How does protium-protium fusion work? How does protium-protium fusion work? As far as I know, a proton turns into a neutron by emitting a positron. How does that work? Shouldn't a proton be slightly lighter than a neutron? This seems to violate the laws of thermodynamics. A positron can't have negative mass either. This produces energy while increasing its own mass and also violates pair production.
 A: You are correct that a single proton cannot decay into a neutron, positron, and neutrino:
$$p\to n+\beta^+ + \bar{\nu}.$$
Based on mass calculations, that is an endothermic reaction. On the other hand, if you can somehow get two protons to fuse and form helium-2 something interesting happens:
$$p + p \to \rm{}^2He.$$
This is a very unstable nucleus due to the Coulomb repulsion, but the nuclear force is great enough to allow enough nucleon-nucleon interaction to allow positron decay. This is possible because the mass of a $\rm^2 He$ nucleus is greater than the mass of a $\rm^2H$ nucleus.
If $\rm^2 He$ didn't form long enough for the nucleons to interact with nuclear force, the decay would not happen. Due to the huge Coulomb repulsion, the protons must have a great amount of center of mass energy so that they get close enough for the nuclear force to be effective. That usually happens only in the extreme temperatures of stellar cores, but even that is not enough. The process requires quantum mechanical tunneling to finish the fusion.
A: 
As far as i know, a proton turns into a neutron by emitting a positron

This is forbidden at ordinary temperature.You are correct, proton plus proton cannot fuse  at ordinary temperatures, this can happen only in the high energy plasma available in star formations and is complicated with a number of interactions. The energy is provided by the high kinetic energy of the hydrogen atoms due to the gravitational collapse into a star.

If you start with a mass of hydrogen gas and bring it together under its own gravity, it will eventually contract once it radiates enough heat away. Bring a few million (or more) Earth masses' worth of hydrogen together, and your molecular cloud will eventually contract so severely that you'll begin to form stars inside. When you pass the critical threshold of about 8% our Sun's mass, you'll ignite nuclear fusion, and form the seeds of a new star. While it's true that stars convert hydrogen into helium, that's neither the greatest number of reactions nor the cause of the greatest energy release from stars

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[Over large amounts of time, hydrogen fuel gets burned through a series of reactions, producing, in the end, large amounts of helium-4.

.........

This occurs because the product of the reaction, helium-4, is lower in mass, by about 0.7%, than the reactants (four hydrogen nuclei) that went into creating it.

Go on in the link provided to see the sequence of reactions
A: I will give a more descriptive answer, two protium nuclei are able to fuse into a Helium-2 nucleus due to the fact that 1.25 MeV of energy is added. If you find the difference between the mass of the Helium-2 nucleus (2.015894 amu), and the mass of the two protium nuclei (2.01455294 amu) you will get 0.00134106 amu. If you plug this value into Einstein's Mass-Energy equivalence formula ($m=E/c^2$) you will get approximately 1.25 MeV of energy, which is what we said was needed to create the additional mass required to turn these protium nuclei into a Helium-2 nucleus.  Here's the formula:

${^1_1H + ^1_1H + 1.25MeV → ^2_2He}$

Now that we have a Helium-2 nucleus, one of the protons can undergo beta decay. This happens when an up quark emits a W+ boson, turning it into a down quark. This W+ boson will decay into a positron, and electron neutrino, as shown in the picture below.

We now have a nucleus consisting of one proton, and one neutron, making this a deuteron (deuterium nucleus). You will also notice that the mass of a Helium-2 nucleus (2.015894 amu) is greater than that of a deuteron (2.013553212745 amu) meaning that during beta decay the difference in the mass of our Helium-2 nucleus and the mass of the deuteron will be converted into approximately 1.67 MeV of energy, as shown below.

${^2_2He → ^2_1D + e^+ + ν_e + 1.67MeV}$

And by combining these two steps into one formula we get the overall formula:

${^1_1H + ^1_1H → ^2_1D + e^ + + ν_e + 0.42MeV}$

