  We present a master formula describing the neutrinoless-double-beta decay
($0\nu\beta\beta$) rate induced by lepton-number-violating (LNV) operators up
to dimension nine in the Standard Model Effective Field Theory. We provide an
end-to-end framework connecting the possibly very high LNV scale to the nuclear
scale, through a chain of effective field theories. Starting at the electroweak
scale, we integrate out the heavy Standard Model degrees of freedom and we
match to an $SU(3)_c\otimes U(1)_{\mathrm{em}}$ effective theory. After
evolving the resulting effective Lagrangian to the QCD scale, we use chiral
perturbation theory to derive the lepton-number-violating chiral Lagrangian.
The chiral Lagrangian is used to derive the two-nucleon $0\nu\beta\beta$
transition operators to leading order in the chiral power counting. Based on
renormalization arguments we show that in various cases short-range two-nucleon
operators need to be enhanced to leading order. We show that all required
nuclear matrix elements can be taken from existing calculations. Our final
result is a master formula that describes the $0\nu\beta\beta$ rate in terms of
phase-space factors, nuclear matrix elements, hadronic low-energy constants,
QCD evolution factors, and high-energy LNV Wilson coefficients, including all
the interference terms. Our master formula can be easily matched to any model
where LNV originates at energy scales above the electroweak scale. As an
explicit example, we match our formula to the minimal left-right-symmetric
model in which contributions of operators of different dimension compete, and
we discuss the resulting phenomenology.
