 
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