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I know that we introduce R-parity to avoid proton decay.

But some papers introduce $U(1)_R$ Lepton Number, e.g claudia, thomas.

I have questions

1.What is the differences of R parity and $U(1)_R$?

2.What is the meaning of $U(1)_R$ Lepton Number?

Thank you

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$R$ Parity is a discrete $Z_2 $ symmetry while an $R$ symmetry is a global continuous symmetry. If you use a $Z_2$ symmetry to build your model then each field can just be either odd and even, that's it. If you impose a continuous symmetry then there are an infinite number of possible choices of $R$ charges. From a model building perspective, a continuous symmetry is equivalent to a discrete symmetries with an infinite number of group elements or a $Z_{\infty}$ (but up to some topological in-equivalence between U(1) and $Z_{\infty}$).

Its been shown that supersymmetry cannot be broken without $R $ symmetry, making it a very natural assumption, while $R$ parity is not as well motivated.

With regards to your second question. This should really be another question in general, but the $U(1)_R$ lepton number is a term used in the paper you link to denote a choice of $R$ charges that simulate lepton number.

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