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From what I read on beta minus decay, when it happens a neutron gets "converted" into a proton, an electron and an electron-antineutrino. I also read that both the neutron and the proton are made of elementary particles named quarks (the neutron is composed of 2 downs and one up, and the proton is composed of 2 ups and one down), unlike the electron which is an elementary particle by itself. So how can a down quark get "converted" into and up quark, while "creating" an electron, which is not composed of quarks?

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A charge -1/3 down quark emits a $W^-$ charge -1 weak-interaction gauge boson and changes into a charge +2/3 up quark. The $W^-$ then decays into a charge -1 electron and a charge neutral electron anti-neutrino.

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As @mikestone noted, $\beta^-$ ($\beta^+$) decay follows $d\to u+W^-$ ($u\to d+W^+$) with $W^-\to e^-+\bar{\nu}_e$ ($W^+\to e^++\nu_e$), with $W^\mp$ virtual. But let's address a more general point that I think is the OP's concern. While composite particles have a nonzero size due to an internal structure of elementary/fundamental particles (which may or may not be point particles, but let's put that aside for now), elementary particles, while not made of smaller ones in theory, can mutate. As the two examples above show, if you assume anything that comes out was inside to begin with, $d$ contains $u$ which contains $d$ which...

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I see from your profile that this is you second question on this site, and last year you had just learned classical mechanics, so maybe the answers to the general questions are still too complicated for the level of your physics background.

This is to illustrate the answer by Mike Stone.

Elementary particle physics, as described in the standard model, is understood at present to be the underlying level of all physics theories up to now, including nuclear physics. The model uses quantum field theory , and the calculations use Feynman diagrams. This is the Feynman diagram for a beta decay:

betadecay

Here you see that a down quark turns into an up quark through the emission of a virtual ( see this answer for the meaning of virtual ) W and it is the W that decays into an electron, to conserve charge, and an electron antineutrino to conserve lepton number. Quantum numbers have to follow rules at the vertices of interactions and decays.

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W. Heisenberg remarked already in 1932 that there is no real difference between a neutron and a proton (if the electromagnetic interaction is neglected) and one could consider it as the same particle in 2 different states. So he associated a quantum number to these 2 states and to the dublett as a whole: Isospin. Later when end of the 1960s the quarks were discovered the concept of isospin was also carried over to the quarks. So a dublett of quarks

$$\left(\begin{array}{c} u \\ d\end{array}\right) $$

is considered as same particle with isospin $I=1/2$ and 2 different states $I_3=+1/2$ and $I_3=-1/2$.

One can imagine a similar process happening in QED. A photon is emitted from a electron with spin $S_z =+1/2$ flips into an electron with spin $S_z=-1/2$. Under certain circumstances such spin flips happen, so if an u-quark changes into a d-quark it is like flip of isospin. One might wonder about this curious process, but this is also an aspect of science: generalisation of concepts (here from spin to isospin) and make abstraction of some "unimportant" details. On first sight different particles are actually the same (or at least behave as if they were the same). Another important aspect appears here is symmetry (well that's already a well-known concept), but in particular in elementary particle physics imperfect symmetries are realized frequently. Yes, one can say the symmetry between an u- and d-quark is not perfect, since they don't have the same mass. But in the context of weak interaction the difference does not matter. A little detail has to annotated: Isospin was first a concept in strong interaction, later upon the establishment of the weak interaction, a weak isospin was also introduced in weak interaction and for the (u,d) dublett the strong isospin agrees with the weak isospin.

Something similar happens amongst outgoing particles. The decay products of the beta-decay can be considered as a weak isospin flip of another dublett of weak isospin $(I=1/2,I_3=\pm 1/2)$ $$\left(\begin{array}{c} \nu_{eL} \\ e_L\end{array}\right)$$

A backwards in time in-going electron-neutrino of $I_3=1/2$ turns into an electron $I_3=-1/2$. Under the perspective of weak interaction there is no real difference between an left-chiral electron and a (left-chiral) electron neutrino. Both are rather (even extremely) light particles whose mass can simply often neglected. Then we can consider (as already done a line above) these particles as chiral. When it comes to weak interaction --- here into play because (u,d) isospin flip is acompanied by the emission of $W^{-}$-boson, one of the exchange particles the carry the weak interaction --- (left-chiral) electron-neutrino and left-chiral electron share a lot of properties. They are like the same particle in different weak isospin states. And --- actually it is out of scope of this post --- right-chiral electrons are actually more different from a left-chiral electron and a left-chiral electron from a left-chiral neutrino (under the weak interaction perspective). But, a left chiral electron can transform into a right-chiral electron by a simple spin flip, for instance caused by EM-interaction.

So particles under the perspective of a single interaction --- here the weak one --- can behave as the same particle. Considering another interaction, for instance electromagnetic interaction they might behave as different particles. That's the bottom message.

Well, I did not explain well chirality and the chiral character of the weak interaction... that is for another post. Neither I explained that the emission of an anti-neutrino can be considered as an absorption of a in time backwards in-going neutrino. Knowing these details help to understand better the answer but crucial is here the concept of unify particles under some underlying symmetry considering them as the same particle so that they can just be considered as simple change of quantum state if it comes to particle processes.

The symmetry which manifests in the common properties of (u,d)- and $(\nu_{eL},e_L)$ dublets is here SU(2). And as it is well-known weak interaction being part of electroweak interaction is a gauge theory based on $SU(2)\times U(1)$-symmetry.

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